book metadata European Mathematical Society Publishing House 2020-02-09 13:23:29 178 https://ems.eye-com.dev/meta/bmeta-stream.php?update_since=2020-02-09 Yann Bugeaud Université de Strasbourg, France Number theory Primary: 11-02, 11J86, 11D; Secondary: 11B37, 11D25, 11D41, 11D59, 11D61, 11D75, 11D88, 11J25, 11J81, 11J82 Baker's theory, linear form in logarithms, Diophantine equation, Thue equation, $abc$-conjecture, primitive divisor, irrationality measure, $p$-adic analysis The aim of this book is to serve as an introductory text to the theory of linear forms in the logarithms of algebraic numbers, with a special emphasis on a large variety of its applications. We wish to help students and researchers to learn what is hidden inside the blackbox ‚Baker's theory of linear forms in logarithms' (in complex or in $p$-adic logarithms) and how this theory applies to many Diophantine problems, including the e ffective resolution of Diophantine equations, the $abc$-conjecture, and upper bounds for the irrationality measure of some real numbers. Written for a broad audience, this accessible and self-contained book can be used for graduate courses (some 30 exercises are supplied). Specialists will appreciate the inclusion of over 30 open problems and the rich bibliography of over 450 references. 2 28 2018 978-3-03719-183-5 978-3-03719-683-0 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/183 http://www.ems-ph.org/doi/10.4171/183 2523-5133 2523-5141 28 Impanga 15 Jarosław Buczyński Polish Academy of Sciences and University of Warsaw, Poland Mateusz Michałek Polish Academy of Sciences, Warsaw, Poland and Max Planck-Institute, Leipzig, Germany Elisa Postinghel Loughborough University, UK Algebraic geometry Several complex variables and analytic spaces Algebraic topology Primary 14-06; secondary 32L10, 14M15, 55N91, 14C17, 14G17 Analytic geometry IMPANGA, vector bundles, Schubert varieties and degeneracy loci, homogeneous spaces, equivariant cohomology, Thom polynomials, characteristic classes, symmetric functions and polynomials, quasi-elliptic surfaces IMPANGA stands for the activities of Algebraic Geometers at the Institute of Mathematics, Polish Academy of Sciences, including one of the most important seminars in algebraic geometry in Poland. The topics of the lectures usually fit within the framework of complex algebraic geometry and neighboring areas of mathematics. This volume is a collection of contributions by the participants of the conference IMPANGA15, organized by participants of the seminar, as well as notes from the major lecture series of the seminar in the period 2010–2015. Both original research papers and self-contained expository surveys can be found here. The articles circulate around a broad range of topics within algebraic geometry such as vector bundles, Schubert varieties, degeneracy loci, homogeneous spaces, equivariant cohomology, Thom polynomials, characteristic classes, symmetric functions and polynomials, and algebraic geometry in positive characteristic. 1 21 2018 978-3-03719-182-8 978-3-03719-682-3 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/182 https://ems.eye-com.dev/doi/10.4171/182 2523-515X 2523-5168

Jarosław Buczyński Polish Academy of Sciences, Warsaw, Poland Mateusz Michałek Polish Academy of Sciences, Warsaw, Poland Elisa Postinghel Loughborough University, UK 1 7 1 10.4171/182-1/1 https://ems.eye-com.dev/doi/10.4171/182-1/1

Piotr Pragacz Polish Academy of Sciences, Warsaw, Poland 9 23 1 10.4171/182-1/2 https://ems.eye-com.dev/doi/10.4171/182-1/2

Soojin Cho Ajou University, Suwon, Republic of Korea Takeshi Ikeda Okayama University of Science, Japan Maximal orthogonal Grassmannian, Pieri rule, factorial Schur $P$-functions Algebraic geometry Combinatorics We prove an identity expressing the product of two factorial Schur $P$-functions as an alternating sum of the factorial Schur $P$-functions with explicitly defined coefficients depending on the deformation parameters. As an application, we derive the Pieri rule for the factorial Schur $P$-functions. 25 48 1 10.4171/182-1/3 https://ems.eye-com.dev/doi/10.4171/182-1/3

Izzet Coskun University of Illinois at Chicago, USA Homogeneous varieties, Schubert varieties, quadratic forms, restriction coefficients Algebraic geometry Commutative rings and algebras This is a survey paper based on the author's lectures given at IMPAN in December 2013. We will discuss recent results on the restriction and rigidity problems. The purpose of the lectures was to develop a more geometric approach to the study of isotropic flag varieties. As an illustration of the techniques, we compute the map induced in cohomology of the inclusion of $OG(k,n)$ and $SG(k,n)$ in $G(k,n)$ via an explicit sequence of rational equivalences. We also discuss applications to classifying representatives of Schubert classes. 49 95 1 10.4171/182-1/4 https://ems.eye-com.dev/doi/10.4171/182-1/4

Letterio Gatto Politecnico di Torino, Italy Parham Salehyan IBILCE/UNESP, São José do Rio Preto, Brazil Algebraic geometry The KP hierarchy (after Kadomtsev and Petshiasvily) is a system of infinitely many PDEs in Lax form defining a universal family of iso-spectral deformation of an ordinary linear differential operator. It is a classical result due to Sato's japanese school that the rational solutions to the KP hierarchy are parametrized by a cone over an infinite-dimensional Grassmann variety. The present survey will revisit this fact from the point of view of Schubert derivations on a Grassmann algebra. These enable to encode the classical Plücker equations of Grassmannians of $r$-dimensional subspaces in a formula whose limit for $r \to \infty$ coincides with the KP hierarchy, phrased in terms of vertex operators, showing in particular how the latter is intimately related to Schubert calculus. 97 125 1 10.4171/182-1/5 https://ems.eye-com.dev/doi/10.4171/182-1/5

Thomas Hudson Bergische Universität Wuppertal, Germany Tomoo Matsumura Okayama University of Science, Japan Grassmann bundles, Schubert varieties, Oriented cohomology theories, Chern classes Algebraic geometry Combinatorics n this paper we prove a generalisation of Kempf–Laksov formula for the degeneracy loci classes in the even infinitesimal cohomology theories of Grassmann bundles and Lagrangian Grassmann bundles. 127 151 1 10.4171/182-1/6 https://ems.eye-com.dev/doi/10.4171/182-1/6

Toshiyuki Katsura Hosei University, Tokyo, Japan Quasi-elliptic surfaces, multicanonical systems, positive characteristic Algebraic geometry We consider the multicanonical systems $\vert mK_S \vert$ of quasi-elliptic surfaces with Kodaira dimension $1$ in characteristic 3. We show that for any $m \geq 5$ $\vert mK_S \vert$ gives the structure of quasi-elliptic fiber space, and the number $5$ is best possible to give the structure for any such surfaces. 153 157 1 10.4171/182-1/7 https://ems.eye-com.dev/doi/10.4171/182-1/7

Laurentiu Maxim University of Wisconsin, Madison, USA Jörg Schürmann Universität Münster, Germany Characteristic classes, Atiyah–Singer classes, mixed Hodge modules, $V$-filtration, nearby and vanishing cycles, singularities, toric varieties, hypersurface, symmetric product, generating series. Manifolds and cell complexes Algebraic geometry Associative rings and algebras Several complex variables and analytic spaces We give an overview, with an emphasis on applications, of recent developments on the interaction between characteristic class theories for singular spaces and Saito's theory of mixed Hodge modules in the complex algebraic context. 159 202 1 10.4171/182-1/8 https://ems.eye-com.dev/doi/10.4171/182-1/8

Piotr Pragacz Polish Academy of Sciences, Warsaw, Poland $U(\mathfrak b)$-module, Demazure module, KP module, KP filtration, cyclic module, character, Schur function, Schur functor, Schubert polynomial, subquotient, positivity, ample bundle Algebraic geometry Group theory and generalizations We report on results of Kraskiewicz and the author, and Watanabe on KP modules materializing Schubert polynomials, and filtrations having KP modules as their sub quotients. We discuss applications of the bundles $S_w(E)$ for filtered ample bundles $E$ and KP filtrations to positivity due to Fulton and Watanabe respectively. 203 224 1 10.4171/182-1/9 https://ems.eye-com.dev/doi/10.4171/182-1/9

Richárd Rimányi University of North Carolina at Chapel Hill, USA Alexander Varchenko University of North Carolina at Chapel Hill, USA Equivariant Chern–Schwartz–MacPherson class, Schubert calculus, weight function Algebraic geometry Nonassociative rings and algebras Consider the natural torus action on a partial flag manifold $\mathcal F}_\lambda$. Let $\Omega_I\subset \mathcal F}_\lambda$ be an open Schubert variety, and let $c^{sm}(\Omega_I)\in H_T^*(\mathcal F}_\lambda)$ be its torus equivariant Chern–Schwartz–MacPherson class. We show a set of interpolation properties that uniquely determine $c^{sm}(\Omega_I)$, as well as a formula, of 'localization type', for $c^{sm}(\Omega_I)$. In fact, we proved similar results for a class $\kappa_I\in H_T^*(\mathcal F}_\lambda)$ – in the context of quantum group actions on the equivariant cohomology groups of partial flag varieties. In this note we show that $c^{sm}(\Omega_I)=\kappa_I$. 225 235 1 10.4171/182-1/10 https://ems.eye-com.dev/doi/10.4171/182-1/10

Takahisa Sasajima Kyoto, Japan Toru Ohmoto Hokkaido University, Sapporo, Japan $\mathcal A$-classification of map-germs, Thom polynomials, classical enumerative geometry, projective surfaces Algebraic geometry Manifolds and cell complexes We study universal polynomials of characteristic classes associated to the $\mathcal A$-classification of map-germs $(\mathbb C^2,0) \to (\mathbb C^n, 0)$ $(n=2,3)$, that enable us to systematically generalize enumerative formulae in classical algebraic geometry of projective surfaces in 3 and 4-spaces. 237 259 1 10.4171/182-1/11 https://ems.eye-com.dev/doi/10.4171/182-1/11

Harry Tamvakis University of Maryland, College Park, USA Schubert polynomials, theta polynomials, symmetric functions, Weyl groups, divided difference operators, flag varieties, degeneracy loci, equivariant cohomology Algebraic geometry Combinatorics In previous work, we employed the approach to Schubert polynomials by Fomin, Stanley, and Kirillov to obtain simple, uniform proofs that the double Schubert polynomials of Lascoux and Schützenberger and Ikeda, Mihalcea, and Naruse represent degeneracy loci for the classical groups in the sense of Fulton. Using this as our starting point, and purely combinatorial methods, we obtain a new proof of the general formulas of [T5], which represent the degeneracy loci coming from any isotropic partial flag variety. Along the way, we also find several new formulas and elucidate the connections between some earlier ones. 261 314 1 10.4171/182-1/12 https://ems.eye-com.dev/doi/10.4171/182-1/12

Shoji Yokura Kagoshima University, Japan 315 330 1 10.4171/182-1/13 https://ems.eye-com.dev/doi/10.4171/182-1/13

Magdalena Zielenkiewicz University of Warsaw, Poland Gysin homomorphism, equivariant cohomology, torus action, homogeneous space Algebraic geometry Differential geometry In this paper we review the results presented during the IMPANGA 15 Conference on the author's approach to equivariant Gysin homomorphism via iterated residues at infinity, with connections to the Jeffrey–Kirwan nonabelian localization theorem in symplectic geometry. We show examples of computations using the formulas of our previous paper, which express the push-forwards in equivariant cohomology as iterated residues at infinity. As an example we consider the equivariant cohomology of the complex Lagrangian Grassmannian $LG(n)$ with the action of the maximal torus in the symplectic group $Sp(n)$. In particular, we obtain, via our methods, analogues in equivariant cohomology of some well-known results due to Pragacz and Ratajski on push-forwards of Schur classes on $LG(n)$. 331 345 1 10.4171/182-1/14 https://ems.eye-com.dev/doi/10.4171/182-1/14

Françoise Michel Université Paul Sabatier, Toulouse, France Claude Weber Université de Genève, Switzerland Manifolds and cell complexes Several complex variables and analytic spaces 57Q45, 57R65, 32S55 Algebraic topology Knots in high dimensions, homotopy spheres, complex singularities Michel Kervaire wrote six papers which can be considered fundamental to the development of higher-dimensional knot theory. They are not only of historical interest but naturally introduce to some of the essential techniques in this fascinating theory. This book is written to provide graduate students with the basic concepts necessary to read texts in higher-dimensional knot theory and its relations with singularities. The first chapters are devoted to a presentation of Pontrjagin’s construction, surgery and the work of Kervaire and Milnor on homotopy spheres. We pursue with Kervaire’s fundamental work on the group of a knot, knot modules and knot cobordism. We add developments due to Levine. Tools (like open books, handlebodies, plumbings, …) often used but hard to find in original articles are presented in appendices. We conclude with a description of the Kervaire invariant and the consequences of the Hill–Hopkins–Ravenel results in knot theory. 7 25 2017 978-3-03719-180-4 978-3-03719-680-9 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/180 https://ems.eye-com.dev/doi/10.4171/180 2523-5176 2523-5184 Second edition Jørn Justesen Technical University of Denmark, Lyngby, Denmark Tom Høholdt Technical University of Denmark, Lyngby, Denmark Information and communication, circuits Field theory and polynomials 94-01;12-01 Mathematical theory of computation Fields + rings Error-correcting codes, Reed–Solomon codes, convolutional codes, product codes, graph codes, algebraic geometry codes This book, updated and enlarged for the second edition, is written as a text for a course aimed at 3rd or 4th year students. Only some familiarity with elementary linear algebra and probability is directly assumed, but some maturity is required. The students may specialize in discrete mathematics, computer science, or communication engineering. The book is also a suitable introduction to coding theory for researchers from related fields or for professionals who want to supplement their theoretical basis. The book gives the coding basics for working on projects in any of the above areas, but material specific to one of these fields has not been included. The chapters cover the codes and decoding methods that are currently of most interest in research, development, and application. They give a relatively brief presentation of the essential results, emphasizing the interrelations between different methods and proofs of all important results. A sequence of problems at the end of each chapter serves to review the results and give the student an appreciation of the concepts. In addition, some problems and suggestions for projects indicate direction for further work. The presentation encourages the use of programming tools for studying codes, implementing decoding methods, and simulating performance. Specific examples of programming exercises are provided on the book's home page. 7 11 2017 978-3-03719-179-8 978-3-03719-679-3 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/179 https://ems.eye-com.dev/doi/10.4171/179 A Geometrical Analysis Antoine Henrot Université de Lorraine, Vandœuvre-lès-Nancy, France Michel Pierre ENS Cachan Bretagne, Bruz, France Calculus of variations and optimal control; optimization Partial differential equations Differential geometry Global analysis, analysis on manifolds 49Q10, 49Q05, 49Q12, 49K20, 49K40, 53A10, 35R35, 35J20, 58E25, 31B15, 65K10, 93B27, 74P20, 74P15, 74G65, 76M30 Calculus + mathematical analysis Differential equations Differential + Riemannian geometry Shape optimization, optimum design, calculus of variations, variations of domains, Hausdorff convergence, continuity with respect to domains, G-convergence, shape derivative, geometry of optimal shapes, Laplace-Dirichlet problem, Neumann problem, overdetermined problems, isoperimetric inequality, capacity, potential theory, spectral theory, homogenization Optimizing the shape of an object to make it the most efficient, resistant, streamlined, lightest, noiseless, stealthy or the cheapest is clearly a very old task. But the recent explosion of modeling and scientific computing have given this topic new life. Many new and interesting questions have been asked. A mathematical topic was born – shape optimization (or optimum design). This book provides a self-contained introduction to modern mathematical approaches to shape optimization, relying only on undergraduate level prerequisite but allowing to tackle open questions in this vibrant field. The analytical and geometrical tools and methods for the study of shapes are developed. In particular, the text presents a systematic treatment of shape variations and optimization associated with the Laplace operator and the classical capacity. Emphasis is also put on differentiation with respect to domains and a FAQ on the usual topologies of domains is provided. The book ends with geometrical properties of optimal shapes, including the case where they do not exist. 2 15 2018 978-3-03719-178-1 978-3-03719-678-6 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/178 https://ems.eye-com.dev/doi/10.4171/178 Martin Raussen Aalborg University, Denmark Christian Skau Norwegian University of Science and Technology (NTNU), Trondheim, Norway History and biography General 01A70, 01A60, 01A61, 01A80, 00A35 History of mathematics Abel prize, laureates, interviews, history of mathematics, appreciation of mathematics The Abel Prize was established in 2002 by the Norwegian Ministry of Education and Research. It has been awarded annually to mathematicians in recognition of pioneering scientific achievements. Since the first occasion in 2003, Martin Raussen and Christian Skau have had the opportunity to conduct extensive interviews with the laureates. The interviews were broadcast by Norwegian television; moreover, they have appeared in the membership journals of several mathematical societies. The interviews from the period 2003 – 2016 have now been collected in this edition. They highlight the mathematical achievements of the laureates in a historical perspective and they try to unravel the way in which the world’s most famous mathematicians conceive and judge their results, how they collaborate with peers and students, and how they perceive the importance of mathematics for society. 9 1 2017 978-3-03719-177-4 978-3-03719-677-9 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/177 https://ems.eye-com.dev/doi/10.4171/177

Martin Raussen Aalborg Universitet, Denmark Christian Skau Norwegian University of Science and Technology (NTNU), Trondheim, Norway History and biography General 1 10 1 10.4171/177-1/1 https://ems.eye-com.dev/doi/10.4171/177-1/1

Martin Raussen Aalborg Universitet, Denmark Christian Skau Norwegian University of Science and Technology (NTNU), Trondheim, Norway History and biography General 11 29 1 10.4171/177-1/2 https://ems.eye-com.dev/doi/10.4171/177-1/2

Martin Raussen Aalborg Universitet, Denmark Christian Skau Norwegian University of Science and Technology (NTNU), Trondheim, Norway History and biography General 31 46 1 10.4171/177-1/3 https://ems.eye-com.dev/doi/10.4171/177-1/3

Martin Raussen Aalborg Universitet, Denmark Christian Skau Norwegian University of Science and Technology (NTNU), Trondheim, Norway History and biography General 47 60 1 10.4171/177-1/4 https://ems.eye-com.dev/doi/10.4171/177-1/4

Martin Raussen Aalborg Universitet, Denmark Christian Skau Norwegian University of Science and Technology (NTNU), Trondheim, Norway History and biography General 61 78 1 10.4171/177-1/5 https://ems.eye-com.dev/doi/10.4171/177-1/5

Martin Raussen Aalborg Universitet, Denmark Christian Skau Norwegian University of Science and Technology (NTNU), Trondheim, Norway History and biography General 79 96 1 10.4171/177-1/6 https://ems.eye-com.dev/doi/10.4171/177-1/6

Martin Raussen Aalborg Universitet, Denmark Christian Skau Norwegian University of Science and Technology (NTNU), Trondheim, Norway History and biography General 97 121 1 10.4171/177-1/7 https://ems.eye-com.dev/doi/10.4171/177-1/7

Martin Raussen Aalborg Universitet, Denmark Christian Skau Norwegian University of Science and Technology (NTNU), Trondheim, Norway History and biography General 123 140 1 10.4171/177-1/8 https://ems.eye-com.dev/doi/10.4171/177-1/8

Martin Raussen Aalborg Universitet, Denmark Christian Skau Norwegian University of Science and Technology (NTNU), Trondheim, Norway History and biography General 141 159 1 10.4171/177-1/9 https://ems.eye-com.dev/doi/10.4171/177-1/9

Martin Raussen Aalborg Universitet, Denmark Christian Skau Norwegian University of Science and Technology (NTNU), Trondheim, Norway History and biography General 161 182 1 10.4171/177-1/10 https://ems.eye-com.dev/doi/10.4171/177-1/10

Martin Raussen Aalborg Universitet, Denmark Christian Skau Norwegian University of Science and Technology (NTNU), Trondheim, Norway History and biography General 183 200 1 10.4171/177-1/11 https://ems.eye-com.dev/doi/10.4171/177-1/11

Martin Raussen Aalborg Universitet, Denmark Christian Skau Norwegian University of Science and Technology (NTNU), Trondheim, Norway History and biography General 201 217 1 10.4171/177-1/12 https://ems.eye-com.dev/doi/10.4171/177-1/12

Martin Raussen Aalborg Universitet, Denmark Christian Skau Norwegian University of Science and Technology (NTNU), Trondheim, Norway History and biography General 219 243 1 10.4171/177-1/13 https://ems.eye-com.dev/doi/10.4171/177-1/13

Martin Raussen Aalborg Universitet, Denmark Christian Skau Norwegian University of Science and Technology (NTNU), Trondheim, Norway History and biography General 245 265 1 10.4171/177-1/14 https://ems.eye-com.dev/doi/10.4171/177-1/14

Martin Raussen Aalborg Universitet, Denmark Christian Skau Norwegian University of Science and Technology (NTNU), Trondheim, Norway History and biography General 267 287 1 10.4171/177-1/15 https://ems.eye-com.dev/doi/10.4171/177-1/15

The Pavel Exner Anniversary Volume Jaroslav Dittrich Czech Academy of Sciences, Rez-Prague, Czech Republic Hynek Kovařík Università degli Studi di Brescia, Italy Ari Laptev Imperial College London, UK Quantum theory Partial differential equations 81Q37, 81Q35, 35P15, 35P25 Quantum physics (quantum mechanics) Differential equations Schrödinger operators, point interactions, metric graphs, quantum waveguides, eigenvalue estimates, operator-valued functions, Cayley–Hamilton theorem, adiabatic theorem This volume is dedicated to Pavel Exner on the occasion of his 70th anniversary. It collects contributions by numerous scientists with expertise in mathematical physics and in particular in problems arising from quantum mechanics. The questions addressed in the contributions cover a large range of topics. A lot of attention was paid to differential operators with zero range interactions, which are often used as models in quantum mechanics. Several authors considered problems related to systems with mixed-dimensions such as quantum waveguides, quantum layers and quantum graphs. Eigenvalues and eigenfunctions of Laplace and Schrödinger operators are discussed too, as well as systems with adiabatic time evolution. Although most of the problems treated in the book have a quantum mechanical background, some contributions deal with issues which go well beyond this framework; for example the Cayley–Hamilton theorem, approximation formulae for contraction semigroups or factorization of analytic operator-valued Fredholm functions. As for the mathematical tools involved, the book provides a wide variety of techniques from functional analysis and operator theory. Altogether the volume presents a collection of research papers which will be of interest to any active scientist working in one of the above mentioned fields. 5 5 2017 978-3-03719-175-0 978-3-03719-675-5 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/175 https://ems.eye-com.dev/doi/10.4171/175 2523-515X 2523-5168

Sergio Albeverio Universität Bonn, Germany Claudio Cacciapuoti Università dell'Insubria, Como, Italy Mauro Spreafico Università del Salento & INFN, Lecce, Italy Relative zeta function, relative partition function, relative spectral measures, Coulomb interaction, point interactions, zeta regularization, finite temperature quantum fields, Casimir effect, asymptotic expansions Global analysis, analysis on manifolds Number theory Quantum theory The relative partition function and the relative zeta function of the perturbation of the Laplace operator by a Coulomb potential plus a point interaction centered in the origin is discussed. Applications to the study of the Casimir effect are indicated. 1 29 1 10.4171/175-1/1 https://ems.eye-com.dev/doi/10.4171/175-1/1

Asao Arai Hokkaido University, Sapporo, Japan Canonical anticommutation relations, fermion Fock space, inequivalent representation, mass, quantum Dirac field Quantum theory Operator theory A family of irreducible representations of the canonical anticommutation relations over an abstract Hilbert space indexed by a set of bounded linear operators is presented and a theorem on the mutual equivalence of them is established. As an application of the theorem, it is proved that quantum Dirac fields of different masses are mutually inequivalent. Moreover, a new class of irreducible representations of the CAR over a Hilbert space, which includes, as a special case, time-zero quantum Dirac fields, is constructed. 31 53 1 10.4171/175-1/2 https://ems.eye-com.dev/doi/10.4171/175-1/2

Diana Barseghyan University of Ostrava, Czech Republic Olga Rossi University of Ostrava, Czech Republic Spectral transition, discrete spectrum, eigenvalue estimates Quantum theory Partial differential equations We show that the operator for $\lambda\frac{\pi^2}{4}$ its spectrum contains the real line. In the critical case $\lambda=\frac{\pi^2}{4}$ we prove that the spectrum coincides with the half line $[0, \infty)$. 55 70 1 10.4171/175-1/3 https://ems.eye-com.dev/doi/10.4171/175-1/3

Giulia Basti Università di Roma La Sapienza, Italy Alessandro Teta Università di Roma La Sapienza, Italy Point interactions, self-adjoint extensions, unitary gas, Thomas effect Quantum theory Functional analysis Operator theory In this note we discuss the quantum mechanical three-body problem with pairwise zero-range interactions in dimension three. We review the state of the art concerning the construction of the corresponding Hamiltonian as a self-adjoint operator in the bosonic and in the fermionic case. Exploiting a quadratic form method, we also prove self-adjointness and boundedness from below in the case of three identical bosons when the Hilbert space is suitably restricted, i.e., excluding the "s-wave“ subspace. 71 93 1 10.4171/175-1/4 https://ems.eye-com.dev/doi/10.4171/175-1/4

Jussi Behrndt TU Graz, Austria Fritz Gesztesy Baylor University, Waco, USA Helge Holden University of Trondheim, Norway Roger Nichols The University of Tennessee at Chattanooga, USA Factorization of operator-valued analytic functions, multiplicity of eigenvalues, index computations for finitely meromorphic operator-valued functions, Birman–Schwinger operators, dual pairs Operator theory We revisit and connect several notions of algebraic multiplicities of zeros of analytic operator-valued functions and discuss the concept of the index of meromorphic operator-valued functions in complex, separable Hilbert spaces. Applications to abstract perturbation theory and associated Birman–Schwinger-type operators and to the operator-valued Weyl–Titchmarsh functions associated to closed extensions of dual pairs of closed operators are provided. 95 127 1 10.4171/175-1/5 https://ems.eye-com.dev/doi/10.4171/175-1/5

Jussi Behrndt TU Graz, Austria Matthias Langer University of Strathclyde, Glasgow, UK Vladimir Lotoreichik Nuclear Physics Institute, Řež - Prague, Czech Republic Trace formula, delta interaction, Schrödinger operator, singular potential Partial differential equations Quantum theory Operator theory Let $\Sigma\subset\mathbb R^d$ be a $C^\infty$-smooth closed compact hypersurface, which splits the Euclidean space $\mathbb R^d$ into two domains $\Omega_\pm$. In this note self-adjoint Schrödinger operators with $\delta$ and $\delta'$-interactions supported on $\Sigma$ are studied. For large enough $m\in\mathbb N$ the difference of $m$th powers of resolvents of such a Schrödinger operator and the free Laplacian is known to belong to the trace class. We prove trace formulae, in which the trace of the resolvent power difference in $L^2(\mathbb R^d)$ is written in terms of Neumann-to-Dirichlet maps on the boundary space $L^2(\Sigma)$. 129 152 1 10.4171/175-1/6 https://ems.eye-com.dev/doi/10.4171/175-1/6

Rafael Benguria Pontificia Universidad Católica de Chile, Santiago de Chile, Chile Soledad Benguria University of Wisconsin, Madison, USA Brezis–Nirenberg problem, hyperbolic space, nonexistence of solutions, Pohozaev identity, Hardy inequality Partial differential equations Using a Rellich–Pohozaev argument and Hardy's inequality, we derive an improved bound on the nonlinear eigenvalue for the non existence of radial solutions of a Brezis–Nirenberg problem, with Dirichlet boundary conditions, on a geodesic ball of $\mathbb{H}^n$, for $2 153 160 1 10.4171/175-1/7 https://ems.eye-com.dev/doi/10.4171/175-1/7

Philippe Briet Université de Toulon, La Garde, France Hiba Hammedi Université de Toulon et du Var, La Garde, France Waveguide, mixed boundary conditions, twisting Quantum theory Operator theory This paper is concerned with the study of the existence/non-existence of the discrete spectrum of the Laplace operator on a domain of $\mathbb R ^3$ which consists in a twisted tube. This operator is defined by means of mixed boundary conditions. Here we impose Neumann Boundary conditions on a bounded open subset of the boundary of the domain (the Neumann window) and Dirichlet boundary conditions elsewhere. 161 175 1 10.4171/175-1/8 https://ems.eye-com.dev/doi/10.4171/175-1/8

Giuseppe Cardone Università del Sannio, Benevento, Italy Andrii Khrabustovskyi Karlsruher Institut für Technologie, Germany Periodic waveguides, spectral gaps, asymptotic analysis Partial differential equations Operator theory In this note we investigate spectral properties of a periodic waveguide $\Omega^\varepsilon$ ($\varepsilon$ is a small parameter) obtained from a straight strip by attaching an array of $\varepsilon$-periodically distributed identical protuberances having "room-and-passage" geometry. In the current work we consider the operator $\mathcal{A}^\varepsilon =-\rho^\varepsilon\Delta_{\Omega^\varepsilon}$, where $\Delta_{\Omega^\varepsilon}$ is the Neumann Laplacian in $\Omega^\varepsilon$, the weight $\rho^\varepsilon$ is equal to $1$ everywhere except the union of the „rooms". We will prove that the spectrum of $\mathcal{A}^\varepsilon$ has at least one gap as $\varepsilon$ is small enough provided certain conditions on the weight $\rho^\varepsilon$ and the sizes of attached protuberances hold. 177 187 1 10.4171/175-1/9 https://ems.eye-com.dev/doi/10.4171/175-1/9

Raffaele Carlone Università degli Studi di Napoli Federico II, Italy Michele Correggi Università degli Studi Roma Tre, Italy Rodolfo Figari Università degli Studi di Napoli Federico II, Italy Partial differential equations Quantum theory We study the time-evolution of a quantum particle subjected to time-dependent zero-range forces in two dimensions. After establishing a conceivable ansatz for the solution to the Schrödinger equation, we prove that the wave packet time-evolution is completely specified by the solutions of a system of Volterra-type equations – the charge equations – involving the coefficients of the singular part of the wave function, thus extending to the two-dimensional case known results in one and three dimensions. 189 211 1 10.4171/175-1/10 https://ems.eye-com.dev/doi/10.4171/175-1/10

Ngoc Do Texas A&M University, College Station, USA Peter Kuchment Texas A&M University, College Station, USA Beng Ong CGG, Houston, USA Quantum graph, spectral gap, resonator Quantum theory Partial differential equations Global analysis, analysis on manifolds In this brief paper we present some results on creating and manipulating spectral gaps for a (regular) quantum graph by inserting appropriate internal structures into its vertices. Complete proofs and extensions of the results are planned for another publication. 213 222 1 10.4171/175-1/11 https://ems.eye-com.dev/doi/10.4171/175-1/11

Martin Fraas Albany, USA Partial differential equations Quantum theory We derive an adiabatic theory for a stochastic differential equation, $$ \varepsilon\, \d X(s) = L_1(s) X(s)\, \d s + \sqrt{\varepsilon} L_2(s) X(s) \, \d B_s, $$ under a condition that instantaneous stationary states of $L_1(s)$ are also stationary states of $L_2(s)$. We use our results to derive the full statistics of tunneling for a driven stochastic Schrödinger equation describing a dephasing process. 223 243 1 10.4171/175-1/12 https://ems.eye-com.dev/doi/10.4171/175-1/12

Rupert Frank Caltech, Pasadena, United States Partial differential equations Quantum theory We consider Schrödinger operators in $\mathbb R^d$ with complex potentials supported on a hyperplane and show that all eigenvalues lie in a disk in the complex plane with radius bounded in terms of the $L^p$ norm of the potential with $d-1 < p \leq d$. We also prove bounds on sums of powers of eigenvalues. 245 259 1 10.4171/175-1/13 https://ems.eye-com.dev/doi/10.4171/175-1/13

Pedro Freitas Universidade de Lisboa, Portugal David Krejčiřík Czech Technical University in Prague, Prague, Czech Republic Dirichlet Laplacian in tubes, quantum waveguides, quantum layers, ground-state energy Number theory Algebraic geometry We derive a lower bound to the spectral threshold of the Dirichlet Laplacian in tubular neighbourhoods of constant radius about complete surfaces. This lower bound is given by the lowest eigenvalue of a one-dimensional operator depending on the radius and principal curvatures of the reference surface. Moreover, we show that it is optimal if the reference surface is non-negatively curved. 261 269 1 10.4171/175-1/14 https://ems.eye-com.dev/doi/10.4171/175-1/14

Yaroslav Granovskyi National Academy of Science of Ukraine, Slavyansk, Ukraine Mark Malamud National Academy of Science of Ukraine, Slavyansk, Ukraine Hagen Neidhardt Karl-Weierstraß-Institut für Mathematik, Berlin, Germany Andrea Posilicano Università dell'Insubria, Como, Italy Vector-valued Sturm–Liouville operator, point interaction, spectrum, absolutely and singular continuous spectrum, eigenvalues, boundary triplets, Weyl function Ordinary differential equations Operator theory The paper is devoted to the spectral theory of vector-valued Sturm–Liouville operators on the half-line with a summable potential and a finite number of point interactions. It is shown that the positive spectrum is purely absolutely continuous and of constant multiplicity. The negative spectrum is either finite or discrete with the only accumulation point at zero. Our approach relies on the thorough investigation of the corresponding Weyl functions and involves technique elaborated in our previous papers. 271 313 1 10.4171/175-1/15 https://ems.eye-com.dev/doi/10.4171/175-1/15

André Hänel Leibniz Universität Hannover, Germany Timo Weidl Universität Stuttgart, Germany Laplacian, Neumann window, spectral asymptotics, Dirichlet-to-Neumann operator, Birman–Schwinger analysis Partial differential equations Quantum theory In the present article we will give a new proof of the ground state asymptotics of the Dirichlet Laplacian with a Neumann window acting on functions which are defined on a two-dimensional infinite strip or a three-dimensional infinite layer. The proof is based on the analysis of the corresponding Dirichlet-to-Neumann operator as a first order classical pseudo-differential operator. Using the explicit representation of its symbol we prove an asymptotic expansion as the window length decreases. 315 352 1 10.4171/175-1/16 https://ems.eye-com.dev/doi/10.4171/175-1/16

Bernard Helffer Université de Nantes, France Rola Kiwan American University in Dubai, United Arab Emirates Courant theorem, nodal lines, nodal domains, Dirichlet, cube Partial differential equations Global analysis, analysis on manifolds This paper is devoted to the refined analysis of Courant's theorem for the Dirichlet Laplacian in a bounded open set. Starting from the work by A. Pleijel in 1956, many papers have investigated in which cases the inequality in Courant's theorem is an equality. All these results were established for open sets in $\mathbb R^2$ or for surfaces like $\mathbb S^2$ or $\mathbb T^2$. The aim of the current paper is to look for the case of the cube in $\mathbb R^3$. We will prove that the only eigenvalues of the Dirichlet Laplacian which are Courant sharp are the two first eigenvalues. 353 371 1 10.4171/175-1/17 https://ems.eye-com.dev/doi/10.4171/175-1/17

Masao Hirokawa Hiroshima University, Japan Electron-phonon interaction, Carleman operator, infrared catastrophe Quantum theory This paper is dedicated to Pavel Exner on the occasion of his 70th birthday by introducing a part of our attempts in material science, that is, a mathematical modeling of electron-phonon interaction for small wave numbers close to zero. We extrapolate the electron-phonon interaction on the presupposition that the phonon dispersion relation and the electron-phonon coupling function are estimated using some experimental data. 373 399 1 10.4171/175-1/18 https://ems.eye-com.dev/doi/10.4171/175-1/18

Takashi Ichinose Kanazawa University, Japan Trotter product formula, Trotter–Kato product formula, Zeno product formula, unitary groups, resolvents, form sum of selfadjoint operators, Feynman path integral Operator theory Quantum theory Modified unitary form-sum Trotter–Kato product formula and Zeno product formula, which replace their short-time unitary groups appearing as factors in their products by their resolvents, are proved by Kato's method. 401 417 1 10.4171/175-1/19 https://ems.eye-com.dev/doi/10.4171/175-1/19

Sylwia Kondej University of Zielona Góra, Poland Schrödinger operator with delta potential, separation of variables, eigenvalues asymptotics Partial differential equations We consider two dimensional system governed by the Hamiltonian with delta interaction supported by two concentric circles separated by distance $d$. We analyze the asymptotics of the discrete eigenvalues for $d \to 0$ as well as for $d\to \infty$. 419 432 1 10.4171/175-1/20 https://ems.eye-com.dev/doi/10.4171/175-1/20

Hynek Kovařík Università degli Studi di Brescia, Italy Bartosch Ruszkowski Universität Stuttgart, Germany Timo Weidl Universität Stuttgart, Germany Heisenberg Laplacian, Berezin–Li–Yau inequality Operator theory Difference and functional equations We study Riesz means of eigenvalues of the Heisenberg Laplacian with Dirichlet boundary conditions on a cylinder in dimension three. We obtain an inequality with a sharp leading term and an additional lower order term. 433 446 1 10.4171/175-1/21 https://ems.eye-com.dev/doi/10.4171/175-1/21

Konstantin Pankrashkin Université Paris-Sud, Orsay, France Singular Schrödinger operator, leaky graph, $\delta$-potential, star graph Ordinary differential equations We provide a purely variational proof of the existence of eigenvalues below the bottom of the essential spectrum for the Schrödinger operator with an attractive $\delta$-potential supported by a star graph, i.e. by a finite union of rays emanating from the same point. In contrast to the previous works, the construction is valid without any additional assumption on the number or the relative position of the rays. The approach is used to obtain an upper bound for the lowest eigenvalue. 447 458 1 10.4171/175-1/22 https://ems.eye-com.dev/doi/10.4171/175-1/22

Yehuda Pinchover Technion - Israel Institute of Technology, Haifa, Israel Green function, ground state, Liouville theorem, positive solution, principal eigenvalue, small perturbation Partial differential equations Operator theory For a given second-order linear elliptic operator $L$ which admits a positive minimal Green function, and a given positive weight function $W$, we introduce a family of weighted Lebesgue spaces $L^p(\phi_p)$ with their dual spaces, where $1\leq p\leq \infty$. We study some fundamental properties of the corresponding (weighted) Green operators on these spaces. In particular, we prove that these Green operators are bounded on $L^p(\phi_p)$ for any $1\leq p\leq \infty$ with a uniform bound. We study the existence of a principal eigenfunction for these operators in these spaces, and the simplicity of the corresponding principal eigenvalue. We also show that such a Green operator is a resolvent of a densely defined closed operator which is equal to $(-W^{-1})L$ on $C_0^\infty$, and that this closed operator generates a strongly continuous contraction semigroup. Finally, we prove that if $W$ is a (semi)small perturbation of $L$, then for any $1\leq p\leq \infty$, the associated Green operator is compact on $L^p(\phi_p)$, and the corresponding spectrum is $p$-independent. 459 489 1 10.4171/175-1/23 https://ems.eye-com.dev/doi/10.4171/175-1/23

Olaf Post Universität Trier, Germany Abstract boundary value problems, Dirichlet-to-Neumann operator, graph Laplacians, coupled spaces Operator theory Combinatorics Dynamical systems and ergodic theory Global analysis, analysis on manifolds In this note we present some abstract ideas how one can construct spaces from building blocks according to a graph. The coupling is expressed via boundary pairs, and can be applied to very different spaces such as discrete graphs, quantum graphs or graph-like manifolds. We show a spectral analysis of graph-like spaces, and consider as a special case vector-valued quantum graphs. Moreover, we provide a prototype of a convergence theorem for shrinking graph-like spaces with Dirichlet boundary conditions. 491 524 1 10.4171/175-1/24 https://ems.eye-com.dev/doi/10.4171/175-1/24

Barry Simon California Institute of Technology, Pasadena, USA Periodic Jacobi matrices, discriminant, magic formula Operator theory Functions of a complex variable Let $K$ be a doubly infinite, self-adjoint matrix which is finite band (i.e. $K_{jk} = 0$ if $|j-k| > m$) and periodic ($KS^n = S^nK$ for some $n$ where $(Su)_j = u_{j+1}$) and non-degenerate (i.e. $K_{j j+m} \ne 0$ for all $j$). Then, there is a polynomial, $p(x,y)$, in two variables with $p(K,S^n) = 0$. This generalizes the tridiagonal case where $p(x,y) = y^2 - y \Delta(x) + 1$ where $\Delta$ is the discriminant. I hope Pavel Exner will enjoy this birthday bouquet. 525 529 1 10.4171/175-1/25 https://ems.eye-com.dev/doi/10.4171/175-1/25

Atushi Tanaka Tokyo Metropolitan University, Japan Taksu Cheon Kochi University of Technology, Japan Adiabatic theorem, eigenspace, homotopy, covering space Quantum theory Algebraic geometry An adiabatic time evolution of a closed quantum system connects eigenspaces of initial and final Hermitian Hamiltonians for slowly driven systems, or, unitary Floquet operators for slowly modulated driven systems. We show that the connection of eigenspaces depends on a topological property of the adiabatic paths for given initial and final points. An example in slowly modulated periodically driven systems is shown. These analysis are based on the topological analysis of the exotic quantum holonomy in adiabatic closed paths. 531 542 1 10.4171/175-1/26 https://ems.eye-com.dev/doi/10.4171/175-1/26

Ondřej Turek Czech Academy of Sciences, Řež - Prague, Czech Republic Quantum graph, vertex coupling, spectral filtering, quantum control Quantum theory We examine transmission through a quantum graph vertex to which auxiliary edges with constant potentials are attached. We find a characterization of vertex couplings for which the transmission probability from a given „input" line to a given „output" line shows a flat passband. The bandwidth is controlled directly by the potential on the auxiliary edges. Vertices with such couplings can thus serve as controllable band-pass filters. The paper extends earlier works on the topic. The result also demonstrates the effectivity of the $ST$-form of boundary conditions for a study of scattering in quantum graph vertices. 543 563 1 10.4171/175-1/27 https://ems.eye-com.dev/doi/10.4171/175-1/27

Valentin Zagrebnov Centre de Mathématiques et Informatique, Marseille, France Chernoff lemma, semigroup theory, product formula, convergence rate Operator theory The Chernoff $\sqrt{n}$-lemma is revised. This concerns two aspects: an improvement of the Chernoff estimate in the strong operator topology and an operator-norm estimate for quasi-sectorial contractions. Applications to the Lie–Trotter product formula approximation for semigroups is presented. 565 573 1 10.4171/175-1/28 https://ems.eye-com.dev/doi/10.4171/175-1/28

Tilted and Hochschild Extension Algebras Andrzej Skowroński Nicolaus Copernicus University, Toruń, Poland Kunio Yamagata Tokyo University of Agriculture and Technology, Japan Associative rings and algebras Commutative rings and algebras Linear and multilinear algebra; matrix theory Category theory; homological algebra (primary; secondary): 16-01; 13E10, 15A63, 15A69, 16Dxx, 16E10, 16E30, 16E40, 16G10, 16G20, 16G60, 16G70, 16S50, 16S70, 18A25, 18E30, 18G15 Mathematics Algebra Algebra, module, bimodule, representation, quiver, ideal, radical, simple module, semisimple module, uniserial module, projective module, injective module, tilting module, hereditary algebra, tilted algebra, Frobenius algebra, symmetric algebra, selfinjective algebra, Hochschild extension algebra, category, functor, torsion pair, projective dimension, injective dimension, global dimension, Euler form, Grothendieck group, irreducible homomorphism, almost split sequence, Auslander-Reiten translation, Auslander-Reiten quiver, stable equivalence, syzygy module, duality bimodule, Hochschild extension This is the second of three volumes which will provide a comprehensive introduction to the modern representation theory of Frobenius algebras. The first part of the book is devoted to fundamental results of the representation theory of finite dimensional hereditary algebras and their tilted algebras, which allow to describe the representation theory of prominent classes of Frobenius algebras. The second part is devoted to basic classical and recent results concerning the Hochschild extensions of finite dimensional algebras by duality bimodules and their module categories. Moreover, the shapes of connected components of the stable Auslander-Reiten quivers of Frobenius algebras are described. The only prerequisite in this volume is a basic knowledge of linear algebra and some results of the fi rst volume. It includes complete proofs of all results presented and provides a rich supply of examples and exercises. The text is primarily addressed to graduate students starting research in the representation theory of algebras as well mathematicians working in other fi elds. 5 12 2017 978-3-03719-174-3 978-3-03719-674-8 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/174 https://ems.eye-com.dev/doi/10.4171/174 Walter Schachermayer Universität Wien, Austria Statistics Probability theory and stochastic processes Game theory, economics, social and behavioral sciences Primary: 62P05, 91G10; Secondary: 60G44 Probability + statistics Portfolio optimization, transaction costs, shadow price, semimartingale, fractional Brownian motion A classical topic in Mathematical Finance is the theory of portfolio optimization. Robert Merton's work from the early seventies had enormous impact on academic research as well as on the paradigms guiding practitioners. One of the ramifications of this topic is the analysis of (small) proportional transaction costs, such as a Tobin tax. The lecture notes present some striking recent results of the asymptotic dependence of the relevant quantities when transaction costs tend to zero. An appealing feature of the consideration of transaction costs is that it allows for the first time to reconcile the no arbitrage paradigm with the use of non-semimartingale models, such as fractional Brownian motion. This leads to the culminating theorem of the present lectures which roughly reads as follows: for a fractional Brownian motion stock price model we always find a shadow price process for given transaction costs. This process is a semimartingale and can therefore be dealt with using the usual machinery of mathematical finance. 3 10 2017 978-3-03719-173-6 978-3-03719-673-1 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/173 https://ems.eye-com.dev/doi/10.4171/173 Hans Triebel University of Jena, Germany Partial differential equations 35–02, 46–02, 76–02, 92–02; 35K05, 35Q30, 35Q92, 42B35, 46E35, 76D05, 92C15, 92C17 Differential equations Function spaces of Besov–Sobolev type, chemotaxis, hydrodynamics, heat equations, Keller–Segel equations, Navier–Stokes equations This book deals with PDE models for chemotaxis (the movement of biological cells or organisms in response of chemical gradients) and hydrodynamics (viscous, homogeneous, and incompressible fluid filling the entire space). The underlying Keller–Segel equations (chemotaxis), Navier–Stokes equations (hydrodynamics), and their numerous modifications and combinations are treated in the context of inhomogeneous spaces of Besov–Sobolev type paying special attention to mapping properties of related nonlinearities. Further models are considered, including (deterministic) Fokker–Planck equations and chemotaxis Navier–Stokes equations. These notes are addressed to graduate students and mathematicians having a working knowledge of basic elements of the theory of function spaces, especially of Besov-Sobolev type and interested in mathematical biology and physics. 3 23 2017 978-3-03719-172-9 978-3-03719-672-4 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/172 https://ems.eye-com.dev/doi/10.4171/172 2523-5176 2523-5184 Henning Krause Universität Bielefeld, Germany Peter Littelmann Universität Köln, Germany Gunter Malle Universität Kaiserslautern, Germany Karl-Hermann Neeb FAU Erlangen-Nürnberg, Germany Christoph Schweigert Universität Hamburg, Germany Algebraic geometry Associative rings and algebras Nonassociative rings and algebras Category theory; homological algebra Primary: 14Mxx,16Gxx, 17Bxx, 18Exx, 20Gxx, 22Exx; secondary: 58Cxx, 81Txx. Algebraic geometry Algebraic groups, bounded and semibounded representations, categorification, character formulae, cluster algebras, Deligne-Lusztig theory, flat degenerations, geometrization, higher representation theory, highest weight categories, infinite dimensional Lie groups, local-global conjectures, special varieties, topological field theory From April 2009 until March 2016, the German Science Foundation supported generously the Priority Program SPP 1388 in Representation Theory. The core principles of the projects realized in the framework of the priority program have been categorification and geometrization, this is also reflected by the contributions to this volume. Apart from the articles by former postdocs supported by the priority program, the volume contains a number of invited research and survey articles, many of them are extended versions of talks given at the last joint meeting of the priority program in Bad Honnef in March 2015. This volume is covering current research topics from the representation theory of finite groups, of algebraic groups, of Lie superalgebras, of finite dimensional algebras and of infinite dimensional Lie groups. Graduate students and researchers in mathematics interested in representation theory will find this volume inspiring. It contains many stimulating contributions to the development of this broad and extremely diverse subject. 1 12 2017 978-3-03719-171-2 978-3-03719-671-7 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/171 https://ems.eye-com.dev/doi/10.4171/171 2523-515X 2523-5168

Alexander Alldridge Universität Köln, Germany Chevalley restriction theorem, differential operator, Harish-Chandra homomorphism, Lie superalgebra, radial part, Riemannian symmetric superspace Global analysis, analysis on manifolds Nonassociative rings and algebras Differential geometry General We study the restriction of invariant polynomials on the tangent space of a Riemannian symmetric supermanifold to a Cartan subspace. We survey known results in the case the symmetric space is a Lie supergroup, and more generally, where the Cartan subspace is even. We then describe an approach to this problem, developed in joint work in progress with K. Coulembier, based on the study of radial parts of di fferential operators. This leads to a characterisation of the invariant functions for an arbitrary linear isometric action, and as a special case, to a Chevalley restriction theorem valid for the isotropy representation of any contragredient Riemannian symmetric superspace. 1 11 1 10.4171/171-1/1 https://ems.eye-com.dev/doi/10.4171/171-1/1

Giovanni Cerulli Irelli Università di Roma La Sapienza, Italy Quiver Grassmannians, Dynkin quivers, cluster algebras Associative rings and algebras Algebraic geometry General The paper includes a new proof of the fact that quiver Grassmannians associated with rigid representations of Dynkin quivers do not have cohomology in odd degrees. Moreover, it is shown that they do not have torsion in homology. A new proof of the Caldero–Chapoton formula is provided. As a consequence a new proof of the positivity of cluster monomials in the acyclic clusters associated with Dynkin quivers is obtained. The methods used here are based on joint works with Markus Reineke and Evgeny Feigin. 13 45 1 10.4171/171-1/2 https://ems.eye-com.dev/doi/10.4171/171-1/2

Stéphanie Cupit-Foutou Ruhr-Universität Bochum, Germany Reductive groups, spherical varieties Algebraic geometry Group theory and generalizations General This paper is a brief overview on recent classi cation results and related problems concerning spherical varieties. The emphasis is made on the representation theoretical aspects of these objects. 47 57 1 10.4171/171-1/3 https://ems.eye-com.dev/doi/10.4171/171-1/3

Olivier Dudas Université Paris Diderot Paris 7, France Michela Varagnolo Université de Cergy-Pontoise, France Éric Vasserot Université Paris Diderot Paris 7, France Finite reductive groups, Deligne–Lusztig theory, higher representation theory Group theory and generalizations General In this note we explain how Lusztig's induction and restriction functors yield categorical actions of Kac–Moody algebras on the derived category of unipotent representations. We focus on the example of finite general linear groups and induction/restriction associated with split Levi subgroups, providing a derived analogue of Harish–Chandra induction/restriction as studied by Chuang–Rouquier in [5]. 59 74 1 10.4171/171-1/4 https://ems.eye-com.dev/doi/10.4171/171-1/4

Wolfgang Ebeling Universität Hannover, Germany Homological mirror symmetry, singularities, strange duality, invertible polynomials, derived categories, weighted projective lines, Coxeter-Dynkin diagrams, group action, orbifold E-function, Burnside ring, unimodal, bimodal Algebraic geometry Associative rings and algebras Several complex variables and analytic spaces Differential geometry We give a survey on results related to the Berglund–Hubsch duality of invertible polynomials and the homological mirror symmetry conjecture for singularities. 75 107 1 10.4171/171-1/5 https://ems.eye-com.dev/doi/10.4171/171-1/5

Michael Ehrig Universität Bonn, Germany Catharina Stroppel Universität Bonn, Germany Orthosymplectic supergroup, finite dimensional representations, Brauer algebra, Deligne category Nonassociative rings and algebras General We study the combinatorics of the category $\mathcal F$ of fi nite-dimensional modules for the orthosymplectic Lie supergroup OSp$(r|2n)$. In particular we present a positive counting formula for the dimension of the space of homomorphisms between two projective modules. This refi nes earlier results of Gruson and Serganova. For each block $\mathcal B$ we construct an algebra $A_\mathcal B$ whose module category shares the combinatorics with $\mathcal B$. It arises as a subquotient of a suitable limit of type D Khovanov algebras. It turns out that $A_\mathcal B$ is isomorphic to the endomorphism algebra of a minimal projective generator of $\mathcal B$. This provides a direct link from $\mathcal F$ to parabolic categories $\mathcal O$ of type B/D, with maximal parabolic of type A, to the geometry of isotropic Grassmannians of types B/D and to Springer fi bres of type C/D. We also indicate why $\mathcal F$ is not highest weight in general. 109 170 1 10.4171/171-1/6 https://ems.eye-com.dev/doi/10.4171/171-1/6

Ben Elias University of Oregon, Eugene, USA Noah Snyder Indiana University, Bloomington, United States Geordie Williamson Max-Planck-Institut für Mathematik, Bonn, Germany Frobenius extensions, diagrammatic algebra, Soergel bimodules $K$-theory Nonassociative rings and algebras Category theory; homological algebra General Given a hypercube of Frobenius extensions between commutative algebras, we provide a diagrammatic description of some natural transformations between compositions of induction and restriction functors, in terms of colored transversely-intersecting planar 1-manifolds. The relations arise in the first and third author’s work on (singular) Soergel bimodules. 171 186 1 10.4171/171-1/7 https://ems.eye-com.dev/doi/10.4171/171-1/7

Xin Fang Universität Köln, Germany Ghislain Fourier University of Glasgow, UK Peter Littelmann Universität Köln, Germany Flag varieties, spherical varieties, cluster algebras, toric degenerations Algebraic geometry Convex and discrete geometry General Following the historical track in pursuing $T$-equivariant flat toric degenerations of flag varieties and spherical varieties, we explain how powerful tools in algebraic geometry and representation theory, such as canonical bases, Newton–Okounkov bodies, PBW- filtrations and cluster algebras come to push the subject forward. 187 232 1 10.4171/171-1/8 https://ems.eye-com.dev/doi/10.4171/171-1/8

Peter Fiebig Universität Erlangen-Nürnberg, Germany Restricted representations, critical level, Kac–Moody algebras, Feigin–Frenkel conjecture Nonassociative rings and algebras Quantum theory General We introduce subquotient categories of the restricted category $\mathcal O$ over an affi ne Kac–Moody algebra at the critical level and show that some of them have a realization in terms of moment graph sheaves. 233 253 1 10.4171/171-1/9 https://ems.eye-com.dev/doi/10.4171/171-1/9

Jürgen Fuchs Karlstads Universitet, Sweden Christoph Schweigert Universität Hamburg, Germany Topological field theory, tensor categories, categorification Quantum theory Manifolds and cell complexes General Structures in low-dimensional topology and low-dimensional geometry – often combined with ideas from (quantum) fi eld theory – can explain and inspire concepts in algebra and in representation theory and their categori ed versions. We present a personal view on some of these instances which have appeared within the Research Priority Programme SPP 1388 "Representation theory". 255 267 1 10.4171/171-1/10 https://ems.eye-com.dev/doi/10.4171/171-1/10

Martin Kalck University of Edinburgh, UK Triangulated categories, quasi-hereditary algebras, exceptional sequences, recollements, gentle algebras, derived equivalences, derived composition series, derived Jordan–Hölder property Category theory; homological algebra Associative rings and algebras General We study composition series of derived module categories in the sense of Angeleri Hügel, König & Liu for quasi-hereditary algebras. More precisely, we show that having a composition series with all factors being derived categories of vector spaces does not characterise derived categories of quasi-hereditay algebras. This gives a negative answer to a question of Liu & Yang and the proof also confi rms part of a conjecture of Bobi nski & Malicki. In another direction, we show that derived categories of quasi-hereditary algebras can have composition series with lots of diff erent lengths and composition factors. In other words, there is no Jordan–Hölder property for composition series of derived categories of quasi-hereditary algebras. 269 308 1 10.4171/171-1/11 https://ems.eye-com.dev/doi/10.4171/171-1/11

Steffen Koenig Universität Stuttgart, Germany Dominant dimension, representation dimension, Schur algebras, gendosymmetric algebras Associative rings and algebras General Dominant dimension is a little known homological dimension, which is, however, crucial in many respects, both for abstractly studying fi nite dimensional algebras and their representation theory, and for applications to group algebras or in algebraic Lie theory. Various aspects and recent applications of dominant dimension will be outlined and illustrated. 309 330 1 10.4171/171-1/12 https://ems.eye-com.dev/doi/10.4171/171-1/12

Henning Krause Universität Bielefeld, Germany Highest weight category, strict polynomial functor, polynomial representation, divided power, Schur algebra, quasi-hereditary algebra, Ringel duality Group theory and generalizations Category theory; homological algebra General Highest weight categories are described in terms of standard objects and recollements of abelian categories, working over an arbitrary commutative base ring. Then the highest weight structure for categories of strict polynomial functors is explained, using the theory of Schur and Weyl functors. A consequence is the well-known fact that Schur algebras are quasi-hereditary. 331 373 1 10.4171/171-1/13 https://ems.eye-com.dev/doi/10.4171/171-1/13

Julian Külshammer Universität Stuttgart, Germany BGG category $\mathcal O$, bocs, Eilenberg–Moore category, exact Borel subalgebra, Kleisli category, $q$-Schur algebras, quasi-hereditary algebras, reduction algorithm, representation type, Schur algebras, tame, wild Associative rings and algebras Nonassociative rings and algebras Category theory; homological algebra General This paper surveys bocses, quasi-hereditary algebras and their relationship which was established in a recent result by Koenig, Ovsienko, and the author. Particular emphasis is placed on applications of this result to the representation type of the category filtered by standard modules for a quasi-hereditary algebra. In this direction, joint work with Thiel is presented showing that the subcategory of modules fi ltered by Weyl modules for tame Schur algebras is of fi nite representation type. The paper also includes a new proof for the classi cation of quasi-hereditary algebras with two simple modules, a result originally obtained by Membrillo–Hernández in [70]. 375 426 1 10.4171/171-1/14 https://ems.eye-com.dev/doi/10.4171/171-1/14

Sefi Ladkani University of Haifa, Israel Algebra of quaternion type, 2-CY-tilted algebra, Brauer graph algebra, derived equivalence, Jacobian algebra, marked surface, periodic modules, quiver with potential, ribbon graph, ribbon quiver, triangulation quiver, triangulation algebra, symmetric algebra Associative rings and algebras Commutative rings and algebras Category theory; homological algebra Group theory and generalizations We construct a new class of symmetric algebras of tame representation type that are also the endomorphism algebras of cluster-tilting objects in 2-Calabi–Yau triangulated categories, hence all their non-projective indecomposable modules are $\Omega$-periodic of period dividing 4. Our construction is based on the combinatorial notion of triangulation quivers, which arise naturally from triangulations of oriented surfaces with marked points. This class of algebras contains the algebras of quaternion type introduced and studied by Erdmann with relation to certain blocks of group algebras. On the other hand, it contains also the Jacobian algebras of the quivers with potentials associated by Fomin–Shapiro–Thurston and Labardini–Fragoso to triangulations of closed surfaces with punctures, hence our construction may serve as a bridge between the modular representation theory of nite groups and the theory of cluster algebras. 427 500 1 10.4171/171-1/15 https://ems.eye-com.dev/doi/10.4171/171-1/15

Martina Lanini Edinburgh University, UK Moment graphs, semi-infinite order, character formulae Nonassociative rings and algebras Group theory and generalizations General In this work we discuss some appearances of semi-infi nite combinatorics in representation theory. We propose a semi-in finite moment graph theory and we motivate it by considering the (not yet rigorously de fined) geometric side of the story. We show that it is possible to compute stalks of the local intersection cohomology of the semi-infi nite flag variety, and hence of spaces of quasi maps, by performing an algorithm due to Braden and MacPherson. 501 518 1 10.4171/171-1/16 https://ems.eye-com.dev/doi/10.4171/171-1/16

Gunter Malle TU Kaiserslautern, Germany Local-global conjectures, McKay conjecture, Alperin–McKay conjecture, Alperin weight conjecture, Brauer's height zero conjecture, Dade conjecture, reduction theorems Group theory and generalizations General We give a survey of recent developments in the investigation of the various local-global conjectures for representations of nite groups. 519 539 1 10.4171/171-1/17 https://ems.eye-com.dev/doi/10.4171/171-1/17

Karl-Hermann Neeb Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany Infinite dimensional Lie group, host algebra, semibounded representation, bounded representation, holomorphic induction Topological groups, Lie groups General In this note we describe the recent progress in the classi fication of bounded and semibounded representations of in finite dimensional Lie groups. We start with a discussion of the semiboundedness condition and how the new concept of a smoothing operator can be used to construct $C*$ -algebras (so called host algebras) whose representations are in one-to-one correspondence with certain semibounded representations of an infi nite dimensional Lie group $G$. This makes the full power of $C*$ -theory available in this context. Then we discuss the classi cation of bounded representations of several types of unitary groups on Hilbert spaces and of gauge groups. After explaining the method of holomorphic induction as a means to pass from bounded representations to semibounded ones, we describe the classifi cation of semibounded representations for hermitian Lie groups of operators, loop groups (with infi nite dimensional targets), the Virasoro group and certain in finite dimensional oscillator groups. 541 563 1 10.4171/171-1/18 https://ems.eye-com.dev/doi/10.4171/171-1/18

Ivan Penkov Jacobs-Universität Bremen, Germany Alexey Petukhov The University of Manchester, UK Primitive ideals, finitary Lie algebras, highest weight modules, $\mathfrak{osp}$-duality Nonassociative rings and algebras General We provide a review of results on two-sided ideals in the enveloping algebra $\operatorname{U}(\mathfrak g(\infty))$ of a locally simple Lie algebra $\mathfrak g(\infty)$. We pay special attention to the case when $\mathfrak g(\infty)$ is one of the finitary Lie algebras $\mathfrak{sl}(\infty), \mathfrak o(\infty), \mathfrak{sp}(\infty)$. The main results include a description of all integrable ideals in $\operatorname{U}(\mathfrak g(\infty))$, as well as a criterion for the annihilator of an arbitrary (not necessarily integrable) simple highest weight module to be nonzero. This criterion is new for $\mathfrak g(\infty)=\mathfrak o(\infty), \mathfrak{sp}(\infty)$. All annihilators of simple highest weight modules are integrable ideals for $\mathfrak g(\infty)=\mathfrak{sl}(\infty),$ $\mathfrak o(\infty)$. Finally, we prove that the lattices of ideals in $\operatorname{U}(\mathfrak o(\infty))$ and $\operatorname{U}(\mathfrak{sp}(\infty))$ are isomorphic. 565 602 1 10.4171/171-1/19 https://ems.eye-com.dev/doi/10.4171/171-1/19

Guido Pezzini Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany Algebraic groups, spherical varieties, representation theory, Kac–Moody groups Algebraic geometry Group theory and generalizations General In this short note we review some applications of the theory of spherical varieties in related fields, some generalizations of this theory, and present some open problems. 603 612 1 10.4171/171-1/20 https://ems.eye-com.dev/doi/10.4171/171-1/20

Markus Reineke Ruhr-Universität Bochum, Germany Quiver moduli, GIT quotients, small desingularizations Associative rings and algebras Algebraic geometry Algebraic topology General We construct small desingularizations of moduli spaces of semistable quiver representations for indivisible dimension vectors using deformations of stabilities and a dimension estimate for nullcones. We apply this construction to several classes of GIT quotients. 613 635 1 10.4171/171-1/21 https://ems.eye-com.dev/doi/10.4171/171-1/21

Henrik Seppänen Georg-August Universität Göttingen, Germany Valdemar Tsanov Georg-August Universität Göttingen, Germany Flag variety, geometric invariant theory, principal SL(2)-subgroup, branching cone, Mori dream space Algebraic geometry Nonassociative rings and algebras General Let $G$ be a semisimple complex Lie group. In this article, we study Geometric Invariant Theory on a flag variety $G/B$ with respect to the action of a principal 3-dimensional simple subgroup $S\subset G$. We determine explicitly the GIT-equivalence classes of $S$-ample line bundles on $G/B$. We show that, under mild assumptions, among the GIT-classes there are chambers, in the sense of Dolgachev-Hu. The GIT-quotients with respect to various chambers form a family of Mori dream spaces, canonically associated with $G$. We are able to determine the three important cones in the Picard group of any of these quotients: the pseudoeffective-, the movable-, and the nef cones. 637 663 1 10.4171/171-1/22 https://ems.eye-com.dev/doi/10.4171/171-1/22

Britta Späth Bergische Universität Wuppertal, Germany Reduction theorems, character triples, counting conjectures Group theory and generalizations General In recent years several global/local conjectures in the representation theory of fi nite groups have been reduced to conditions on quasi-simple groups. We reformulate the inductive conditions for the conjectures by Alperin and McKay using (new) order relations between ordinary, respectively modular character triples. This allows to clarify the similarities and di fferences between those conditions. 665 680 1 10.4171/171-1/23 https://ems.eye-com.dev/doi/10.4171/171-1/23

Ulrich Thiel Universität Stuttgart, Germany Rational Cherednik algebras, Calogero–Moser spaces, reflection groups Associative rings and algebras Group theory and generalizations General We give an overview of the representation theory of restricted rational Cherednik algebras. These are certain finite-dimensional quotients of rational Cherednik algebras at $t = 0$. Their representation theory is connected to the geometry of the Calogero–Moser space, and there is a lot of evidence that they contain certain information about Hecke algebras even though the precise connection is so far unclear. We outline the basic theory along with some open problems and conjectures, and give explicit results in the cyclic and dihedral cases. 681 745 1 10.4171/171-1/24 https://ems.eye-com.dev/doi/10.4171/171-1/24

Christoph Zellner Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany infinite dimensional Lie group, smooth vector, analytic vector, semibounded representation, continuous representation Topological groups, Lie groups General Let $G$ be a locally convex Lie group and $\pi:G \to \mathrm{U}(\mathcal H)$ be a continuous unitary representation. $\pi$ is called smooth if the space of $\pi$-smooth vectors $\mathcal H^\infty\subset \mathcal H$ is dense. In this article we show that under certain conditions, concerning in particular the structure of the Lie algebra $\mathfrak{g}$ of $G$, a continuous unitary representation of $G$ is automatically smooth. As an application, this yields a dense space of smooth vectors for continuous positive energy representations of oscillator groups, double extensions of loop groups and the Virasoro group. Moreover we show the existence of a dense space of analytic vectors for the class of semibounded representations of Banach–Lie groups. Here $\pi$ is called semibounded, if $\pi$ is smooth and there exists a non-empty open subset $U\subset\mathfrak{g}$ such that the operators $i \mathrm d\ pi(x)$ from the derived representation are uniformly bounded from above for $x \in U$. 747 763 1 10.4171/171-1/25 https://ems.eye-com.dev/doi/10.4171/171-1/25

Alessio Figalli ETH Zürich, Switzerland Partial differential equations Differential geometry Primary: 35J96; Secondary: 35B65, 35J60, 35J66, 35B45, 35B50, 35D05, 35D10, 35J65, 53A15, 53C45 Monge–Ampère equation, weak and strong solutions, existence, uniqueness, regularity The Monge-Ampère equation is one of the most important partial differential equations, appearing in many problems in analysis and geometry. This monograph is a comprehensive introduction to the existence and regularity theory of the Monge-Ampère equation and some selected applications; the main goal is to provide the reader with a wealth of results and techniques he or she can draw from to understand current research related to this beautiful equation. The presentation is essentially self-contained, with an appendix wherein one can find precise statements of all the results used from different areas (linear algebra, convex geometry, measure theory, nonlinear analysis, and PDEs). This book is intended for graduate students and researchers interested in nonlinear PDEs: explanatory figures, detailed proofs, and heuristic arguments make this book suitable for self-study and also as a reference. 1 31 2017 978-3-03719-170-5 978-3-03719-670-0 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/170 http://www.ems-ph.org/doi/10.4171/170 Nicolas Raymond Université de Rennes, France Partial differential equations Calculus of variations and optimal control; optimization Quantum theory 35P15, 35P20, 49R05, 81Q10, 81Q20 Magnetic Schrödinger equation, discrete spectrum, semiclassical analysis, magnetic harmonic approximation This book is a synthesis of recent advances in the spectral theory of the magnetic Schrödinger operator. It can be considered a catalog of concrete examples of magnetic spectral asymptotics. Since the presentation involves many notions of spectral theory and semiclassical analysis, it begins with a concise account of concepts and methods used in the book and is illustrated by many elementary examples. Assuming various points of view (power series expansions, Feshbach–Grushin reductions, WKB constructions, coherent states decompositions, normal forms) a theory of Magnetic Harmonic Approximation is then established which allows, in particular, accurate descriptions of the magnetic eigenvalues and eigenfunctions. Some parts of this theory, such as those related to spectral reductions or waveguides, are still accessible to advanced students while others (e.g., the discussion of the Birkhoff normal form and its spectral consequences, or the results related to boundary magnetic wells in dimension three) are intended for seasoned researchers. 1 31 2017 978-3-03719-169-9 978-3-03719-669-4 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/169 http://www.ems-ph.org/doi/10.4171/169 27 Lecture notes by Diana Davis, Bryce Weaver, Roland K. W. Roeder, Pablo Lessa Françoise Dal’Bo Université de Rennes I, France François Ledrappier University of Notre Dame, USA Amie Wilkinson University of Chicago, USA Dynamical systems and ergodic theory Differential geometry 37A, 37B, 37D,37F, 37H, 53A Dynamical systems, geometry, ergodic theory, billards, complex dynamics, random walk, group theory This book arose from 4 lectures given at the Undergraduate Summer School of the Thematic Program Dynamics and Boundaries held at the University of Notre Dame. It is intended to introduce (under)graduate students to the field of dynamical systems by emphasizing elementary examples, exercises and bare hands constructions. The lecture of Diana Davis is devoted to billiard flows on polygons, a simple-sounding class of continuous time dynamical system for which many problems remain open. Bryce Weaver focuses on the dynamics of a 2x2 matrix acting on the flat torus. This example introduced by Vladimir Arnold illustrates the wide class of uniformly hyperbolic dynamical systems, including the geodesic flow for negatively curved, compact manifolds. Roland Roeder considers a dynamical system on the complex plane governed by a quadratic map with a complex parameter. These maps exhibit complicated dynamics related to the Mandelbrot set defined as the set of parameters for which the orbit remains bounded. Pablo Lessa deals with a type of non-deterministic dynamical system: a simple walk on an infinite graph, obtained by starting at a vertex and choosing a random neighbor at each step. The central question concerns the recurrence property. When the graph is a Cayley graph of a group, the behavior of the walk is deeply related to algebraic properties of the group. 11 30 2016 978-3-03719-168-2 978-3-03719-668-7 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/168 http://www.ems-ph.org/doi/10.4171/168

Diana Davis Williams College, WILLIAMSTOWN, UNITED STATES Dynamical systems and ergodic theory General 1 55 1 10.4171/168-1/1 http://www.ems-ph.org/doi/10.4171/168-1/1

Bryce Weaver James Madison University, HARRISONBURG, UNITED STATES Dynamical systems and ergodic theory General 57 100 1 10.4171/168-1/2 http://www.ems-ph.org/doi/10.4171/168-1/2

Roland Roeder Indiana University Purdue University Indianapolis, INDIANAPOLIS, UNITED STATES Dynamical systems and ergodic theory General 101 155 1 10.4171/168-1/3 http://www.ems-ph.org/doi/10.4171/168-1/3

Pablo Lessa Universidad de la República, MONTEVIDEO, URUGUAY Dynamical systems and ergodic theory General 157 201 1 10.4171/168-1/4 http://www.ems-ph.org/doi/10.4171/168-1/4

Vincent Guedj Université Paul Sabatier, Toulouse, France Ahmed Zeriahi Université Paul Sabatier, Toulouse, France Several complex variables and analytic spaces 32W20, 32U20, 32Q20, 35D30, 53C55, 32U15, 32U40, 35B65 Complex analysis Pluripotential theory, plurisubharmonic functions, complex Monge–Ampère operators, generalized capacities, weak solutions, a priori estimates, canonical Kähler metrics, singular varieties Winner of the 2016 EMS Monograph Award! Complex Monge–Ampère equations have been one of the most powerful tools in Kähler geometry since Aubin and Yau’s classical works, culminating in Yau’s solution to the Calabi conjecture. A notable application is the construction of Kähler-Einstein metrics on some compact Kähler manifolds. In recent years degenerate complex Monge–Ampère equations have been intensively studied, requiring more advanced tools. The main goal of this book is to give a self-contained presentation of the recent developments of pluripotential theory on compact Kähler manifolds and its application to Kähler–Einstein metrics on mildly singular varieties. After reviewing basic properties of plurisubharmonic functions, Bedford–Taylor’s local theory of complex Monge–Ampère measures is developed. In order to solve degenerate complex Monge–Ampère equations on compact Kähler manifolds, fine properties of quasi-plurisubharmonic functions are explored, classes of finite energies defined and various maximum principles established. After proving Yau’s celebrated theorem as well as its recent generalizations, the results are then used to solve the (singular) Calabi conjecture and to construct (singular) Kähler–Einstein metrics on some varieties with mild singularities. The book is accessible to advanced students and researchers of complex analysis and differential geometry. 1 12 2017 978-3-03719-167-5 978-3-03719-667-0 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/167 https://ems.eye-com.dev/doi/10.4171/167 26 Yves Cornulier Université Paris-Sud, Orsay, France Pierre de la Harpe Université de Genève, Switzerland Group theory and generalizations Topological groups, Lie groups Geometry Manifolds and cell complexes Primary: 20F65; Secondary: 20F05, 22D05, 51F99, 54E35, 57M07, 57T20 Locally compact groups, left-invariant metrics, $\sigma$-compactness, second countability, compact generation, compact presentation, metric coarse equivalence, quasi-isometry, coarse connectedness, coarse simple connectedness, growth, amenability The main aim of this book is the study of locally compact groups from a geometric perspective, with an emphasis on appropriate metrics that can be defined on them. The approach has been successful for finitely generated groups, and can favourably be extended to locally compact groups. Parts of the book address the coarse geometry of metric spaces, where ‘coarse’ refers to that part of geometry concerning properties that can be formulated in terms of large distances only. This point of view is instrumental in studying locally compact groups. Basic results in the subject are exposed with complete proofs, others are stated with appropriate references. Most importantly, the development of the theory is illustrated by numerous examples, including matrix groups with entries in the the field of real or complex numbers, or other locally compact fields such as p-adic fields, isometry groups of various metric spaces, and, last but not least, discrete group themselves. The book is aimed at graduate students and advanced undergraduate students, as well as mathematicians who wish some introduction to coarse geometry and locally compact groups. 9 30 2016 978-3-03719-166-8 978-3-03719-666-3 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/166 http://www.ems-ph.org/doi/10.4171/166 25 Dan-Virgil Voiculescu University of California, Berkeley, USA Nicolai Stammeier University of Oslo, Norway Moritz Weber Universität Saarbrücken, Germany Functional analysis Group theory and generalizations Operator theory Probability theory and stochastic processes Primary 46L54; secondary 60B20, 47C15, 20G42 Free probability, operator algebras, random matrices, free monotone transport, free group factors, free convolution, compact quantum groups, easy quantum groups, noncrossing partitions, free independence, entropy, max-stable laws, exchangeability Free probability is a probability theory dealing with variables having the highest degree of noncommutativity, an aspect found in many areas (quantum mechanics, free group algebras, random matrices etc). Thirty years after its foundation, it is a well-established and very active field of mathematics. Originating from Voiculescu’s attempt to solve the free group factor problem in operator algebras, free probability has important connections with random matrix theory, combinatorics, harmonic analysis, representation theory of large groups, and wireless communication. These lecture notes arose from a masterclass in Münster, Germany and present the state of free probability from an operator algebraic perspective. This volume includes introductory lectures on random matrices and combinatorics of free probability (Speicher), free monotone transport (Shlyakhtenko), free group factors (Dykema), free convolution (Bercovici), easy quantum groups (Weber), and a historical review with an outlook (Voiculescu). In order to make it more accessible, the exposition features a chapter on basics in free probability, and exercises for each part. This book is aimed at master students to early career researchers familiar with basic notions and concepts from operator algebras. 7 31 2016 978-3-03719-165-1 978-3-03719-665-6 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/165 http://www.ems-ph.org/doi/10.4171/165

Dan-Virgil Voiculescu University of California, BERKELEY, UNITED STATES Functional analysis General 1 6 1 10.4171/165-1/1 http://www.ems-ph.org/doi/10.4171/165-1/1

Moritz Weber Universität des Saarlandes, SAARBRÜCKEN, GERMANY Probability theory and stochastic processes General 7 16 1 10.4171/165-1/2 http://www.ems-ph.org/doi/10.4171/165-1/2

Roland Speicher Universität des Saarlandes, SAARBRÜCKEN, GERMANY Probability theory and stochastic processes General 17 37 1 10.4171/165-1/3 http://www.ems-ph.org/doi/10.4171/165-1/3

Dimitri Shlyakhtenko University of California Los Angeles, LOS ANGELES, UNITED STATES Probability theory and stochastic processes General 39 56 1 10.4171/165-1/4 http://www.ems-ph.org/doi/10.4171/165-1/4

Ken Dykema Texas A&M University, COLLEGE STATION, UNITED STATES Functional analysis General 57 72 1 10.4171/165-1/5 http://www.ems-ph.org/doi/10.4171/165-1/5

Hari Bercovici Indiana University, BLOOMINGTON, UNITED STATES Operator theory General 73 93 1 10.4171/165-1/6 http://www.ems-ph.org/doi/10.4171/165-1/6

Moritz Weber Universität des Saarlandes, SAARBRÜCKEN, GERMANY Operator theory General 95 121 1 10.4171/165-1/7 http://www.ems-ph.org/doi/10.4171/165-1/7

Wolfgang König WIAS Berlin and Technical University Berlin, Germany General 00-XX Mathematics in the public; mathematics in architecture, biology, climate, encryption, engineering, finance, industry, nature shapes, physics, telecommunication, and voting systems; experimental mathematics, mathematics museums, mathematics for complex data The ubiquity and importance of mathematics in our complex society is generally not in doubt. However, even a scientifically interested layman would be hard pressed to point out aspects of our society where contemporary mathematical research is essential. Most popular examples are finance, engineering, wheather and industry, but the way mathematics comes into play is widely unknown in the public. And who thinks of application fields like biology, encryption, architecture, or voting systems? This volume comprises a number of success stories of mathematics in our society – important areas being shaped by cutting edge mathematical research. The authors are eminent mathematicians with a high sense for public presentation, addressing scientifically interested laymen as well as professionals in mathematics and its application disciplines. 7 4 2016 978-3-03719-164-4 978-3-03719-664-9 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/164 http://www.ems-ph.org/doi/10.4171/164

George Szpiro , NEW YORK, UNITED STATES General 1 5 1 10.4171/164-1/1 http://www.ems-ph.org/doi/10.4171/164-1/1

David Bailey University of California at Davis, DAVIS, UNITED STATES Jonathan Borwein The University of Newcastle, CALLAGHAN, AUSTRALIA General Computer-based tools for mathematics are changing how mathematics is researched, taught and communicated to society. Future technology trends point to ever-more powerful tools in the future. Computation in mathematics is thus giving rise to a new mode of mathematical research, where algorithms, datasets and public databases are as significant as the resulting theorems, and even the definition of what constitutes secure mathematical knowledge is seen in a new light. 7 25 1 10.4171/164-1/2 http://www.ems-ph.org/doi/10.4171/164-1/2

Albrecht Beutelspacher Mathematikum Giessen e.V., GIESSEN, GERMANY General In this article we look at mathematical experiments, in particular those shown in mathematical science centers. Although some mathematicians have the sneaking suspicion that such experiments are far too superficial and do not correspond to proper mathematics, we try to show that in fact mathematical experiments provide an ideal first step for the general public into mathematics. 27 35 1 10.4171/164-1/3 http://www.ems-ph.org/doi/10.4171/164-1/3

Walter Schachermayer Universität Wien, WIEN, AUSTRIA General This article consists of two parts. The first briefly discusses the history and the basic ideas of option pricing. Based on this background, in the second part we critically analyze the role of academic research in Mathematical Finance relating to the emergence of the 2007–2008 financial crisis. 37 50 1 10.4171/164-1/4 http://www.ems-ph.org/doi/10.4171/164-1/4

Aad van der Vaart Universiteit Leiden, LEIDEN, NETHERLANDS Wessel van Wieringen VU University Medical Center, AMSTERDAM, NETHERLANDS Statistics General High-dimensional data and models are central to modern statistics. We review some key concepts, some intriguing connections to ideas of the past, and some methods and theoretical results developed in the past decade. We illustrate these results in the context of genomic data from cancer research. 51 70 1 10.4171/164-1/5 http://www.ems-ph.org/doi/10.4171/164-1/5

Ofer Zeitouni Weizmann Institute of Science, REHOVOT, ISRAEL General The evolution of engineering needs, especially in the areas of estimation and system theory, has triggered the development of mathematical tools which, in turns, have had a profound influence on engineering practice. We describe this interaction through one example, the evolution of filtering theory. 71 80 1 10.4171/164-1/6 http://www.ems-ph.org/doi/10.4171/164-1/6

Jean Bertoin Universität Zürich, ZÜRICH, SWITZERLAND General Mathematical models are used more and more frequently in Life Sciences. These may be deterministic, or stochastic. We present some classical models for population dynamics and discuss in particular the averaging effect in the setting of large populations, to point at circumstances where randomness prevails nonetheless. 81 99 1 10.4171/164-1/7 http://www.ems-ph.org/doi/10.4171/164-1/7

Jürg Fröhlich ETH Zürich, ZÜRICH, SWITZERLAND General The purpose of this paper is to illustrate, on some concrete examples, the quest of theoretical physicists for new laws of Nature and for appropriate mathematical structures that enables them to formulate and analyze new laws in as simple terms as possible and to derive consequences therefrom. The examples are taken from thermodynamics, atomism and quantum theory. 101 129 1 10.4171/164-1/8 http://www.ems-ph.org/doi/10.4171/164-1/8

Helmut Pottmann Technische Universität Wien, WIEN, AUSTRIA Johannes Wallner Technische Universität Graz, GRAZ, AUSTRIA Geometry General During the last decade, the geometric aspects of freeform architecture have defined a field of applications which is systematically explored and which conversely serves as inspiration for new mathematical research. This paper discusses topics relevant to the realization of freeform skins by various means (flat and curved panels, straight and curved members, masonry, etc.) and illuminates the interrelations of those questions with theory, in particular discrete differential geometry and discrete conformal geometry. 131 151 1 10.4171/164-1/9 http://www.ems-ph.org/doi/10.4171/164-1/9

Christiane Rousseau University of Montreal, MONTRÉAL, CANADA Geometry General Since ancient times, the development of mathematics has been inspired, at least in part, by the need to provide models in other sciences, and that of describing and understanding the world around us. In this note, we concentrate on the shapes of nature and introduce two related geometries that play an important role in contemporary science. Fractal geometry allows describing a wide range of shapes in nature. In 1973, Harry Blum introduced a new geometry well suited to describe animal morphology. 153 165 1 10.4171/164-1/10 http://www.ems-ph.org/doi/10.4171/164-1/10

Helmut Neunzert ITWM, KAISERSLAUTERN, GERMANY General Industrial mathematics has become a fashionable subject in the last three decades. Today, there are many university groups and even research institutes dedicated to industrial and applied mathematics worldwide, proving that mathematics has indeed become a key technology. In this paper we first give a short account of the history of industrial mathematics. Using experiences from the Fraunhofer Institute for Industrial Mathematics (ITWM), we try to characterize the specific problem driven work of industrial mathematicians and take a look at future challenges. 167 183 1 10.4171/164-1/11 http://www.ems-ph.org/doi/10.4171/164-1/11

Holger Boche Technische Universität München, MÜNCHEN, GERMANY Ezra Tampubolon Technische Universität München, MÜNCHEN, GERMANY General Orthogonal transmission schemes constitute the foundations of both our present and future communication standards. One of the major drawback of orthogonal transmission schemes is their high dynamical behaviour, which can be measured by the so called Peakto–Average power value – the ratio between the peak value (i.e. $L^\infty$-norm) and the average power (i.e., $L^2$-norm) of a signal. This undesired behaviour of orthogonal schemes has remarkable negative impacts on the performance, the energy-efficiency, and the maintain cost of the transmission systems. In this work, we give some discussions concerning to the problem of reduction of the high dynamics of an orthogonal transmission scheme. We show that this problem is connected with some mathematical fields, such as functional analysis (Hahn-Banach Theorem and Baire Category), additive combinatorics (Szeméredi Theorem, Green-Tao Theorem on arithmetic progressions in the primes, sparse Szeméredi type Theorems by Conlon and Gowers, and the famous Erd˝os problem on arithmetic progressions), and both trigonometric and non-trigonometric harmonic analysis. 185 220 1 10.4171/164-1/12 http://www.ems-ph.org/doi/10.4171/164-1/12

Claus Diem Universität Leipzig, LEIPZIG, GERMANY General Information processing by electronic devices leads to a multitude of security-relevant challenges. With the help of cryptography, many of these challenges can be solved and new applications can be made possible. What methods are hereby used? On which mathematical foundations do they rest? How did the prevailing ideas and methods come about? What are the current developments, what challenges exist and which future challenges can be predicted? 221 250 1 10.4171/164-1/13 http://www.ems-ph.org/doi/10.4171/164-1/13

Werner Kirsch FernUniversität Hagen, HAGEN, GERMANY General In this article we describe some concepts, ideas and results from the mathematical theory of voting. We give a mathematical description of voting systems and introduce concepts to measure the power of a voter. We also describe and investigate two-tier voting systems, for example the Council of the European Union. In particular, we prove criteria which give the optimal voting weights in such systems. 251 279 1 10.4171/164-1/14 http://www.ems-ph.org/doi/10.4171/164-1/14

Jörn Behrens Universität Hamburg, HAMBURG, GERMANY General Studying the climate, weather or other geoscientific phenomena is strongly related to simulation based knowledge gain, since the climate system, for example, is not assessable by laboratory experiments. In these simulations, mathematical models as well as numerical methods play a crucial role in many aspects of the knowledge work-flow. We will describe the general set-up of geoscientific models, and explore some of the applied mathematical methods involved in solving such models. One of the paramount problems of geoscientific simulation applications is the large span of scales that interact. 281 293 1 10.4171/164-1/15 http://www.ems-ph.org/doi/10.4171/164-1/15

Volume II Davide Barilari Université Paris 7 Denis Diderot, Paris, France Ugo Boscain École Polytechnique, Palaiseau, France Mario Sigalotti École Polytechnique, Palaiseau, France Differential geometry Partial differential equations Calculus of variations and optimal control; optimization Probability theory and stochastic processes 53C17, 35H10, 60H30, 49J15 Sub-Riemannian geometry, hypoelliptic operators, non-holonomic constraints, optimal control, rough paths Sub-Riemannian manifolds model media with constrained dynamics: motion at any point is only allowed along a limited set of directions, which are prescribed by the physical problem. From the theoretical point of view, sub-Riemannian geometry is the geometry underlying the theory of hypoelliptic operators and degenerate diffusions on manifolds. In the last twenty years, sub-Riemannian geometry has emerged as an independent research domain, with extremely rich motivations and ramifications in several parts of pure and applied mathematics, such as geometric analysis, geometric measure theory, stochastic calculus and evolution equations together with applications in mechanics, optimal control and biology. The aim of the lectures collected here is to present sub-Riemannian structures for the use of both researchers and graduate students. 10 25 2016 978-3-03719-163-7 978-3-03719-663-2 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/163 http://www.ems-ph.org/doi/10.4171/163

Andrei Agrachev SISSA, TRIESTE, ITALY Davide Barilari Université Paris 7, Denis Diderot, PARIS 13 CEDEX, FRANCE Ugo Boscain Ecole Polytechnique, PALAISEAU CEDEX, FRANCE General 1 83 1 10.4171/163-1/1 http://www.ems-ph.org/doi/10.4171/163-1/1

Anton Thalmaier Université du Luxembourg, LUXEMBOURG, LUXEMBOURG General 85 169 1 10.4171/163-1/2 http://www.ems-ph.org/doi/10.4171/163-1/2

Peter Friz Technische Universität Berlin, BERLIN, GERMANY Paul Gassiat Université Paris-Dauphine, PARIS CEDEX 16, FRANCE General 171 210 1 10.4171/163-1/3 http://www.ems-ph.org/doi/10.4171/163-1/3

Luigi Ambrosio Scuola Normale Superiore, PISA, ITALY Roberta Ghezzi Université de Bourgogne, DIJON CEDEX, FRANCE General 211 273 1 10.4171/163-1/4 http://www.ems-ph.org/doi/10.4171/163-1/4

Michail Zhitomirskii Technion - Israel Institute of Technology, HAIFA, ISRAEL General 275 295 1 10.4171/163-1/5 http://www.ems-ph.org/doi/10.4171/163-1/5

Volume I Davide Barilari Université Paris 7 Denis Diderot, Paris, France Ugo Boscain École Polytechnique, Palaiseau, France Mario Sigalotti École Polytechnique, Palaiseau, France Differential geometry Partial differential equations Calculus of variations and optimal control; optimization Probability theory and stochastic processes 53C17, 35H10, 60H30, 49J15 Sub-Riemannian geometry, hypoelliptic operators, non-holonomic constraints, optimal control, rough paths Sub-Riemannian manifolds model media with constrained dynamics: motion at any point is only allowed along a limited set of directions, which are prescribed by the physical problem. From the theoretical point of view, sub-Riemannian geometry is the geometry underlying the theory of hypoelliptic operators and degenerate diffusions on manifolds. In the last twenty years, sub-Riemannian geometry has emerged as an independent research domain, with extremely rich motivations and ramifications in several parts of pure and applied mathematics, such as geometric analysis, geometric measure theory, stochastic calculus and evolution equations together with applications in mechanics, optimal control and biology. The aim of the lectures collected here is to present sub-Riemannian structures for the use of both researchers and graduate students. 6 30 2016 978-3-03719-162-0 978-3-03719-662-5 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/162 http://www.ems-ph.org/doi/10.4171/162

Francesco Serra Cassano Università di Trento, TRENTO (TN), ITALY Differential geometry General 1 121 1 10.4171/162-1/1 http://www.ems-ph.org/doi/10.4171/162-1/1

Nicola Garofalo Università di Padova, PADOVA, ITALY Differential geometry General 123 257 1 10.4171/162-1/2 http://www.ems-ph.org/doi/10.4171/162-1/2

Fabrice Baudoin Purdue University, WEST LAFAYETTE, UNITED STATES Differential geometry General 259 321 1 10.4171/162-1/3 http://www.ems-ph.org/doi/10.4171/162-1/3

Athanase Papadopoulos Functions of a complex variable Primary 30-00, 32-00, 57-00, 32G13, 32G15, 30F60. Secondary 11F06, 11F75, 14D20, 11G32, 14C05, 14H15, 14H30, 14H15, 14H60, 14H55, 14J60, 18A22, 20F14, 20F28, 20F38, 20F65, 20F67, 20H10, 22E46, 30-03, 30C62, 30F20, 30F25, 30F10, 30F15, 30F30, 30F35, 30F40, 30F45, 32-03, 32S30, 32G13, 32G15, 37-99, 53A35, 53B35, 53C35, 53C50, 53C80, 53D55, 53Z05, 57M07, 57M20, 57M27, 57M50, 57M60, 57N16 This volume is the sixth in a series dedicated to Teichmüller theory in a broad sense, including various moduli and deformation spaces, and the study of mapping class groups. It is divided into five parts: Part A: The metric and the analytic theory. Part B: The group theory. Part C: Representation theory and generalized structures. Part D: The Grothendieck–Teichmüller theory. Part D: Sources. The topics surveyed include Grothendieck’s construction of the analytic structure of Teichmüller space, identities on the geodesic length spectrum of hyperbolic surfaces (including Mirzakhani’s application to the computation of Weil–Petersson volumes), moduli spaces of configurations spaces, the Teichmüller tower with the action of the Galois group on dessins d’enfants, and several others actions related to surfaces. The last part contains three papers by Teichmüller, translated into English with mathematical commentaries, and a document that contains H. Grötzsch’s comments on Teichmüller’s famous paper Extremale quasikonforme Abbildungen und quadratische Differentiale. The handbook is addressed to researchers and to graduate students. 5 31 2016 978-3-03719-161-3 978-3-03719-661-8 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/161 http://www.ems-ph.org/doi/10.4171/161 27

Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE General 1 29 1 10.4171/161-1/1 http://www.ems-ph.org/doi/10.4171/161-1/1

Valentin Poenaru Le Bois-Marie, BURES-SUR-YVETTE, FRANCE History and biography General 31 32 1 10.4171/161-1/2 http://www.ems-ph.org/doi/10.4171/161-1/2

Norbert A’Campo Universität Basel, BASEL, SWITZERLAND Lizhen Ji University of Michigan, ANN ARBOR, UNITED STATES Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE General 35 69 1 10.4171/161-1/3 http://www.ems-ph.org/doi/10.4171/161-1/3

Vincent Alberge Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Hideki Miyachi Osaka University, OSAKA, JAPAN Ken’ichi Ohshika Osaka University Graduate School of Science, OSAKA, JAPAN General 71 94 1 10.4171/161-1/4 http://www.ems-ph.org/doi/10.4171/161-1/4

Yi Huang University of Melbourne, MELBOURNE, Victoria, AUSTRALIA General 95 127 1 10.4171/161-1/5 http://www.ems-ph.org/doi/10.4171/161-1/5

Javier Aramayona Universidad Autónoma de Madrid. , MADRID, SPAIN Juan Souto Université de Rennes 1, RENNES CEDEX, FRANCE General 131 165 1 10.4171/161-1/6 http://www.ems-ph.org/doi/10.4171/161-1/6

Yuuki Tadokoro Kisarazu National College of Technology, CHIBA, JAPAN General 167 193 1 10.4171/161-1/7 http://www.ems-ph.org/doi/10.4171/161-1/7

Robin de Jong Universiteit Leiden, LEIDEN, NETHERLANDS General 195 227 1 10.4171/161-1/8 http://www.ems-ph.org/doi/10.4171/161-1/8

John Loftin Rutgers University, NEWARK, UNITED STATES Ian McIntosh University of York, YORK, UNITED KINGDOM General 231 274 1 10.4171/161-1/9 http://www.ems-ph.org/doi/10.4171/161-1/9

Pierre Will Université de Grenoble I, SAINT-MARTIN D'HÈRES, FRANCE Functions of a complex variable General 275 334 1 10.4171/161-1/10 http://www.ems-ph.org/doi/10.4171/161-1/10

Alexey Sossinsky Independent University of Moscow, MOSCOW, RUSSIAN FEDERATION Functions of a complex variable General 335 373 1 10.4171/161-1/11 http://www.ems-ph.org/doi/10.4171/161-1/11

Ioannis Platis University of Crete, HERAKLION, GREECE Functions of a complex variable General 375 393 1 10.4171/161-1/12 http://www.ems-ph.org/doi/10.4171/161-1/12

Norbert A’Campo Universität Basel, BASEL, SWITZERLAND Lizhen Ji University of Michigan, ANN ARBOR, UNITED STATES Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Functions of a complex variable General 397 435 1 10.4171/161-1/13 http://www.ems-ph.org/doi/10.4171/161-1/13

Pierre Guillot Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Functions of a complex variable General 437 466 1 10.4171/161-1/14 http://www.ems-ph.org/doi/10.4171/161-1/14

A. Muhammed Uludağ Galatasaray University, ISTANBUL, TURKEY İsmail Sağlam Galatasaray University, ISTANBUL, TURKEY Functions of a complex variable General 467 500 1 10.4171/161-1/15 http://www.ems-ph.org/doi/10.4171/161-1/15

A. Muhammed Uludağ Galatasaray University, ISTANBUL, TURKEY Ayberk Zeytin Galatasaray University, ISTANBUL, TURKEY Functions of a complex variable General 501 519 1 10.4171/161-1/16 http://www.ems-ph.org/doi/10.4171/161-1/16

Norbert A’Campo Universität Basel, BASEL, SWITZERLAND Lizhen Ji University of Michigan, ANN ARBOR, UNITED STATES Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Functions of a complex variable General 521 533 1 10.4171/161-1/17 http://www.ems-ph.org/doi/10.4171/161-1/17

Reiner Kühnau Martin-Luther-Universität Halle-Wittenberg, HALLE, GERMANY Functions of a complex variable General 537 546 1 10.4171/161-1/18 http://www.ems-ph.org/doi/10.4171/161-1/18

Oswald Teichmüller , Berlin, GERMANY Functions of a complex variable General 547 560 1 10.4171/161-1/19 http://www.ems-ph.org/doi/10.4171/161-1/19

Vincent Alberge Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Functions of a complex variable General 561 567 1 10.4171/161-1/20 http://www.ems-ph.org/doi/10.4171/161-1/20

Oswald Teichmüller , Berlin, GERMANY Functions of a complex variable General 569 596 1 10.4171/161-1/21 http://www.ems-ph.org/doi/10.4171/161-1/21

Norbert A’Campo Universität Basel, BASEL, SWITZERLAND Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Functions of a complex variable General 597 603 1 10.4171/161-1/22 http://www.ems-ph.org/doi/10.4171/161-1/22

Oswald Teichmüller , Berlin, GERMANY Functions of a complex variable General 605 612 1 10.4171/161-1/23 http://www.ems-ph.org/doi/10.4171/161-1/23

Vincent Alberge Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Functions of a complex variable General 613 629 1 10.4171/161-1/24 http://www.ems-ph.org/doi/10.4171/161-1/24

Athanase Papadopoulos Université de Strasbourg, France Functions of a complex variable Primary 30-00, 32-00, 57-00, 32G13, 32G15, 30F60; Secondary 11F06, 11F75, 14D20, 14H15, 14H60, 14H55, 14J60, 20F14, 20F28, 20F38, 20F65, 20F67, 20H10, 22E46, 30-03, 30C62, 30F20, 30F25, 30F10, 30F15, 30F30, 30F35, 30F40, 30F45, 32-03, 32S30, 37-99, 53A35, 53B35, 53C35, 53C50, 53C80, 53D55, 53Z05, 57M07, 57M20, 57M27, 57M50, 57M60, 57N16 This volume is the fifth in a series dedicated to Teichmüller theory in a broad sense, including the study of various deformation spaces and of mapping class group actions. It is divided into four parts: Part A: The metric and the analytic theory Part B: The group theory Part C: Representation theory and generalized structures Part D: Sources The topics that are covered include identities for the hyperbolic geodesic length spectrum, Thurston's metric, the cohomology of moduli space and mapping class groups, the Johnson homomorphisms, Higgs bundles, dynamics on character varieties, and there are many others. Besides surveying important parts of the theory, several chapters contain conjectures and open problems. The last part contains two fundamental papers by Teichmüller, translated into English and accompanied by mathematical commentaries. The chapters, like those of the other volumes in this collection, are written by experts who have a broad view on the subject. They have an expository character (which fits with the original purpose of this handbook), but some of them also contain original and new material. The Handbook is addressed to researchers and to graduate students. 1 11 2016 978-3-03719-160-6 978-3-03719-660-1 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/160 http://www.ems-ph.org/doi/10.4171/160 26

Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE General 1 16 1 10.4171/160-1/1 http://www.ems-ph.org/doi/10.4171/160-1/1

Martin Bridgeman Boston College, CHESTNUT HILL, UNITED STATES Ser Peow Tan National University of Singapore, SINGAPORE, SINGAPORE Manifolds and cell complexes Group theory and generalizations Functions of a complex variable General In this survey, we discuss four classes of identities due principally to Basmajian, McShane, Bridgeman-Kahn and Luo-Tan on hyperbolic manifolds and provide a unified approach for proving them. We also elucidate on the connections between the various identities. 19 53 1 10.4171/160-1/2 http://www.ems-ph.org/doi/10.4171/160-1/2

Weixu Su Fudan University, SHANGHAI, CHINA Several complex variables and analytic spaces Functions of a complex variable General We present a list of problems related to the study of Thurston metric on Teichmüller space. The problems originate in discussions by the participants at the 2012 AIM workshop “Lipschitz metric on Teichmüller space”. Some of the problems were worked on by small groups, and they remain open. Some others were suggested by various participants at the closing problem session of the conference or added after the conference. We have updated the list in order to include it in the present Handbook. 55 72 1 10.4171/160-1/3 http://www.ems-ph.org/doi/10.4171/160-1/3

Yusuke Kuno Tsuda College, TOKYO, JAPAN Algebraic geometry Group theory and generalizations Several complex variables and analytic spaces Manifolds and cell complexes We give a survey on Meyer functions, with emphasis on their application to the signature of fibered 4-manifolds. 75 96 1 10.4171/160-1/4 http://www.ems-ph.org/doi/10.4171/160-1/4

Nariya Kawazumi University of Tokyo, TOKYO, JAPAN Yusuke Kuno Tsuda College, TOKYO, JAPAN Group theory and generalizations Several complex variables and analytic spaces Manifolds and cell complexes General We survey a geometric approach to the Johnson homomorphisms using the Goldman–Turaev Lie bialgebra. 97 165 1 10.4171/160-1/5 http://www.ems-ph.org/doi/10.4171/160-1/5

Takao Satoh Tokyo University of Science, TOKYO, JAPAN Group theory and generalizations General This is a survey on the Johnson homomorphisms of the automorphism groups of free groups. We exposit some well known facts and recent developments for the Johnson homomorphisms and its related topics. 167 209 1 10.4171/160-1/6 http://www.ems-ph.org/doi/10.4171/160-1/6

Inkang Kim KIAS, SEOUL, SOUTH KOREA Topological groups, Lie groups Differential geometry Manifolds and cell complexes General We survey recent progress on character varieties. We give a general definition of a character variety, yet focus on rigidity and flexibility issues of a given representation. On the other hand, we study dynamical aspects on the character variety, the relation to bounded cohomology theory and several topological issues using different branches of mathematics. We try to give as many familiar examples as possible to motivate the readers. 213 235 1 10.4171/160-1/7 http://www.ems-ph.org/doi/10.4171/160-1/7

Lizhen Ji University of Michigan, ANN ARBOR, UNITED STATES Topological groups, Lie groups Algebraic geometry General There are two closely related classes of groups arising from Fuchsian groups and their actions on the hyperbolic plane $\H^2$: discrete subgroups of semisimple Lie groups acting on symmetric spaces, and mapping class groups and their subgroups acting on Teichmüller space. Convex cocompact Fuchsian groups have been generalized to Anosov subgroups of semisimple Lie groups of higher rank, which have played an important role in higher Teichmüller theory. They have also been generalized to convex cocompact subgroups of mapping class groups. Due to the fact that Teichmüller space shares some similarities with symmetric spaces of rank one, the analogy between convex cocompact Fuchsian groups and convex cocompact subgroups of mapping class groups is rather complete. But less is known about actions of Anosov subgroups on symmetric spaces of higher rank. In this chapter, we discuss three conjectures on compactifications and coarse fundamental domains for locally symmetric spaces associated with Anosov subgroups of noncompact semisimple Lie groups, and describe results, motivations, and evidence for these conjectures. One conjecture deals with the existence of maximal Satake compactifications of Anosov locally symmetric spaces, and other two are concerned with the characterization of Anosov subgroups. By comparing these locally symmetric spaces of infinite volume with symmetric spaces and locally symmetric spaces of finite volume, and by examining applications of the maximal Satake compactification of symmetric spaces and the Borel-Serre compactification of locally symmetric spaces of finite volume, we conclude that the conjectural maximal Satake compactifications of Anosov locally symmetric spaces arising from the maximal Satake compactification of symmetric spaces are the natural compactifications. In general, there is more than one maximal Satake compactification for Anosov locally symmetric spaces. To explain this non-uniqueness, we develop a reduction theory for Anosov subgroups by introducing the crucial notion of anti-Siegel sets. We also explain how geometric boundaries of the maximal Satake compactifications are related to problems in the spectral theory of Laplace operators of the Riemannian manifolds under consideration, and conclude with some questions on the Martin compactification of Anosov locally symmetric spaces, which is related to positive harmonic functions. 237 305 1 10.4171/160-1/8 http://www.ems-ph.org/doi/10.4171/160-1/8

Lisa Jeffrey University of Toronto, TORONTO, ONTARIO, CANADA Global analysis, analysis on manifolds Partial differential equations General This chapter aims to provide a survey on the subject of representations of fundamental groups of 2-manifolds, or in other guises flat connections on orientable 2-manifolds or moduli spaces parametrizing holomorphic vector bundles on Riemann surfaces. It emphasizes the relationships between the different descriptions of these spaces. 307 317 1 10.4171/160-1/9 http://www.ems-ph.org/doi/10.4171/160-1/9

Oswald Teichmüller , Berlin, GERMANY General 321 483 1 10.4171/160-1/10 http://www.ems-ph.org/doi/10.4171/160-1/10

Vincent Alberge Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Weixu Su Fudan University, SHANGHAI, CHINA General We provide a mathematical commentary on Teichmüller’s paper Extremale quasikonforme Abbildungen und quadratische Differentiale (Extremal quasiconformal mappings of closed oriented Riemann surfaces), Abh. Preuss. Akad. Wiss., Math.- Naturw. Kl. 1940, No.22, 1–197 (1940). The paper is quoted in several works, although it was read by very few people. Some of the results it contains were re- discovered later on and published without any reference to Teichmüller. In this commentary, we highlight the main results and the main ideas contained in that paper and we describe some of the important developments they gave rise to. 485 531 1 10.4171/160-1/11 http://www.ems-ph.org/doi/10.4171/160-1/11

Oswald Teichmüller , Berlin, GERMANY General 533 567 1 10.4171/160-1/12 http://www.ems-ph.org/doi/10.4171/160-1/12

Annette A’Campo-Neuen Universität Basel, BASEL, SWITZERLAND Norbert A’Campo Universität Basel, BASEL, SWITZERLAND Vincent Alberge Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE General This is a mathematical commentary on Teichmüller’s paper Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen (Determination of extremal quasiconformal maps of closed oriented Riemann surfaces) [24], (1943). This paper is among the last (and may be the last one) that Teichmüller wrote on the theory of moduli. It contains the proof of the so-called Teichmüller existence theorem for a closed surface of genus $g ≥ 2$. For this proof, the author defines a mapping between a space of equivalence classes of marked Riemann surfaces (the Teichmüller space) and a space of equivalence classes of certain Fuchsian groups (the so-called Fricke space). After that, he defines a map between the latter and the Euclidean space of dimension $6g−6$. Using Brouwer’s theorem of invariance of domain, he shows that this map is a home- omorphism. This involves in particular a careful definition of the topologies of Fricke space, the computation of its dimension, and comparison results between hyperbolic distance and quasiconformal dilatation. The use of the invariance of domain theorem is in the spirit of Poincar ́e and Klein’s use of the so-called “continuity principle” in their attempts to prove the uniformization theorem. 569 580 1 10.4171/160-1/13 http://www.ems-ph.org/doi/10.4171/160-1/13

Dietmar Salamon ETH Zürich, Switzerland Measure and integration Primary: 28-01; Secondary: 28C05, 28C10, 28C15, 35J05, 43A05, 44A35, 46B22, 46C05, 46E27, 46E30 sigma-Algebra, Lebesgue monotone convergence, Caratheodory criterion, Lebesgue measure, Borel measure, Dieudonné’s measure, Riesz Representation Theorem, Alexandrov Double Arrow Space, Sorgenfrey Line, separability, Cauchy–Schwarz inequality, Jensen’s inequality, Egoroff’s theorem, Hardy’s inequality, absolutely continuous measure, truly continuous measure, singular measure, signed measure, Radon–Nikodym Theorem, Lebesgue Decomposition Theorem, Hahn Decomposition Theorem, Jordan Decomposition Theorem, Hardy–Littlewood maximal inequality, Vitali’s Covering Lemma, Lebesgue point, Lebesgue Differentiation Theorem, Banach-Zarecki Theorem, Vitali–Caratheodory Theorem, Cantor function, product sigma-algebra, Fubini’s Theorem, convolution, Young’s inequality, mollifier, Marcinciewicz interpolation, Poisson identity, Green’s formula, Calderon–Zygmund inequality, Haar measure, modular character The book is intended as a companion to a one semester introductory lecture course on measure and integration. After an introduction to abstract measure theory it proceeds to the construction of the Lebesgue measure and of Borel measures on locally compact Hausdorff spaces, $L^p$ spaces and their dual spaces and elementary Hilbert space theory. Special features include the formulation of the Riesz Representation Theorem in terms of both inner and outer regularity, the proofs of the Marcinkiewicz Interpolation Theorem and the Calderon–Zygmund inequality as applications of Fubini’s theorem and Lebesgue differentiation, the treatment of the generalized Radon–Nikodym theorem due to Fremlin, and the existence proof for Haar measures. Three appendices deal with Urysohn’s Lemma, product topologies, and the inverse function theorem. The book assumes familiarity with first year analysis and linear algebra. It is suitable for second year undergraduate students of mathematics or anyone desiring an introduction to the concepts of measure and integration. 3 29 2016 978-3-03719-159-0 978-3-03719-659-5 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/159 http://www.ems-ph.org/doi/10.4171/159 Gromov’s Theory of Convergence and Concentration of Metrics and Measures Takashi Shioya Tohoku University, Sendai, Japan Differential geometry Measure and integration Functions of a complex variable Probability theory and stochastic processes Primary: 53C23; Secondary: 28A33, 30Lxx, 35P15, 53C20, 54Exx, 58C40, 58J50, 60B10 Metric measure space, concentration of measure phenomenon, observable distance, pyramid, convergence of spaces, curvature-dimension condition, Laplacian, dissipation This book studies a new theory of metric geometry on metric measure spaces, originally developed by M. Gromov in his book “Metric Structures for Riemannian and Non-Riemannian Spaces” and based on the idea of the concentration of measure phenomenon due to Lévy and Milman. A central theme in this text is the study of the observable distance between metric measure spaces, defined by the difference between 1-Lipschitz functions on one space and those on the other. The topology on the set of metric measure spaces induced by the observable distance function is weaker than the measured Gromov–Hausdorff topology and allows to investigate a sequence of Riemannian manifolds with unbounded dimensions. One of the main parts of this presentation is the discussion of a natural compactification of the completion of the space of metric measure spaces. The stability of the curvature-dimension condition is also discussed. This book makes advanced material accessible to researchers and graduate students interested in metric measure spaces. 1 5 2016 978-3-03719-158-3 978-3-03719-658-8 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/158 http://www.ems-ph.org/doi/10.4171/158 25 Koen Thas University of Gent, Belgium Combinatorics Number theory Commutative rings and algebras Algebraic geometry Primary: 05E18, 11M26, 13F35, 13K05, 14A15, 14A20, 14A22, 14G15, 14G40, 14H10, 18A05, 19E08, 20B25, 20G05, 20G35, 20M25, 51E24; Secondary: 05E05, 06B10, 11G20, 11G25, 11R18, 11T55, 13A35, 13C60, 14C40, 14F05, 14L15, 14M25, 14M26, 14P10, 15B48, 16G20, 16Y60, 18D50, 18F20, 20E42, 20F36, 20M14, 20M32, 20N20, 51B25, 55N30, 55P42, 55Q45 The field with one element, $\mathbb F_1$-geometry, combinatorial $\mathbb F_1$-geometry, non-additive category, Deitmar scheme, graph, monoid, motive, zeta function, automorphism group, blueprint, Euler characteristic, K-theory, Grassmannian, Witt ring, noncommutative geometry, Witt vector, total positivity, moduli space of curves, operad, torificiation, Absolute Arithmetic, counting function, Weil conjectures, Riemann Hypothesis It has been known for some time that geometries over finite fields, their automorphism groups and certain counting formulae involving these geometries have interesting guises when one lets the size of the field go to 1. On the other hand, the nonexistent field with one element, $\mathbb F_1$, presents itself as a ghost candidate for an absolute basis in Algebraic Geometry to perform the Deninger–Manin program, which aims at solving the classical Riemann Hypothesis. This book, which is the first of its kind in the $\mathbb F_1$-world, covers several areas in $\mathbb F_1$-theory, and is divided into four main parts – Combinatorial Theory, Homological Algebra, Algebraic Geometry and Absolute Arithmetic. Topics treated include the combinatorial theory and geometry behind $\mathbb F_1$, categorical foundations, the blend of different scheme theories over $\mathbb F_1$ which are presently available, motives and zeta functions, the Habiro topology, Witt vectors and total positivity, moduli operads, and at the end, even some arithmetic. Each chapter is carefully written by experts, and besides elaborating on known results, brand new results, open problems and conjectures are also met along the way. The diversity of the contents, together with the mystery surrounding the field with one element, should attract any mathematician, regardless of speciality. 7 25 2016 978-3-03719-157-6 978-3-03719-657-1 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/157 http://www.ems-ph.org/doi/10.4171/157

Koen Thas Universiteit Gent, GENT, BELGIUM Combinatorics Number theory Commutative rings and algebras Algebraic geometry 3 36 1 10.4171/157-1/1 http://www.ems-ph.org/doi/10.4171/157-1/1

Anton Deitmar Universität Tübingen, TÜBINGEN, GERMANY Combinatorics Number theory Commutative rings and algebras Algebraic geometry 39 80 1 10.4171/157-1/2 http://www.ems-ph.org/doi/10.4171/157-1/2

Koen Thas Universiteit Gent, GENT, BELGIUM Combinatorics Number theory Commutative rings and algebras Algebraic geometry 83 159 1 10.4171/157-1/3 http://www.ems-ph.org/doi/10.4171/157-1/3

Oliver Lorscheid , RIO DE JANEIRO, BRAZIL Combinatorics Number theory Commutative rings and algebras Algebraic geometry 161 219 1 10.4171/157-1/4 http://www.ems-ph.org/doi/10.4171/157-1/4

Lieven Le Bruyn University of Antwerp, ANTWERP, BELGIUM Combinatorics Number theory Commutative rings and algebras Algebraic geometry 221 271 1 10.4171/157-1/5 http://www.ems-ph.org/doi/10.4171/157-1/5

James Borger Australian National University, CANBERRA, AUSTRALIA Combinatorics Number theory Commutative rings and algebras Algebraic geometry 273 329 1 10.4171/157-1/6 http://www.ems-ph.org/doi/10.4171/157-1/6

Yuri Manin , BONN, GERMANY Matilde Marcolli California Institute of Technology, PASADENA, UNITED STATES Combinatorics Number theory Commutative rings and algebras Algebraic geometry 331 361 1 10.4171/157-1/7 http://www.ems-ph.org/doi/10.4171/157-1/7

Koen Thas Universiteit Gent, GENT, BELGIUM Combinatorics Number theory Commutative rings and algebras Algebraic geometry 365 386 1 10.4171/157-1/8 http://www.ems-ph.org/doi/10.4171/157-1/8

Alberto Cavicchioli Università degli Studi di Modena e Reggio Emilia, Italy Friedrich Hegenbarth Università degli Studi di Milano, Italy Dušan Repovš University of Ljubljana, Slovenia Manifolds and cell complexes Category theory; homological algebra $K$-theory Primary: 57P05, 57P10, 57P99, 57R65, 57R67; Secondary: 18F15, 19J25, 57N15, 57N60, 57N65 Homology manifold, Poincaré duality, degree 1 normal map, boundedly controlled surgey, surgery spectrum, assembly map, Quinn index, Euclidean neighborhood retract, cell-like resolution, disjoint disks property, manifold recognition problem Generalized manifolds are a most fascinating subject to study. They were introduced in the 1930s, when topologists tried to detect topological manifolds among more general spaces (this is nowadays called the manifold recognition problem). As such, generalized manifolds have served to understand the nature of genuine manifolds. However, it soon became more important to study the category of generalized manifolds itself. A breakthrough was made in the 1990s, when several topologists discovered a systematic way of constructing higher-dimensional generalized manifolds, based on advanced surgery techniques. In fact, the development of controlled surgery theory and the study of generalized manifolds developed in parallel. In this process, earlier studies of geometric surgery turned out to be very helpful. Generalized manifolds will continue to be an attractive subject to study, for there remain several unsolved fundamental problems. Moreover, they hold promise for new research, e.g. for finding appropriate structures on these spaces which could bring to light geometric (or even analytic) aspects of higher-dimensional generalized manifolds. This is the first book to systematically collect the most important material on higher-dimensional generalized manifolds and controlled surgery. It is self-contained and its extensive list of references reflects the historic development. The book is based on our graduate courses and seminars, as well as our talks given at various meetings, and is suitable for advanced graduate students and researchers in algebraic and geometric topology. 5 31 2016 978-3-03719-156-9 978-3-03719-656-4 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/156 http://www.ems-ph.org/doi/10.4171/156 Hans Triebel University of Jena, Germany Functional analysis Fourier analysis If one tries to transfer assertions for the inhomogeneous spaces $A^s_{p,q} (\mathbb R^n)$, $A \in \{B,F \}$, appropriately to their homogeneous counterparts ${\overset {\, \ast}{A}}{}^s_{p,q} (\mathbb R^n)$ within the framework of the dual pairing $\big( S(\mathbb R^n), S'(\mathbb R^n) \big)$ then it is hard to make a mistake as long as the parameters $p,q,s$ are restricted by $0 < p,q \le \infty$ and, in particular, $n(\frac {1}{p} – 1) < s < \frac {n}{p}$. It is the main aim of these notes to say what this means. This book is addressed to graduate students and mathematicians having a working knowledge of basic elements of the theory of function spaces, especially of type $B^s_{p,q}$ and $F^s_{p,q}$. 9 30 2015 978-3-03719-155-2 978-3-03719-655-7 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/155 http://www.ems-ph.org/doi/10.4171/155 Matthias Aschenbrenner University of California Los Angeles, USA Stefan Friedl Universität Regensburg, Germany Henry Wilton University of Cambridge, UK Manifolds and cell complexes Group theory and generalizations Primary: 57M05, 57M27; Secondary: 20E26 3-manifolds, fundamental groups, Geometrization theorem, hyperbolic 3-manifolds, virtually special groups The field of 3-manifold topology has made great strides forward since 1982, when Thurston articulated his influential list of questions. Primary among these is Perelman's proof of the Geometrization Conjecture, but other highlights include the Tameness Theorem of Agol and Calegari–Gabai, the Surface Subgroup Theorem of Kahn–Markovic, the work of Wise and others on special cube complexes, and finally Agol's proof of the Virtual Haken Conjecture. This book summarizes all these developments and provides an exhaustive account of the current state of the art of 3-manifold topology, especially focussing on the consequences for fundamental groups of 3-manifolds. As the first book on 3-manifold topology that incorporates the exciting progress of the last two decades, it will be an invaluable resource for researchers in the field who need a reference for these developments. It also gives a fast-paced introduction to this material – although some familiarity with the fundamental group is recommended, little other previous knowledge is assumed, and the book is accessible to graduate students. The book closes with an extensive list of open questions, which will also be of interest to graduate students and established researchers alike. 8 20 2015 978-3-03719-154-5 978-3-03719-654-0 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/154 http://www.ems-ph.org/doi/10.4171/154 Including the monograph Symplectic cohomology and Viterbo’s theorem by Mohammed Abouzaid Janko Latschev Universität Hamburg, Germany Alexandru Oancea Sorbonne Universités, Paris, France Differential geometry Commutative rings and algebras Associative rings and algebras Algebraic topology Primary: 53D40, 53D12, 53D25, 53D35, 53D37, 55P35, 55P50, 55P62, 55P92, 13D03, 13D07, 13D09, 13D10, 16E40, 16E45; Secondary: 57R15, 57R19, 57R56, 57R70, 57R91 Loop space, symplectic geometry, symplectic topology, string topology, Morse theory, Hochschild and cyclic homology, operations on Hochschild and cyclic homology, rational homotopy theory, minimal models, Lagrangian embeddings, pseudo-holomorphic curves In the late 1990s two initially unrelated developments brought free loop spaces into renewed focus. In 1999, Chas and Sullivan introduced a wealth of new algebraic operations on the homology of these spaces under the name of string topology, the full scope of which is still not completely understood. A few years earlier, Viterbo had discovered a first deep link between the symplectic topology of cotangent bundles and the topology of their free loop space. In the past 15 years, many exciting connections between these two viewpoints have been found. Still, researchers working on one side of the story often know quite little about the other. One of the main purposes of this book is to facilitate communication between topologists and symplectic geometers thinking about free loop spaces. It was written by active researchers coming to the topic from both perspectives and provides a concise overview of many of the classical results, while also beginning to explore the new directions of research that have emerged recently. As one highlight, it contains a research monograph by M. Abouzaid which proves a strengthened version of Viterbo’s isomorphism between the homology of the free loop space of a manifold and the symplectic cohomology of its cotangent bundle, following a new strategy. The book grew out of a learning seminar on free loop spaces held at Strasbourg University in 2008–2009, and should be accessible to a graduate student with a general interest in the topic. It focuses on introducing and explaining the most important aspects rather than offering encyclopedic coverage, while providing the interested reader with a broad basis for further studies and research. 9 30 2015 978-3-03719-153-8 978-3-03719-653-3 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/153 http://www.ems-ph.org/doi/10.4171/153 24 Sylvia Serfaty Université Pierre et Marie Curie (Paris VI), France Statistical mechanics, structure of matter Linear and multilinear algebra; matrix theory Partial differential equations 82B05, 82B21, 82B26, 15B52, 82D55; 35A15, 35J20, 35J60 Coulomb gas, Log gas, one-component plasma, statistical mechanics, Ginzburg-Landau, superconductivity, vortices, Abrikosov lattice, crystallization, random matrices, renormalized energy, mean-field limit, large deviations The topic of this book is systems of points in Coulomb interaction, in particular, the classical Coulomb gas, and vortices in the Ginzburg–Landau model of superconductivity. The classical Coulomb and Log gases are classical statistical mechanics models, which have seen important developments in the mathematical literature due to their connection with random matrices and approximation theory. At low temperature, these systems are expected to “cristallize” to so-called Fekete sets, which exhibit microscopically a lattice structure. The Ginzburg–Landau model, on the other hand, describes superconductors. In superconducting materials subjected to an external magnetic field, densely packed point vortices emerge, forming perfect triangular lattice patterns, so-called Abrikosov lattices. This book describes these two systems and explores the similarity between them. It presents the mathematical tools developed to analyze the interaction between the Coulomb particles or the vortices, at the microscopic scale, and describes a “renormalized energy” governing the point patterns. This is believed to measure the disorder of a point configuration, and to be minimized by the Abrikosov lattice in dimension 2. The book gives a self-contained presentation of results on the mean field limit of the Coulomb gas system, with or without temperature, and of the derivation of the renormalized energy. It also provides a streamlined presentation of the similar analysis that can be performed for the Ginzburg–Landau model, including a review of the vortex-specific tools and the derivation of the critical fields, the mean-field limit and the renormalized energy. 3 20 2015 978-3-03719-152-1 978-3-03719-652-6 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/152 http://www.ems-ph.org/doi/10.4171/152 Olivier Lablée Université Joseph Fourier Grenoble 1, Saint Martin d’Hères, France Global analysis, analysis on manifolds Partial differential equations Operator theory Calculus of variations and optimal control; optimization 58J35, 58J37, 58J50, 58J53, 35P05, 35P15, 35P20, 47A05, 47A10, 47A12, 47A60, 47A75, 49R05, 53C21 Spectral theory, linear operators, spectrum of operators, spectral geometry, eigenvalues, Laplacian, inverse problems, Riemannian geometry, analysis on manifolds Spectral theory is a diverse area of mathematics that derives its motivations, goals and impetus from several sources. In particular, the spectral theory of the Laplacian on a compact Riemannian manifold is a central object in differential geometry. From a physical point a view, the Laplacian on a compact Riemannian manifold is a fundamental linear operator which describes numerous propagation phenomena: heat propagation, wave propagation, quantum dynamics, etc. Moreover, the spectrum of the Laplacian contains vast information about the geometry of the manifold. This book gives a self-containded introduction to spectral geometry on compact Riemannian manifolds. Starting with an overview of spectral theory on Hilbert spaces, the book proceeds to a description of the basic notions in Riemannian geometry. Then its makes its way to topics of main interests in spectral geometry. The topics presented include direct and inverse problems. Direct problems are concerned with computing or finding properties on the eigenvalues while the main issue in inverse problems is “knowing the spectrum of the Laplacian, can we determine the geometry of the manifold?” Addressed to students or young researchers, the present book is a first introduction in spectral theory applied to geometry. For readers interested in pursuing the subject further, this book will provide a basis for understanding principles, concepts and developments of spectral geometry. 2 18 2015 978-3-03719-151-4 978-3-03719-651-9 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/151 http://www.ems-ph.org/doi/10.4171/151 Hans Triebel University of Jena, Germany Functional analysis Partial differential equations Fourier analysis Integral equations 46-02, 46E35, 42B35, 42C40, 35K05, 35Q30, 76D03, 76D05 Function spaces, Morrey spaces, heat equations, Navier-Stokes equations This book is the continuation of Local Function Spaces, Heat and Navier–Stokes Equations (Tracts in Mathematics 20, 2013) by the author. A new approach is presented to exhibit relations between Sobolev spaces, Besov spaces, and Hölder–Zygmund spaces on the one hand and Morrey–Campanato spaces on the other. Morrey–Campanato spaces extend the notion of functions of bounded mean oscillation. These spaces play a crucial role in the theory of linear and nonlinear PDEs. Chapter 1 (Introduction) describes the main motivations and intentions of this book. Chapter 2 is a selfcontained introduction into Morrey spaces. Chapter 3 deals with hybrid smoothness spaces (which are between local and global spaces) in Euclidean n-space based on the Morrey–Campanato refinement of the Lebesgue spaces. The presented approach relies on wavelet decompositions. This is applied in Chapter 4 to linear and nonlinear heat equations in global and hybrid spaces. The obtained assertions about function spaces and nonlinear heat equations are used in the Chapters 5 and 6 to study Navier–Stokes equations in hybrid and global spaces. This book is addressed to graduate students and mathematicians having a working knowledge of basic elements of (global) function spaces, and who are interested in applications to nonlinear PDEs with heat and Navier–Stokes equations as prototypes. 1 15 2015 978-3-03719-150-7 978-3-03719-650-2 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/150 http://www.ems-ph.org/doi/10.4171/150 24 Antonio Campillo Universidad de Valladolid, Spain Franz-Viktor Kuhlmann University of Saskatchewan, Saskatoon, Canada Bernard Teissier Institut de Mathématiques de Jussieu, Paris, France Field theory and polynomials Order, lattices, ordered algebraic structures Commutative rings and algebras Algebraic geometry Primary: 03CXX, 12JXX, 12E30, 12F10, 13A18, 14H20, 14M25; secondary: 06FXX, 11SXX, 11U09, 12DXX, 12E05, 12F05, 12GXX, 12L12, 13D40, 13F30, 13H05, 13JXX, 13N15, 14BXX, 14C20, 14EXX, 14F10, 14J17, 14HXX, 14PXX, 16W60, 32P05, 32SXX, 37A05, 54F50 Valuation, defect, Abhyankar valuation, divisorial valuation, completion, local uniformization, toric geometry, key polynomial, excellent ring, local ring, valuation centered at a local domain, valuative tree, dicritical divisor, Rees valuation, Izumi’s theorem, plane curve singularity, Newton tree, rational surface singularity, Whitney stratification, jet scheme, embedded Nash problem, higher local field, wild ramification, dynamical system, irreducible polynomial, additive polynomial, Hilbertian field, large field, Galois theory, C-minimality, cell decomposition, imaginary element, formally real field, R-place, Hardy field, exponential-logarithmic series field, asymptotic integration, Hahn field, truncation, quasi-valuation. Having its classical roots, since more than a century, in algebraic number theory, algebraic geometry and the theory of ordered fields and groups, valuation theory has seen an amazing expansion into many other areas in recent decades. Moreover, having been dormant for a while in algebraic geometry, it has now been reintroduced as a tool to attack the open problem of resolution of singularities in positive characteristic and to analyse the structure of singularities. Driven by this topic, and by its many new applications in other areas, also the research in valuation theory itself has been intensified, with a particular emphasis on the deep open problems in positive characteristic. As important examples for the expansion of valuation theory, it has become extremely useful in the theory of complex dynamical systems, and in the study of non-oscillating trajectories of real analytic vector fields in three dimensions. Analogues of the Riemann-Zariski valuation spaces have been found to be the proper framework for questions of intersection theory in algebraic geometry and in the analysis of singularities of complex plurisubharmonic functions. In a different direction, the relation between Berkovich geometry, tropical geometry and valuation spaces, on the one hand, and the geometry of arc spaces and valuation spaces, on the other, have begun to deepen and clarify. Ever since its beginnings, valuation theory and Galois theory have grown closely together and influenced each other. Arguably, studying and understanding the extensions of valuations in algebraic field extensions is one of the most important questions in valuation theory, whereas using valuation theory is one of he most important tools in studying Galois extensions of fields, as well as constructing field extensions with given properties. The well established topic of the model theory of valued fields is also being transformed, in particular through the study of valued fields with functions and operators, and through the study of types over valued fields. The multifaceted development of valuation theory has been monitored by two International Conferences and Workshops: the first in 1999 in Saskatoon, Canada, and the second in 2011 in Segovia and El Escorial in Spain. This book grew out of the second conference and presents high quality papers on recent research together with survey papers that illustrate the state of the art in several areas and applications of valuation theory. The book is addressed to researchers and graduate students who work in valuation theory or the areas where it is applied, as well as a general mathematical audience interested in the expansion and usefulness of the valuation theoretical approach, which has been called the “most analytic” form of algebraic reasoning. For young mathematicians who want to enter these areas of research, it provides a valuable source of up-to-date information. 9 1 2014 978-3-03719-149-1 978-3-03719-649-6 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/149 http://www.ems-ph.org/doi/10.4171/149

Kamal Aghigh K.N. Toosi University of Technology, Tehran, IRAN Anuj Bishnoi Panjab University, Chandigarh, INDIA Sudesh Khanduja (IISER, Mohali), SAS Nagar, Punjab, INDIA Sanjeev Kumar Panjab University, Chandigarh, INDIA Field theory and polynomials General In this paper, we give an introduction of the phenomenon of lifting with respect to residually transcendental extensions, the notion of distinguished pairs and complete distinguished chains which lead to the study of certain invariants associated to irreducible polynomials over valued fields. We give an overview of various results regarding these concepts and their applications. 1 10 1 10.4171/149-1/1 http://www.ems-ph.org/doi/10.4171/149-1/1

Lior Bary-Soroker Tel Aviv University, Tel Aviv, ISRAEL Arno Fehm University of Konstanz, Konstanz, GERMANY Field theory and polynomials General Given a finite set $\mathfrak{S}$ of places of a number field, we prove that the field of totally $\mathfrak{S}$-adic algebraic numbers is not Hilbertian. 11 15 1 10.4171/149-1/2 http://www.ems-ph.org/doi/10.4171/149-1/2

Anna Blaszczok University of Silesia, Katowice, POLAND Field theory and polynomials Associative rings and algebras General We consider Artin-Scheier defect extensions of rational function fields in two variables over fields of positive characteristic. We study the problem of constructing infinite towers of such extensions. We classify Artin-Schreier defect extensions into "dependent" and "independent" ones, according to whether they are connected with purely inseparable defect extensions, or not. To understand the meaning of the classification for the issue of local uniformization, we consider various valuations of the rational function field and investigate for which it admits an infinite tower of dependent or independent Artin-Schreier defect extensions. We give also a criterion for a valued field of positive characteristic $p$ with $p$-divisible value group and perfect residue field to admit infinitely many parallel dependent Artin-Schreier defect extensions or an infinite tower of such extensions. 16 54 1 10.4171/149-1/3 http://www.ems-ph.org/doi/10.4171/149-1/3

Sébastien Boucksom CNRS-Université Paris 6, PARIS CEDEX 05, FRANCE Charles Favre École Polytechnique, Palaiseau Cedex, FRANCE Mattias Jonsson University of Michigan, Ann Arbor, UNITED STATES Commutative rings and algebras Associative rings and algebras Several complex variables and analytic spaces General We improve Izumi's inequality, which states that any divisorial valuation $v$ centered at a closed point $0$ on a normal algebraic variety $Y$ is controlled by the order of vanishing at $0$. More precisely, as $v$ ranges through valuations that are monomial with respect to coordinates in a fixed birational model $X$ dominating $Y$, we show that for any regular function $f$ on $Y$ at $0$, the function $v\mapsto v(f) / {\rm ord}_0(f)$ $d_0$ is uniformly Lipschitz continuous as a function of the weight defining $v$. As a consequence, the volume of $v$ is also a Lipschitz continuous function. Our proof uses toroidal techniques as well as positivity properties of the images of suitable nef divisors under birational morphisms. 55 81 1 10.4171/149-1/4 http://www.ems-ph.org/doi/10.4171/149-1/4

Pierrette Cassou-Noguès Université Bordeaux I, Talence Cedex, FRANCE Anatoly Libgober University of Illinois at Chicago, CHICAGO, UNITED STATES Algebraic geometry Several complex variables and analytic spaces General The paper describes several invariants of plane curve singularities in terms of the data of associated Newton trees. Newton trees of singularities are discussed in detail also. The invariants which we study include the constants and faces of quasi-adjunction, log-canonical walls and Arnold-Steenbrink spectrum. As one of the consequences of these calculations we show the failure of ACC for the set of constants of quasi-adjunction of all plane curve singularities, which contains the set of log-canonical thresholds as a subset. 82 135 1 10.4171/149-1/5 http://www.ems-ph.org/doi/10.4171/149-1/5

Vincent Cossart Université de Versailles Saint-Quentin, Versailles Cedex, FRANCE Mickaël Matusinski Université Bordeaux 1, Talence Cedex, FRANCE Guillermo Moreno-Socías CNRS/UVSQ, bt. Fermat, VERSAILLES CEDEX, FRANCE Algebraic geometry General In this article, there are new proofs of the existence and unicity of dicritical divisors of a pencil of plane curves of $\langle F,G\rangle$ Incidentally, we prove the equivalence between dicritical divisors and Rees valuations. Furthermore, in the case where $G_{\mathrm{red}}$ is regular at the base points of $\langle F,G\rangle$, we have that $F/G$ is residually a polynomial along any dicritical divisor; this reproves geometrically [2, Theorem (7.1)]. As a corollary of the latter proof, we get a generalization of the connectedness theorem of [8]. 136 147 1 10.4171/149-1/6 http://www.ems-ph.org/doi/10.4171/149-1/6

Vincent Cossart Université de Versailles Saint-Quentin, Versailles Cedex, FRANCE Olivier Piltant Université de Versailles Saint-Quentin, Versailles Cedex, FRANCE Ana Reguera Universidad de Valladolid, VALLADOLID, SPAIN Algebraic geometry General Let $(R,M)$ be a rational surface singularity and $\nu_E$ be a prime divisor of the second kind for $R$. Then $gr_{\nu_E} R$ is finitely generated over $R/M$. We recover information about the dual graph of the minimal resolution $\widetilde X$ of $\text{Spec } R$ from the set of all $gr_{\nu_E} R$. In particular we characterize those graded algebras corresponding to the exceptional curves in $\widetilde X$. 148 166 1 10.4171/149-1/7 http://www.ems-ph.org/doi/10.4171/149-1/7

Pablo Cubides Kovacsics Université Paris Diderot, PARIS CEDEX 13, FRANCE Mathematical logic and foundations Commutative rings and algebras General Developments in valuation theory, especially the study of algebraically closed valued fields, have used the model theory of $C$-minimal structures in different places, e.g., the work of Hrushovski-Kazdhan in [5] and Haskell-Hrushovski-Macpherson in [3]. We intend with this text both to promulgate a basic comprehension of $C$-minimality for mathematicians interested in valuation theory (equipped with a basic knowledge of model theory), and to provide a slightly different presentation of the cell decomposition theorem proved by Haskell and Macpherson in [6]. 167 207 1 10.4171/149-1/8 http://www.ems-ph.org/doi/10.4171/149-1/8

Steven Cutkosky University of Missouri, COLUMBIA, UNITED STATES Commutative rings and algebras Algebraic geometry General In this article we consider the problem of determining the valuation semigroup of a valuation dominating a Noetherian local ring. We give some general results and examples. 208 218 1 10.4171/149-1/9 http://www.ems-ph.org/doi/10.4171/149-1/9

Salih Durhan Middle East Technical University, Mersin 10, TURKEY Number theory General Additive polynomials in one variable over valued fields of positive characteristic are sufficiently well understood in terms of their approximation properties. We extend results in this direction to multi-variable additive polynomials over perfect valued fields. 219 225 1 10.4171/149-1/10 http://www.ems-ph.org/doi/10.4171/149-1/10

Danielle Gondard-Cozette Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE Commutative rings and algebras Mathematical logic and foundations Field theory and polynomials Algebraic geometry In this survey $K$ will be a formally real fi eld, which means that $-1$ is not a finite sum of squares of elements of $K,$ hence $K$ has characteristic $0$. As often in the literature, we shall write real field insteadof formally real fieldIt is well known from Artin-Schreier theory that such fields are exactly those admitting at least one total order compatible with the field structure. After some background in Real Algebra, we introduce and study the space of $\mathbb{R}$-places. Thereafter, we present other mathematical notions, such as valuation fans, orderings of higher level and the real holomorphy ring. By use of these tools we obtain an outstanding result in Real Algebraic Geometry. Finally we provide some steps towards an abstract theory of $\mathbb{R}$-places. 226 251 1 10.4171/149-1/11 http://www.ems-ph.org/doi/10.4171/149-1/11

Francisco Javier Herrera Govantes Universidad de Sevilla, SEVILLA, SPAIN Miguel Ángel Olalla Acosta Universidad de Sevilla, SEVILLA, SPAIN Mark Spivakovsky Université Paul Sabatier, Toulouse Cedex 9, FRANCE Bernard Teissier UMR 7586 du CNRS, PARIS CEDEX 13, FRANCE Field theory and polynomials Commutative rings and algebras Algebraic geometry Associative rings and algebras This paper is an extended version of the talk given by Miguel Angel Olalla Ácosta at the International Conference on Valuation Theory in El Escorial in July 2011. Its purpose is to provide an introduction to our joint paper [5] without grinding through all of its technical details. We refer the reader to [5] for details and proofs; only a few proofs are given in the present paper. 252 265 1 10.4171/149-1/12 http://www.ems-ph.org/doi/10.4171/149-1/12

Ángel Granja Universidad de León, León, SPAIN M. Martínez Universidad de Valladolid, Valladolid, SPAIN C. Rodríguez Universidad de Léon, Léon, SPAIN Commutative rings and algebras Field theory and polynomials General topology General Let $D$ be a division ring, $T$ be a variable over $D$, $\sigma $ be an endomorphism of $D$, $\delta $ be a $\sigma$-derivation on $D$ and $R=D[T; \sigma , \delta]$ the left skew polynomial ring over $D$. We show that the partially ordered set $(Val_\nu(R),\preceq)$ of $\sigma$-compatible real valuations on $R$ extending a fixed proper real valuation $\nu $ on $D$ has a natural structure of complete parameterized non-metric tree. 266 287 1 10.4171/149-1/13 http://www.ems-ph.org/doi/10.4171/149-1/13

Immanuel Halupczok University of Leeds, Leeds, UNITED KINGDOM Field theory and polynomials Several complex variables and analytic spaces General In these notes a new, strong notion of stratifications which describe singularities of sets in Henselian valued fields is given. The first part presents the definition, some examples, and the main result about their existence. The second part explains how such a stratification in a valued field induces a classical Whitney stratification in $\mathbb{R}$. 288 296 1 10.4171/149-1/14 http://www.ems-ph.org/doi/10.4171/149-1/14

Ehud Hrushovski Hebrew University, JERUSALEM, ISRAEL Mathematical logic and foundations Field theory and polynomials General We give an exposition of material from [1], [2] and [3], regarding definable types in the model completion of the theory of valued fields, and the classification of imaginary sorts. The latter is given a new proof, based on definable types rather than invariant types, and on the notion of generic reparametrization. I also try to bring out the relation to the geometry of [3] - stably dominated definable types as the model theoretic incarnation of a Berkovich point. 297 319 1 10.4171/149-1/15 http://www.ems-ph.org/doi/10.4171/149-1/15

Franz-Viktor Kuhlmann University of Saskatchewan, SASKATOON SASK., CANADA Asim Naseem GC University, Lahore, PAKISTAN Field theory and polynomials Algebraic geometry General The defect (also called ramification deficiency) of valued field extensions is a major stumbling block in deep open problems of valuation theory in positive characteristic. For a detailed analysis, we define and investigate two finer notions of defect: the completion defect and the defect quotient. We define all three defects for finite valued field extensions as well as for certain valued function fields (those with Abhyankar valuations that are allowed to be nontrivial on the ground field). These defects of valued function fields have played an important role in genus reduction formulas that were presented by several authors. We prove the most general known form of the Finiteness and Independence Theorem for the defect of valued function fields. Further, we investigate the completion defect and the defect quotient in detail and present analogues of the results that hold for the usual defect. 320 349 1 10.4171/149-1/16 http://www.ems-ph.org/doi/10.4171/149-1/16

Mickaël Matusinski Université Bordeaux 1, Talence Cedex, FRANCE Field theory and polynomials Commutative rings and algebras General We survey some important properties of fields of generalized series and of exponential-logarithmic series, with particular emphasis on their possible di fferential structure, based on a joint work of the author with S. Kuhlmann [40, 39]. 350 372 1 10.4171/149-1/17 http://www.ems-ph.org/doi/10.4171/149-1/17

Hussein Mourtada Institut Mathématique de Jussieu-Paris Rive Gauche, PARIS CEDEX 13, FRANCE Algebraic geometry Commutative rings and algebras General We prove that for $m\in \mathbb{N},~m$ large enough, the number of irreducible components of the schemes of $m-$jets centered at a point which is a double point singularity is independent of $m$ and is equal to the number of exceptional curves on the minimal resolution of the singularity. We also relate some irreducible components of the jet schemes of an $E_6$ singularity to its "minimal" embedded resolutions of singularities. 373 388 1 10.4171/149-1/18 http://www.ems-ph.org/doi/10.4171/149-1/18

Josnei Novacoski University of Silesia, Katowice, POLAND Commutative rings and algebras Algebraic geometry General Take a two-dimensional regular local domain $R$. The space of all valuations centered at $R$ has a non-metric tree structure, called the valuative tree of $R$. However, the notion of non-metric tree appearing in the literature does not guarantee the existence of infimum for a non-empty set of valuations. We give a more general definition of a rooted non-metric tree and prove that the valuative tree has this more general property. We also generalize some topological results related to a non-metric tree. For instance, we show that the weak tree topology is always coarser than the metric topology given by any parametrization. 389 403 1 10.4171/149-1/19 http://www.ems-ph.org/doi/10.4171/149-1/19

Josnei Novacoski University of Silesia, Katowice, POLAND Mark Spivakovsky Université Paul Sabatier, Toulouse Cedex 9, FRANCE Commutative rings and algebras Algebraic geometry General The main result of this paper is that in order to prove the local uniformization theorem for local domains, it is enough to prove it for rank one valuations. Our proof does not depend on the nature of the class of local domains for which we want to prove local uniformization. We prove also the reductions for diff erent versions of the local uniformization theorem. 404 431 1 10.4171/149-1/20 http://www.ems-ph.org/doi/10.4171/149-1/20

Florian Pop University of Pennsylvania, PHILADELPHIA, UNITED STATES Field theory and polynomials General The large elds were introduced by the author in [60] and subsequently acquired several other names. This little survey includes earlier and new developments, and at the end of each section we mention a few open questions. 432 463 1 10.4171/149-1/21 http://www.ems-ph.org/doi/10.4171/149-1/21

Shai Sarussi Sce College, Ashdod, ISRAEL Commutative rings and algebras General topology General Suppose $F$ is a field with a nontrivial valuation $v$ and valuation ring $O_{v}$, $E$ is a finite field extension and $w$ is a quasi-valuation on $E$ extending $v$. We study the topology induced by $w$. We prove that the quasi-valuation ring determines the topology, independent of the choice of its quasi-valuation. Moreover, we prove the weak approximation theorem for quasi-valuations. 464 473 1 10.4171/149-1/22 http://www.ems-ph.org/doi/10.4171/149-1/22

Bernard Teissier UMR 7586 du CNRS, PARIS CEDEX 13, FRANCE Algebraic geometry General Given an equicharacteristic complete noetherian local ring $R$ with algebraically closed residue field $k$, we first present a combinatorial proof of \emph{embedded} local uniformization for zero-dimensional valuations of $R$ whose associated graded ring ${\rm gr}_\nu R$ with respect to the filtration defined by the valuation is a finitely generated $k$-algebra. The main idea here is that some of the birational toric maps which provide embedded pseudo-resolutions for the affine toric variety corresponding to ${\rm gr}_\nu R$ also provide local uniformizations for $\nu$ on $R$. These valuations are necessarily Abhyankar (for zero-dimensional valuations this means that the value group is $\mathbf Z^r$ with $r={\rm dim}R$).\Par In a second part we show that conversely, given an excellent noetherian equicharacteristic local domain $R$ with algebraically closed residue field, if the zero-dimensional valuation $\nu$ of $R$ is Abhyankar, there are local domains $R'$ which are essentially of finite type over $R$ and dominated by the valuation ring $R_\nu$ ($\nu$-modifications of $R$) such that the semigroup of values of $\nu$ on $R'$ is finitely generated, and therefore so is the $k$-algebra ${\rm gr}_\nu R'$. Combining the two results and using the fact that Abhyankar valuations behave well under completion gives a proof of local uniformization for rational Abhyankar valuations and, by a specialization argument, for all Abhyankar valuations. \par As a by-product we obtain a description of the valuation ring of a rational Abhyankar valuation as an inductive limit indexed by $\mathbf N$ of birational toric maps of regular local rings. One of our main tools, the valuative Cohen theorem, is then used to study the extensions of rational monomial Abhyankar valuations of the ring $k[[x_1,\ldots ,x_r]]$ to monogenous integral extensions and the nature of their key polynomials. In the conclusion we place the results in the perspective of local embedded resolution of singularities by a single toric modification after an appropriate re-embedding. 474 565 1 10.4171/149-1/23 http://www.ems-ph.org/doi/10.4171/149-1/23

Adam Topaz University of California, BERKELEY, UNITED STATES Field theory and polynomials General In this note we show how to detect valuations using almost-abelian pro-$\ell$ Galois groups of a field. In particular, we show that "commuting-liftable" subgroups of Galois groups arise, in a controlled way, from Kummer-duals of (principal-)units of valuations. 568 578 1 10.4171/149-1/24 http://www.ems-ph.org/doi/10.4171/149-1/24

Lou van den Dries University of Illinois at Urbana-Champaign, URBANA, UNITED STATES Order, lattices, ordered algebraic structures Field theory and polynomials General We consider truncation closed subgroups, subrings, and subfi elds of Hahn fields, and show that the property of being truncation closed is preserved under various natural ways of extending these substructures. 579 595 1 10.4171/149-1/25 http://www.ems-ph.org/doi/10.4171/149-1/25

Ekaterina Yurova Linnaeus University, Vaxjo, SWEDEN Dynamical systems and ergodic theory General In this paper we present results about ergodicity of dynamical systems on $2$-adic spheres for 1-Lipschitz maps $f:\mathbb Z_2\rightarrow \mathbb Z_2$ announced in [8], and extension of Theorem 3 from [8] for the case of spheres of radii greater than $\frac{1}{8}.$ We propose a new approach to study ergodic properties of 1-Lipschitz transformations of $2$-adic spheres. We use a representation of continuous functions $f$ via its van der Put series. This technique allows us to go beyond the classes of smooth 1-Lipschitz transformations which were studied earlier. 596 599 1 10.4171/149-1/26 http://www.ems-ph.org/doi/10.4171/149-1/26

Liang Xiao University of California at Irvine, Irvine, UNITED STATES Igor Zhukov St. Petersburg University, Peterhof, St. Petersburg, RUSSIAN FEDERATION Algebraic geometry Number theory General A survey paper contains facts, ideas and problems related to ramifi cation in fi nite extensions of complete discrete valuation fi elds with arbitrary residue fields. Some new results are included. 600 656 1 10.4171/149-1/27 http://www.ems-ph.org/doi/10.4171/149-1/27

Lizhen Ji University of Michigan, Ann Arbor, USA Athanase Papadopoulos Université de Strasbourg, France History and biography Topological groups, Lie groups Geometry Differential geometry 01-00, 01-02, 01A05, 01A55, 01A70, 22-00, 22-02, 22-03, 51N15, 51P05, 53A20, 53A35, 53B50, 54H15, 58E40 Sophus Lie, Felix Klein, the Erlangen program, group action, Lie group action, symmetry, projective geometry, non-Euclidean geometry, spherical geometry, hyperbolic geometry, transitional geometry, discrete geometry, transformation group The Erlangen program expresses a fundamental point of view on the use of groups and transformation groups in mathematics and physics. The present volume is the first modern comprehensive book on that program and its impact in contemporary mathematics and physics. Klein spelled out the program, and Lie, who contributed to its formulation, is the first mathematician who made it effective in his work. The theories that these two authors developed are also linked to their personal history and to their relations with each other and with other mathematicians, incuding Hermann Weyl, Élie Cartan, Henri Poincaré, and many others. All these facets of the Erlangen program appear in the present volume. The book is written by well-known experts in geometry, physics and history of mathematics and physics. It is addressed to mathematicians, to graduate students, and to all those interested in the development of mathematical ideas. 4 30 2015 978-3-03719-148-4 978-3-03719-648-9 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/148 http://www.ems-ph.org/doi/10.4171/148 23

Lizhen Ji University of Michigan, ANN ARBOR, UNITED STATES History and biography General 1 26 1 10.4171/148-1/1 http://www.ems-ph.org/doi/10.4171/148-1/1

Lizhen Ji University of Michigan, ANN ARBOR, UNITED STATES History and biography General 27 58 1 10.4171/148-1/2 http://www.ems-ph.org/doi/10.4171/148-1/2

Jeremy Gray The Open University, MILTON KEYNES, UNITED KINGDOM History and biography General 59 75 1 10.4171/148-1/3 http://www.ems-ph.org/doi/10.4171/148-1/3

Hubert Goenner Universität Göttingen, GÖTTINGEN, GERMANY History and biography General 77 90 1 10.4171/148-1/4 http://www.ems-ph.org/doi/10.4171/148-1/4

Norbert A’Campo Universität Basel, BASEL, SWITZERLAND Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE History and biography General 91 136 1 10.4171/148-1/5 http://www.ems-ph.org/doi/10.4171/148-1/5

Alexandre Vinogradov , LIZZANO IN BELVEDERE (BO), ITALY History and biography General 137 190 1 10.4171/148-1/6 http://www.ems-ph.org/doi/10.4171/148-1/6

Charles Frances Université Paris Sud, ORSAY CEDEX, FRANCE History and biography General 191 216 1 10.4171/148-1/7 http://www.ems-ph.org/doi/10.4171/148-1/7

Norbert A’Campo Universität Basel, BASEL, SWITZERLAND Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE History and biography General 217 235 1 10.4171/148-1/8 http://www.ems-ph.org/doi/10.4171/148-1/8

Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Sumio Yamada Gakushuin University, TOKYO, JAPAN History and biography General 237 245 1 10.4171/148-1/9 http://www.ems-ph.org/doi/10.4171/148-1/9

Yuri Suris Technische Universität Berlin, BERLIN, GERMANY History and biography General 247 281 1 10.4171/148-1/10 http://www.ems-ph.org/doi/10.4171/148-1/10

Catherine Meusburger Universität Erlangen-Nürnberg, ERLANGEN, GERMANY History and biography General 283 306 1 10.4171/148-1/11 http://www.ems-ph.org/doi/10.4171/148-1/11

Jean-Bernard Zuber Universite Pierre et Marie Curie, PARIS CEDEX 05, FRANCE History and biography General 307 324 1 10.4171/148-1/12 http://www.ems-ph.org/doi/10.4171/148-1/12

Athanase Papadopoulos Université de Strasbourg, France Marc Troyanov École Polytechnique Fédérale de Lausanne, Switzerland Differential geometry Geometry Convex and discrete geometry Global analysis, analysis on manifolds 01A55, 01-99, 35Q53, 37D25, 37D20, 37D40, 47H09, 51-00, 51-02, 51-03, 51A05, 51B20, 51F99, 51K05, 51K10, 51K99, 51M10, 52A07, 52A20, 52A99, 53A20, 53A35, 53B40, 53C22, 53C24, 53C60, 53C70, 53B40, 54H20, 57S25, 58-00, 58-02, 58-03, 58B20, 58D05, 58F07. Hilbert metric, Funk metric, non-symmetric metric, Finsler geometry, Minkowski space, Minkowski functional, convexity, Cayley-Klein-Beltrami model, projective manifold, projective volume, Busemann curvature, Busemann volume, horofunction, geodesic flow, Teichmüller space, Hilbert fourth problem, entropy, geodesic, Perron-Frobenius theory, geometric structure, holonomy homomorphism This volume presents surveys, written by experts in the field, on various classical and the modern aspects of Hilbert geometry. They are assuming several points of view: Finsler geometry, calculus of variations, projective geometry, dynamical systems, and others. Some fruitful relations between Hilbert geometry and other subjects in mathematics are emphasized, including Teichmüller spaces, convexity theory, Perron–Frobenius theory, representation theory, partial differential equations, coarse geometry, ergodic theory, algebraic groups, Coxeter groups, geometric group theory, Lie groups and discrete group actions. The Handbook is addressed to both students who want to learn the theory and researchers working in the area. 12 1 2014 978-3-03719-147-7 978-3-03719-647-2 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/147 http://www.ems-ph.org/doi/10.4171/147 22

Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Marc Troyanov Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND Geometry Differential geometry General We define the notion of weak Minkowski metric and prove some basic properties of such metrics. We also highlight some of the important analogies between Minkowski geometry and the Funk and Hilbert geometries. 11 32 1 10.4171/147-1/1 http://www.ems-ph.org/doi/10.4171/147-1/1

Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Marc Troyanov Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND Geometry Differential geometry General We survey some basic geometric properties of the Funk metric of a convex set in $\mathbb{R}^n$. In particular, we study its geodesics, its topology, its metric balls, its convexity properties, its perpendicularity theory and its isometries. The Hilbert metric is a symmetrization of the Funk metric, and we show some properties of the Hilbert metric that follow directly from the properties we prove for the Funk metric. 33 67 1 10.4171/147-1/2 http://www.ems-ph.org/doi/10.4171/147-1/2

Marc Troyanov Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND Differential geometry General In 1929, Paul Funk and Ludwig Berwald gave a characterization of Hilbert geometries from the Finslerian viewpoint. They showed that a smooth Finsler metric in a strongly convex bounded domain of $\mathbb{R}^n$ is the Hilbert geometry in that domain if and only if it is complete, if its geodesics are straight lines and if its flag curvature is equal to $-1$. The goal of this chapter is to explain these notions in details, to illustrate the relation between Hilbert geometry, Finsler geometry and the calculus of variations, and to prove the Funk–Berwald characterization theorem. 69 110 1 10.4171/147-1/3 http://www.ems-ph.org/doi/10.4171/147-1/3

Constantin Vernicos Université Montpellier 2, MONTPELLIER CEDEX, FRANCE Differential geometry Geometry General We survey the Hilbert geometry of convex polytopes. In particular we present two important characterizations of these geometries, the first one in terms of the volume growth of their metric balls, the second one as a metric bilipschitzly equivalent to the Hilbert geometry in the simplex. 111 125 1 10.4171/147-1/4 http://www.ems-ph.org/doi/10.4171/147-1/4

Cormac Walsh Ecole Polytechnique, PALAISEAU CEDEX, FRANCE Convex and discrete geometry General The horofunction boundary is a means of compactifying metric spaces that was introduced by Gromov in the 1970s. We describe explicitly the horofunction boundary of the Hilbert geometry, and sketch how it may be used to study the isometry group of this space. 127 146 1 10.4171/147-1/5 http://www.ems-ph.org/doi/10.4171/147-1/5

Ren Guo Oregon State University, CORVALLIS, UNITED STATES Geometry Convex and discrete geometry Differential geometry General This chapter is a survey of different characterizations of hyperbolic geometry among Hilbert geometries. 147 158 1 10.4171/147-1/6 http://www.ems-ph.org/doi/10.4171/147-1/6

Ludovic Marquis Université de Rennes I, RENNES CEDEX, FRANCE Global analysis, analysis on manifolds Partial differential equations General In this chapter, we survey groups of projective transformations acting on a Hilbert geometry. 207 261 1 10.4171/147-1/7 http://www.ems-ph.org/doi/10.4171/147-1/7

Mickaël Crampon Universidad de Santiago de Chile, Santiago, CHILE Dynamical systems and ergodic theory Differential geometry General This is a survey of the dynamics of the geodesic flow of Hilbert geometries. The main idea is to compare this flow with the geodesic flow of negatively curved Finsler or Riemannian manifolds, by making links between various useful objects and by comparing results and questions. 161 206 1 10.4171/147-1/8 http://www.ems-ph.org/doi/10.4171/147-1/8

Anders Karlsson Université de Genève, GENÈVE, SWITZERLAND Operator theory Geometry General In his work on the foundations of geometry, David Hilbert observed that a formula which appeared in works by Klein gives rise to a complete metric on any bounded convex domain. Some decades later, Garrett Birkhoff and Hans Samelson noticed that this metric has interesting applications, when considering certain maps of convex cones that contract the metric. Such situations have since arisen in many contexts, pure and applied, and could be called nonlinear Perron–Frobenius theory. This note centers around one dynamical aspect of this theory. 263 273 1 10.4171/147-1/9 http://www.ems-ph.org/doi/10.4171/147-1/9

Bas Lemmens University of Kent, CANTERBURY, UNITED KINGDOM Roger Nussbaum Rutgers University, PISCATAWAY, UNITED STATES Operator theory Differential geometry General topology General Birkhoff's version of Hilbert's metric is a distance between pairs of rays in a closed cone, and is closely related to Hilbert's classical cross-ratio metric. The version we discuss here was popularized by Bushell and can be traced back to the work of Garrett Birkhoff and Hans Samelson. It has found numerous applications in mathematical analysis, especially in the analysis of linear, and nonlinear, mappings on cones. Some of these applications are discussed in this chapter. Birkhoff's version of Hilbert's metric provides a different perspective on Hilbert geometries and naturally leads to infinite-dimensional generalizations. We illustrate this by showing some of its uses in the geometric analysis of Hilbert geometries. 275 303 1 10.4171/147-1/10 http://www.ems-ph.org/doi/10.4171/147-1/10

Inkang Kim KIAS, SEOUL, SOUTH KOREA Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Geometry Manifolds and cell complexes General We review some basic concepts related to convex real projective structures from the differential geometry point of view. We start by recalling a Riemannian metric which originates in the study of affine spheres using the Blaschke connection (work of Calabi and of Cheng--Yau) mentioning its relation with the Hilbert metric. We then survey some of the deformation theory of convex real projective structures on surfaces. We describe in particular how the set of (Hilbert) lengths of simple closed curves is used in a parametrization of the deformation space in analogy with the classical Fenchel–Nielsen parameters of Teichmüller space (work of Goldman). We then mention parameters of this deformation space that arise in the work of Hitchin on the character variety of representations of the fundamental group of the surface in $\mathrm{SL}(3,\mathbb{R})$. In this character variety, the component of the character variety that corresponds to projective structures is identified with the vector space of pairs of holomorphic quadratic and cubic differentials over a fixed Riemann surface. Labourie and Loftin (independently) obtained parameter spaces that use the cubic differentials and affine spheres. We then display some similarities and differences between Hilbert geometry and hyperbolic geometry using geodesic currents and topological entropy. Finally, we discuss geodesic flows associated to Hilbert metrics and compactifications of spaces of convex real projective structures on surfaces. This makes another analogy with works done on the Teichmüller space of the surface. 307 338 1 10.4171/147-1/11 http://www.ems-ph.org/doi/10.4171/147-1/11

Hideki Miyachi Osaka University, OSAKA, JAPAN Ken’ichi Ohshika Osaka University Graduate School of Science, OSAKA, JAPAN Sumio Yamada Gakushuin University, TOKYO, JAPAN Global analysis, analysis on manifolds Partial differential equations General As a deformation space of hyperbolic metrics defined on a closed surface of genus $g \geq 2$, Teichmüller space can be regarded as a convex set with respect to the Weil--Petersson geometry. The convexity is used to construct a new distance function, called the Weil–Petersson Funk metric, which is a weak metric, lacking the symmetry and non-degeneracy conditions. The distance between two points is defined as a supremum of functions on the pair of points indexed by the set of the simple closed curves of the given surface. This set acts as the index set of the supporting hyperplanes of Teichmüller space regarded as a Weil–Petersson convex body. This Funk-type construction also appears in defining the two well-known distance functions on Teichmüller space: the Teichmüller metric and the Thurston metric. We emphasize in this chapter that the underlying idea for the three distance functions is to treat the Teichmüller space as a convex body. 339 352 1 10.4171/147-1/12 http://www.ems-ph.org/doi/10.4171/147-1/12

Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Sumio Yamada Gakushuin University, TOKYO, JAPAN Global analysis, analysis on manifolds Geometry Differential geometry General We survey the Funk and Hilbert geometries of open convex sets in the sphere $S^n$ and in the hyperbolic space $\mathbb{H}^n$. The theories are developed in analogy with the classical theory in Euclidean space. It is rather unexpected that the Funk geometry, whose definition and development use the affine structure of Euclidean space, has analogues in the non-linear spaces $S^n$ and $\mathbb{H}^n$, where there are no analogies. As we shall see, the existence of a Funk geometry in these non-linear spaces is based on some non-Euclidean trigonometric formulae which display some kind of similarity between (the hyperbolic sine of the lengths of) sides of right triangles. The Hilbert metric in each of the constant curvature settings is a symmetrization of the Funk metric. We show that the Hilbert metric of a convex subset in a space of constant curvature can also be defined using a notion of a cross ratio which is proper to that space. 353 379 1 10.4171/147-1/13 http://www.ems-ph.org/doi/10.4171/147-1/13

Marc Troyanov Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND Geometry History and biography General In this brief chapter we succinctly comment on the historical origin of Hilbert geometry. In particular, we give a summary of the letter in which David Hilbert informs his friend and colleague Felix Klein about his discovery of this geometry. 383 389 1 10.4171/147-1/14 http://www.ems-ph.org/doi/10.4171/147-1/14

Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE History and biography Geometry Differential geometry Global analysis, analysis on manifolds Hilbert's fourth problem asks for the construction and the study of metrics on subsets of projective space for which the projective line segments are geodesics. Several solutions of the problem were given so far, depending on more precise interpretations of this problem, with various additional conditions satisfied. The most interesting solutions are probably those inspired by an integral formula that was first introduced in this theory by Herbert Busemann. Besides that, Busemann and his school made a thorough investigation of metrics defined on subsets of projective space for which the projective lines are geodesics and they obtained several results, characterizing several classes of such metrics. We review some of the developments and important results related to Hilbert's problem, especially those that arose from Busemann's work, mentioning recent results and connections with several branches of mathematics, including Riemannian geometry, the foundations of mathematics, the calculus of variations, metric geometry and Finsler geometry. Hilbert metrics – the subject of this handbook – constitute a basic class of metrics that satisfy the requirements of Hilbert's problem. 391 431 1 10.4171/147-1/15 http://www.ems-ph.org/doi/10.4171/147-1/15

General 433 442 1 10.4171/147-1/16 http://www.ems-ph.org/doi/10.4171/147-1/16

Della Dumbaugh University of Richmond, USA Joachim Schwermer University of Vienna, Austria History and biography Number theory 01A60, 01A70, 11R37, 11R39, 11S37, 11S39, 11Fxx, 11Mxx Number theory, class field theory, L-functions, automorphic L-functions; history of mathematics, Emil Artin This book explores the development of number theory, and class field theory in particular, as it passed through the hands of Emil Artin, Claude Chevalley and Robert Langlands in the middle of the twentieth century. Claude Chevalley’s presence in Artin’s 1931 Hamburg lectures on class field theory serves as the starting point for this volume. From there, it is traced how class field theory advanced in the 1930s and how Artin’s contributions influenced other mathematicians at the time and in subsequent years. Given the difficult political climate and his forced emigration as it were, the question of how Artin created a life in America within the existing institutional framework, and especially of how he continued his education of and close connection with graduate students, is considered. In particular, Artin’s collaboration in algebraic number theory with George Whaples and his student Margaret Matchett’s thesis work “On the zeta-function for ideles” in the 1940s are investigated. A (first) study of the influence of Artin on present day work on a non-abelian class field theory finishes the book. The volume consists of individual essays by the authors and two contributors, James Cogdell and Robert Langlands, and contains relevant archival material. Among these, the letter from Chevalley to Helmut Hasse in 1935 is included, where he introduces the notion of ideles and explores their significance, along with the previously unpublished thesis by Matchett and the seminal letter of Langlands to André Weil of 1967 where he lays out his ideas regarding a non-abelian class field theory. Taken together, these chapters offer a view of both the life of Artin in the 1930s and 1940s and the development of class field theory at that time. They also provide insight into the transmission of mathematical ideas, the careful steps required to preserve a life in mathematics at a difficult moment in history, and the interplay between mathematics and politics (in more ways than one). Some of the technical points in this volume require a sophisticated understanding of algebra and number theory. The broader topics, however, will appeal to a wider audience that extends beyond mathematicians and historians of mathematics to include historically minded individuals, particularly those with an interest in the time period. 3 31 2015 978-3-03719-146-0 978-3-03719-646-5 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/146 http://www.ems-ph.org/doi/10.4171/146 Revisiting a hundred-year-old theoremTranslated from the French by Robert G. Burns Henri Paul de Saint-Gervais Paris History and biography General (primary; secondary): 30F10; 01A55, 14H55, 30-01, 30-03, 30Fxx Riemann surfaces, uniformization, Fuchsian groups, continuity method, Gauss, Riemann, Schwarz, Poincaré, Klein, Koebe In 1907 Paul Koebe and Henri Poincar é almost simultaneously proved the uniformization theorem: Every simply connected Riemann surface is isomorphic to the plane, the open unit disc, or the sphere. It took a whole century to get to the point of stating this theorem and providing a convincing proof of it, relying as it did on prior work of Gauss, Riemann, Schwarz, Klein, Poincar é, and Koebe, among others. The present book o ffers an overview of the maturation process of this theorem. The evolution of the uniformization theorem took place in parallel with the emergence of modern algebraic geometry, the creation of complex analysis, the fi rst stirrings of functional analysis, and with the flowering of the theory of di fferential equations and the birth of topology. The uniformization theorem was thus one of the lightning rods of 19th century mathematics. Rather than describe the history of a single theorem, our aim is to return to the original proofs, to look at these through the eyes of modern mathematicians, to enquire as to their correctness, and to attempt to make them rigorous while respecting insofar as possible the state of mathematical knowledge at the time, or, if this should prove impossible, then using modern mathematical tools not available to their authors. This book will be useful to today's mathematicians wishing to cast a glance back at the history of their discipline. It should also provide graduate students with a non-standard approach to concepts of great importance for modern research. 1 31 2016 978-3-03719-145-3 978-3-03719-645-8 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/145 http://www.ems-ph.org/doi/10.4171/145 Martina Bečvářová Czech Technical University, Prague, Czech Republic Ivan Netuka Charles University, Prague, Czech Republic History and biography Number theory 01A60, 01A70, 11R37, 11R39, 11S37, 11S39, 11Fxx, 11Mxx Mathematical analysis, matrix functions, geometric function theory, potential theory, 20th century history of mathematics This monograph is devoted to two distinguished mathematicians, Karel Löwner (1893–1968) and Lipman Bers (1914–1993), whose lives are dramatically interlinked with key historical events of the 20th century. K. Löwner, Professor of Mathematics at the German University in Prague (Czechoslovakia), was dismissed from his position because he was a Jew, and emigrated to the USA in 1939 (where he changed his name to Charles Loewner). Earlier, he had published several outstanding papers in complex analysis and a masterpiece on matrix functions. In particular, his ground-breaking parametric method in geometric function theory from 1923, which led to Löwner’s celebrated differential equation, brought him world-wide fame and turned out to be a cornerstone in de Branges’ proof of the Bieberbach conjecture. Unexpectedly, Löwner’s differential equation has gained recent prominence with the introduction of a conformally invariant stochastic process called stochastic Loewner evolution (SLE) by O. Schramm in 2000. SLE features in two Fields Medal citations from 2006 and 2010. L. Bers was the final Prague Ph.D. student of K. Löwner. His dissertation on potential theory (1938), completed shortly before his emigration and long thought to be irretrievably lost, was found in 2006. It is here made accessible for the first time, with an extensive commentary, to the mathematical community. This monograph presents an in-depth account of the lives of both mathematicians, with special emphasis on the pre-war period. Löwner’s teaching activities and professional achievements are presented in the context of the prevailing complex political situation and against the background of the wider development of mathematics in Europe. Each of his publications is accompanied by an extensive commentary, tracing the origin and motivation of the problem studied, and describing the state-of-art at the time of the corresponding mathematical field. Special attention is paid to the impact of the results obtained and to the later development of the underlying ideas, thus connecting Löwner’s achievements to current research activity. The text is based on an extensive archival search, and most of the archival findings appear here for the first time. Anyone with an interest in mathematics and the history of mathematics will enjoy reading this book about two famous mathematicians of the 20th century. 4 10 2015 978-3-03719-144-6 978-3-03719-644-1 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/144 http://www.ems-ph.org/doi/10.4171/144 Kurusch Ebrahimi-Fard Universidad Autónoma de Madrid, Spain Frédéric Fauvet Université de Strasbourg, France Combinatorics Order, lattices, ordered algebraic structures Associative rings and algebras Approximations and expansions Primary: 05E15, 06A07, 16T05, 41A58, 58D05, 93C10; Secondary: 05C05, 81T18, 34A25, 34M25, 47H20, 65L05, 81T15, 81T16 Faà di Bruno formula, Dyson–Schwinger equations, geometric numerical integration, Butcher series, Lie–Butcher series, nonlinear control systems, nonlinear operators, combinatorial Hopf algebras, pre-Lie algebras, Lie algebras, trees, Ecalle’s mould calcul Since the early works of G.-C. Rota and his school, Hopf algebras have been instrumental in algebraic combinatorics. In a seminal 1998 paper, A. Connes and D. Kreimer presented a Hopf algebraic approach to renormalization in perturbative Quantum Field Theory (QFT). This work triggered an abundance of new research on applications of Hopf algebraic techniques in QFT as well as other areas of theoretical physics. Furthermore, these new developments were complemented by progress made in other domains of applications, such as control theory, dynamical systems, and numerical integration methods. Especially in the latter context, it became clear that J. Butcher’s work from the early 1970s was well ahead of its time. The present volume emanated from a conference hosted in June 2011 by IRMA at Strasbourg University in France. Researchers from different scientific communities who share similar techniques and objectives gathered at this meeting to discuss new ideas and results on Faà di Bruno algebras, Dyson–Schwinger equations, and Butcher series. The purpose of this book is to present a coherent set of lectures reflecting the state of the art of research on combinatorial Hopf algebras relevant to high energy physics, control theory, dynamical systems, and numerical integration methods. More specifically, connections between Dyson–Schwinger equations, Faà di Bruno algebras, and Butcher series are examined in great detail. This volume is aimed at researchers and graduate students interested in combinatorial and algebraic aspects of QFT, control theory, dynamical systems and numerical analysis of integration methods. It contains introductory lectures on the various constructions that are emerging and developing in these domains. 6 30 2015 978-3-03719-143-9 978-3-03719-643-4 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/143 http://www.ems-ph.org/doi/10.4171/143 21

José Gracia-Bondí­a Universidad Complutense, ZARAGOZA, SPAIN Combinatorics General 1 8 1 10.4171/143-1/1 http://www.ems-ph.org/doi/10.4171/143-1/1

Loïc Foissy Centre Universitaire de la Mi-Voix, CALAIS CEDEX, FRANCE General These lecture notes contain a review of the results of [15], [16], [17], and [19] about combinatorial Dyson–Schwinger equations and systems. Such an equation or system generates a subalgebra of a Connes–Kreimer Hopf algebra of decorated trees, and we shall say that the equation or the system is Hopf if the associated subalgebra is Hopf. We first give a classi cation of the Hopf combinatorial Dyson–Schwinger equations. The proof of the existence of the Hopf subalgebra uses pre-Lie structures and is different from the proof of [15] and [17]. We consider afterwards systems of Dyson-Schwinger equations. We give a description of Hopf systems, with the help of two families of special systems (quasi-cyclic and fundamental) and four operations on systems (change of variables, dilatation, extension, concatenation). We also give a few result on the dual Lie algebras. Again, the proof of the existence of these Hopf subalgebras uses pre-Lie structures and is different from the proof of [16]. 9 89 1 10.4171/143-1/2 http://www.ems-ph.org/doi/10.4171/143-1/2

Alessandra Frabetti Université Claude Bernard Lyon 1, VILLEURBANNE CEDEX, FRANCE Dominique Manchon Université Blaise Pascal, AUBIÈRE CEDEX, FRANCE Associative rings and algebras Combinatorics Group theory and generalizations Topological groups, Lie groups In these lectures we present five interpretations of the Faà di Bruno formula which computes the $n$-th derivative of the composition of two functions of one variable: in terms of groups, Lie algebras and Hopf algebras, in combinatorics and within operads. 91 147 1 10.4171/143-1/3 http://www.ems-ph.org/doi/10.4171/143-1/3

W. Steven Gray Old Dominion University, NORFOLK, UNITED STATES Luis Duffaut Espinosa University of NSW at the Australian Defence Force Academy, CANBERRA, AUSTRALIA Systems theory; control Associative rings and algebras Operator theory General In many applications, nonlinear input-output systems are interconnected in various ways to model complex systems. If a component system is analytic, meaning it can be described in terms of a Chen–Fliess functional series expansion, then it can be represented uniquely by a formal power series over a noncommutative alphabet. System interconnections are then characterized in terms of operations on formal power series. This paper provides an introduction to this methodology with an emphasis on feedback systems, which are ubiquitous in modern technology. In this case, a Faà di Bruno type Hopf algebra is de ned for a group of integral operators, where operator composition is the group product. Using a series expansion for the antipode, an explicit formula for the generating series of the compositional inverse operator is derived. This result produces an explicit formula for the generating series of a feedback system, which had been an open problem until recently. 149 217 1 10.4171/143-1/4 http://www.ems-ph.org/doi/10.4171/143-1/4

Alexander Lundervold Bergen University College, BERGEN, NORWAY Hans Munthe-Kaas University of Bergen, BERGEN, NORWAY Global analysis, analysis on manifolds Partial differential equations General Numerical analysis of time-integration algorithms has applied advanced algebraic techniques for more than fourty years. An explicit description of the group of characters in the Butcher–Connes–Kreimer Hopf algebra fi rst appeared in Butcher’s work on composition of integration methods in 1972. In more recent years, the analysis of structure preserving algorithms, geometric integration techniques and integration algorithms on manifolds have motivated the incorporation of other algebraic structures in numerical analysis. In this paper we will survey algebraic structures that have found applications within these areas. This includes pre-Lie structures for the geometry of flat and torsion free connections appearing in the analysis of numerical fl ows on vector spaces. The much more recent post-Lie and D-algebras appear in the analysis of flows on manifolds with flat connections with constant torsion. Dynkin and Eulerian idempotents appear in the analysis of non-autonomous flows and in backward error analysis. Non-commutative Bell polynomials and a non-commutative Faà di Bruno Hopf algebra are other examples of structures appearing naturally in the numerical analysis of integration on manifolds. 219 263 1 10.4171/143-1/5 http://www.ems-ph.org/doi/10.4171/143-1/5

Emmanuel Vieillard-Baron Université de Bourgogne, DIJON CEDEX, FRANCE Combinatorics Order, lattices, ordered algebraic structures Ordinary differential equations General We present a complete exposition of Ecalle’s arbori fication–coarborifi cation formalism, which is an essential component of his Mould Calculus, and we include in particular original results on the composition of arbori fied moulds. The connections with recent works regarding combinatorial Hopf algebras on trees are made but we give all the proofs in a self contained way, and we treat numerous examples. 265 353 1 10.4171/143-1/6 http://www.ems-ph.org/doi/10.4171/143-1/6

Craig Roberts Argonne National Laboratory, ARGONNE, UNITED STATES Quantum theory General The real-world properties of quantum chromodynamics (QCD) – the strongly interacting piece of the Standard Model – are dominated by two emergent phenomena: con finement; namely, the theory’s elementary degrees-of-freedom – quarks and gluons – have never been detected in isolation; and dynamical chiral symmetry breaking (DCSB), which is a remarkably effective mass generating mechanism, responsible for the mass of more than 98% of visible matter in the Universe. These phenomena are not apparent in the formulae that de fine QCD, yet they play a principal role in determining Nature’s observable characteristics. Much remains to be learnt before con finement can properly be understood. On the other hand, the last decade has seen important progress in the use of relativistic quantum field theory, so that we can now explain the origin of DCSB and are beginning to demonstrate its far-reaching consequences. Dyson–Schwinger equations have played a critical role in these advances. These lecture notes provide an introduction to Dyson–Schwinger equations (DSEs), QCD and hadron physics, and illustrate the use of DSEs to predict observable phenomena. 355 458 1 10.4171/143-1/7 http://www.ems-ph.org/doi/10.4171/143-1/7

Kathrin Bringmann Universität zu Köln, Germany Yann Bugeaud IRMA Strasbourg, France Titus Hilberdink University of Reading, UK Jürgen Sander Universität Hildesheim, Germany Number theory Computer science Primary: 11-02, 11J81; Secondary 11A63, 11J04, 11J13, 11J68, 11J70, 11J87, 68R15 Transcendence, algebraic number, Schmidt Subspace Theorem, continued fraction, digital expansion, Diophantine approximation This book arose from courses given at the International Summer School organized in August 2012 by the number theory group of the Department of Mathematics at the University of Würzburg. It consists of four essentially self-contained chapters and presents recent research results highlighting the strong interplay between number theory and other fields of mathematics, such as combinatorics, functional analysis and graph theory. The book is addressed to (under)graduate students who wish to discover various aspects of number theory. Remarkably, it demonstrates how easily one can approach frontiers of current research in number theory by elementary and basic analytic methods. Kathrin Bringmann gives an introduction to the theory of modular forms and, in particular, so-called Mock theta-functions, a topic which had been untouched for decades but has obtained much attention in the last years. Yann Bugeaud is concerned with expansions of algebraic numbers. Here combinatorics on words and transcendence theory are combined to derive new information on the sequence of decimals of algebraic numbers and on their continued fraction expansions. Titus Hilberdink reports on a recent and rather unexpected approach to extreme values of the Riemann zeta-function by use of (multiplicative) Toeplitz matrices and functional analysis. Finally, Jürgen Sander gives an introduction to algebraic graph theory and the impact of number theoretical methods on fundamental questions about the spectra of graphs and the analogue of the Riemann hypothesis. 11 23 2015 978-3-03719-142-2 978-3-03719-642-7 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/142 http://www.ems-ph.org/doi/10.4171/142 Armen Sergeev Steklov Mathematical Institute, Moscow, Russia Global analysis, analysis on manifolds Differential geometry Primary: 58B20, 58B25, 58B34; Secondary: 53C55, 53D50 Teichmüller spaces, conformal maps, quasisymmetric homeomorphisms, Kähler manifolds, geometric quantization, noncommutative geometry This book is based on a lecture course given by the author at the Educational Center of Steklov Mathematical Institute in 2011. It is designed for a one semester course for undergraduate students, familiar with basic differential geometry, complex and functional analysis. The universal Teichmüller space $\mathcal T$ is the quotient of the space of quasisymmetric homeomorphisms of the unit circle modulo Möbius transformations. The first part of the book is devoted to the study of geometric and analytic properties of $\mathcal T$. It is an infinite-dimensional Kähler manifold which contains all classical Teichmüller spaces of compact Riemann surfaces as complex submanifolds which explains the name “universal Teichmüller space”. Apart from classical Teichmüller spaces, $\mathcal T$ contains the space $\mathcal S$ of diffeomorphisms of the circle modulo Möbius transformations. The latter space plays an important role in the quantization of the theory of smooth strings. The quantization of $\mathcal T$ is presented in the second part of the book. In contrast with the case of diffeomorphism space $\mathcal S$, which can be quantized in frames of the conventional Dirac scheme, the quantization of $\mathcal T$ requires an absolutely different approach based on the noncommutative geometry methods. The book concludes with a list of 24 problems and exercises which can be used during the examinations. 8 12 2014 978-3-03719-141-5 978-3-03719-641-0 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/141 http://www.ems-ph.org/doi/10.4171/141 From the Poisson Equation to Nonlinear Thomas–Fermi Problems Augusto Ponce Université catholique de Louvain, Belgium Measure and integration Potential theory Primary: 28-02, 31-01, 35-02, 35R06; Secondary: 26B20, 26B35, 26D10, 28A12, 28A25, 28A33, 28A78, 28C05, 31B05, 31B10, 31B15, 31B20, 31B35, 35A01, 35A02, 35A08, 35A15, 35A23, 35A35, 35B05, 35B33, 35B45, 35B50, 35B51, 35B60, 35B65, 35C15, 35D30, 35J05, 35J10, 35J15, 35J20, 35J25, 35J60, 35J61, 35J86, 35J91, 35Q40, 35Q75, 35R05, 46E27, 46E30, 46E35, 49J40, 49J45, 46N20, 49S05 Balayage method, Borel measure, Chern–Simons equation, continuous potential, diffuse measure, Dirichlet problem, elliptic PDE, Euler–Lagrange equation, extremum solution, fractional Sobolev inequality, Frostman’s lemma, Hausdorff measure, Hausdorff content, Kato’s inequality, Laplacian, Lebesgue set, Lebesgue space, Marcinkiewicz space, maximum principle, minimization problem, Morrey’s imbedding, obstacle problem, Perron’s method, Poisson equation, potential theory, precise representative, reduced measure, regularity theory, removable singularity, Riesz representation theorem, Schrödinger operator, semilinear equation, Sobolev capacity, Sobolev space, subharmonic, superharmonic, sweeping-out method, Thomas–Fermi equation, trace inequality, Weyl’s lemma Partial differential equations (PDEs) and geometric measure theory (GMT) are branches of analysis whose connections are usually not emphasized in introductory graduate courses. Yet, one cannot dissociate the notions of mass or electric charge, naturally described in terms of measures, from the physical potential they generate. Having such a principle in mind, this book illustrates the beautiful interplay between tools from PDEs and GMT in a simple and elegant way by investigating properties like existence and regularity of solutions of linear and nonlinear elliptic PDEs. Inspired by a variety of sources, from the pioneer balayage scheme of Poincaré to more recent results related to the Thomas–Fermi and the Chern–Simons models, the problems covered in this book follow an original presentation, intended to emphasize the main ideas in the proofs. Classical techniques like regularity theory, maximum principles and the method of sub- and supersolutions are adapted to the setting where merely integrability or density assumptions on the data are available. The distinguished role played by capacities and precise representatives is also explained. Other special features are: • the remarkable equivalence between Sobolev capacities and Hausdorff contents in terms of trace inequalities; • the strong approximation of measures in terms of capacities or densities, normally absent from GMT books; • the rescue of the strong maximum principle for the Schrödinger operator involving singular potentials. This book invites the reader to a trip through modern techniques in the frontier of elliptic PDEs and GMT, and is addressed to graduate students and researchers having some deep interest in analysis. Most of the chapters can be read independently, and only basic knowledge of measure theory, functional analysis and Sobolev spaces is required. 10 14 2016 978-3-03719-140-8 978-3-03719-640-3 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/140 http://www.ems-ph.org/doi/10.4171/140 23 Patrick Dehornoy Université de Caen, France François Digne Université de Picardie Jules-Verne, Amiens, France Eddy Godelle Université de Caen, France Daan Krammer University of Warwick, Coventry, UK Jean Michel Université Denis Diderot Paris 7, France Group theory and generalizations Combinatorics Category theory; homological algebra Computer science Primary: 20Fxx, 20F05, 20F10, 20F36, 20F60, 20F65, 20M05, 20M10; Secondary: 05A05, 18B40, 18G35, 20B30, 20F55, 20L05, 20M50, 20N02, 68Q17 Group, monoid, category, greedy decomposition, normal decomposition, symmetric normal decomposition, Garside family, Garside map, Garside element, Garside monoid, Garside group, word problem, conjugacy problem, braid group, Artin–Tits group, Deligne–Luzstig variety, self-distributivity, ordered group, Yang–Baxter equation, cell decomposition Winner of the 2014 EMS Monograph Award! This text is a monograph in algebra, with connections toward geometry and low-dimensional topology. It mainly involves groups, monoids, and categories, and aims at providing a unified treatment for those situations in which one can find distinguished decompositions by iteratively extracting a maximal fragment lying in a prescribed family. Initiated in 1969 by F. A. Garside in the case of Artin’s braid groups, this approach turned out to lead to interesting results in a number of cases, the central notion being what the authors call a Garside family. At the moment, the study is far from complete, and the purpose of this book is both to present the current state of the theory and to be an invitation for further research. There are two parts: the bases of a general theory, including many easy examples, are developed in Part A, whereas various more sophisticated examples are specifically addressed in Part B. In order to make the content accessible to a wide audience of nonspecialists, exposition is essentially self-contained and very few prerequisites are needed. In particular, it should be easy to use the current text as a textbook both for Garside theory and for the more specialized topics investigated in Part B: Artin–Tits groups, Deligne-Lusztig varieties, groups of algebraic laws, ordered groups, structure groups of set-theoretic solutions of the Yang–Baxter equation. The first part of the book can be used as the basis for a graduate or advanced undergraduate course. 6 1 2015 978-3-03719-139-2 978-3-03719-639-7 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/139 http://www.ems-ph.org/doi/10.4171/139 22 Eduardo Casas-Alvero Universitat de Barcelona, Spain Geometry 51-01, 51N15; 51N10, 51N20 Projective geometry, affine geometry, Euclidean geometry, linear varieties, cross ratio, projectivities, quadrics, pencils of quadrics, correlations Projective geometry is concerned with the properties of figures that are invariant by projecting and taking sections. It is considered one of the most beautiful parts of geometry and plays a central role because its specializations cover the whole of the affine, Euclidean and non-Euclidean geometries. The natural extension of projective geometry is projective algebraic geometry, a rich and active field of research. Regarding its applications, results and techniques of projective geometry are today intensively used in computer vision. This book contains a comprehensive presentation of projective geometry, over the real and complex number fields, and its applications to affine and Euclidean geometries. It covers central topics such as linear varieties, cross ratio, duality, projective transformations, quadrics and their classifications – projective, affine and metric –, as well as the more advanced and less usual spaces of quadrics, rational normal curves, line complexes and the classifications of collineations, pencils of quadrics and correlations. Two appendices are devoted to the projective foundations of perspective and to the projective models of plane non-Euclidean geometries. The presentation uses modern language, is based on linear algebra and provides complete proofs. Exercises are proposed at the end of each chapter; many of them are beautiful classical results. The material in this book is suitable for courses on projective geometry for undergraduate students, with a working knowledge of a standard first course on linear algebra. The text is a valuable guide to graduate students and researchers working in areas using or related to projective geometry, such as algebraic geometry and computer vision, and to anyone wishing to gain an advanced view on geometry as a whole. 5 10 2014 978-3-03719-138-5 978-3-03719-638-0 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/138 http://www.ems-ph.org/doi/10.4171/138 Peter Deuflhard Konrad-Zuse-Zentrum, Berlin, Germany Martin Grötschel Konrad-Zuse-Zentrum; Berlin, Germany Dietmar Hömberg Technische Universität Berlin, Germany Ulrich Horst Humboldt-Universität zu Berlin, Germany Jürg Kramer Humboldt-Universität zu Berlin, Germany Volker Mehrmann Technische Universität Berlin, Germany Konrad Polthier Freie Universität Berlin, Germany Frank Schmidt Konrad-Zuse-Zentrum, Berlin, Germany Christof Schütte Freie Universität Berlin, Germany Martin Skutella Technische Universität Berlin, Germany Jürgen Sprekels Weierstraß Institut für Angewandte Analysis und Stochastik, Berlin, Germany History and biography General 00-02, 01-02 Mathematics for life sciences, mathematics for networks, mathematics for production, mathematics for electronic and optical devices, mathematics for finance, mathematics for visualization, mathematical education Mathematics: intellectual endeavor, production factor, key technology, key to key technologies? Mathematics is all of these! The last three of its facets have been the focus of the research and development in the Berlin-based DFG Research Center MATHEON in the last twelve years. Through these activities MATHEON has become an international trademark for carrying out creative, application-driven research in mathematics and for cooperating with industrial partners in the solution of complex problems in key technologies. Modern key technologies have become highly sophisticated, integrating aspects of engineering, computer, business and other sciences. Flexible mathematical models, as well as fast and accurate methods for numerical simulation and optimization open new possibilities to handle the indicated complexities, to react quickly, and to explore new options. Researchers in mathematical fields such as Optimization, Discrete Mathematics, Numerical Analysis, Scientific Computing, Applied Analysis and Stochastic Analysis have to work hand in hand with scientists and engineers to fully exploit this potential and to strengthen the transversal role of mathematics in the solution of the challenging problems in key technologies. This book presents in seven chapters the highlights of the research work carried out in the MATHEON application areas: Life Sciences, Networks, Production, Electronic and Photonic Devices, Finance, Visualization, and Education. The chapters summarize many of the contributions, put them in the context of current mathematical research activities and outline their impact in various key technologies. To make some of the results more easily accessible to the general public, a large number of “showcases” are presented that illustrate a few success stories. 4 28 2014 978-3-03719-137-8 978-3-03719-637-3 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/137 http://www.ems-ph.org/doi/10.4171/137 1 Jean-Pierre Bourguignon IHÉS, Bures-sur-Yvette, France Oussama Hijazi Université de Lorraine, Nancy, France Jean-Louis Milhorat Université de Nantes, France Andrei Moroianu Université de Versailles-St Quentin, France Sergiu Moroianu Institutul de Matematică al Academiei Române, București, Romania Differential geometry Nonassociative rings and algebras Ordinary differential equations Partial differential equations Primary: 53C27, 53A30, Secondary: 53C26, 53C55, 53C80, 17B10, 34L40, 35S05 Dirac operator, Penrose operator, Spin geometry, Spinc geometry, conformal geometry, Kähler manifolds, Quaternion-Kähler manifolds, Weyl geometry, representation theory, Killing spinors, eigenvalues The book gives an elementary and comprehensive introduction to Spin Geometry, with particular emphasis on the Dirac operator which plays a fundamental role in differential geometry and mathematical physics. After a self-contained presentation of the basic algebraic, geometrical, analytical and topological ingredients, a systematic study of the spectral properties of the Dirac operator on compact spin manifolds is carried out. The classical estimates on eigenvalues and their limiting cases are discussed next, highlighting the subtle interplay of spinors and special geometric structures. Several applications of these ideas are presented, including spinorial proofs of the Positive Mass Theorem or the classification of positive Kähler–Einstein contact manifolds. Representation theory is used to explicitly compute the Dirac spectrum of compact symmetric spaces. The special features of the book include a unified treatment of Spin$^\mathrm c$ and conformal spin geometry (with special emphasis on the conformal covariance of the Dirac operator), an overview with proofs of the theory of elliptic differential operators on compact manifolds based on pseudodifferential calculus, a spinorial characterization of special geometries, and a self-contained presentation of the representation-theoretical tools needed in order to apprehend spinors. This book will help advanced graduate students and researchers to get more familiar with this beautiful, though not sufficiently known, domain of mathematics with great relevance to both theoretical physics and geometry. 6 29 2015 978-3-03719-136-1 978-3-03719-636-6 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/136 http://www.ems-ph.org/doi/10.4171/136 Kazuhiro Fujiwara Nagoya University, Japan Fumiharu Kato Tokyo Institute of Technology, Japan Number theory Order, lattices, ordered algebraic structures Commutative rings and algebras Algebraic geometry Primary: 11G99, Secondary: 06E99, 13F30, 13J07, 14A15, 14A20 Number theory Rigid geometry, formal geometry, birational geometry Rigid geometry is one of the modern branches of algebraic and arithmetic geometry. It has its historical origin in J. Tate’s rigid analytic geometry, which aimed at developing an analytic geometry over non-archimedean valued fields. Nowadays, rigid geometry is a discipline in its own right and has acquired vast and rich structures, based on discoveries of its relationship with birational and formal geometries. In this research monograph, foundational aspects of rigid geometry are discussed, putting emphasis on birational and topological features of rigid spaces. Besides the rigid geometry itself, topics include the general theory of formal schemes and formal algebraic spaces, based on a theory of complete rings which are not necessarily Noetherian. Also included is a discussion on the relationship with Tate‘s original rigid analytic geometry, V.G. Berkovich‘s analytic geometry and R. Huber‘s adic spaces. As a model example of applications, a proof of Nagata‘s compactification theorem for schemes is given in the appendix. The book is encyclopedic and almost self-contained. 1 22 2018 978-3-03719-135-4 978-3-03719-635-9 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/135 https://ems.eye-com.dev/doi/10.4171/135 2523-5192 2523-5206 Emmanuel Hebey Université de Cergy-Pontoise, France Global analysis, analysis on manifolds Partial differential equations 58J05, 35J15 Blow-up theory, compactness, critical nonlinear elliptic equations, stability The book offers an expanded version of lectures given at ETH Zürich in the framework of a Nachdiplomvorlesung. Compactness and stability for nonlinear elliptic equations in the inhomogeneous context of closed Riemannian manifolds are investigated, a field presently undergoing great development. The author describes blow-up phenomena and presents the progress made over the past years on the subject, giving an up-to-date description of the new ideas, concepts, methods, and theories in the field. Special attention is devoted to the nonlinear stationary Schrödinger equation and to its critical formulation. Intended to be as self-contained as possible, the book is accessible to a broad audience of readers, including graduate students and researchers. 7 1 2014 978-3-03719-134-7 978-3-03719-634-2 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/134 http://www.ems-ph.org/doi/10.4171/134 Fabrice Baudoin Purdue University, West Lafayette, USA Probability theory and stochastic processes 60-01, 60G07, 60J60, 60J65, 60H05, 60H07 Brownian motion, diffusion processes, Malliavin calculus, rough paths theory, semigroup theory, stochastic calculus, stochastic processes The main purpose of the book is to present at a graduate level and in a self-contained way the most important aspects of the theory of continuous stochastic processes in continuous time and to introduce to some of its ramifications like the theory of semigroups, the Malliavin calculus and the Lyons’ rough paths. It is intended for students, or even researchers, who wish to learn the basics in a concise but complete and rigorous manner. Several exercises are distributed throughout the text to test the understanding of the reader and each chapter ends up with bibliographic comments aimed to those interested in exploring further the materials. The stochastic calculus has been developed in the 1950s and the range of its applications is huge and still growing today. Besides being a fundamental component of modern probability theory, domains of applications include but are not limited to: mathematical finance, biology, physics, and engineering sciences. The first part of the text is devoted the general theory of stochastic processes, we focus on existence and regularity results for processes and on the theory of martingales. This allows to quickly introduce the Brownian motion and to study its most fundamental properties. The second part deals with the study of Markov processes, in particular diffusions. Our goal is to stress the connections between these processes and the theory of evolution semigroups. The third part deals with stochastic integrals, stochastic differential equations and Malliavin calculus. Finally, in the fourth and final part we present an introduction to the very new theory of rough paths by Terry Lyons. 7 7 2014 978-3-03719-133-0 978-3-03719-633-5 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/133 http://www.ems-ph.org/doi/10.4171/133 Second edition Athanase Papadopoulos IRMA, Strasbourg, France Real functions Functions of a complex variable Several complex variables and analytic spaces Geometry 26-01, 30F25, 30F45, 30F60, 32G15, 32Q45, 51-01, 51K05, 51K10, 51M09, 51M10, 51F99, 52-01, 52A07, 52A41, 53-01, 53C70, 54-01, 54E35 This book is about metric spaces of nonpositive curvature in the sense of Busemann, that is, metric spaces whose distance function satisfies a convexity condition. It also contains a systematic introduction to metric geometry, as well as a detailed presentation of some facets of convexity theory that are useful in the study of nonpositive curvature. The concepts and the techniques are illustrated by many examples, in particular from hyperbolic geometry, Hilbert geometry and Teichmüller theory. For the second edition, some corrections and a few additions have been made, and the bibliography has been updated. 1 18 2014 978-3-03719-132-3 978-3-03719-632-8 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/132 http://www.ems-ph.org/doi/10.4171/132 6 Benoît Grébert Université de Nantes, France Thomas Kappeler University of Zürich, Switzerland Partial differential equations Ordinary differential equations Dynamical systems and ergodic theory 35Q55, 37K15, 37K10, 34L40, 34L20 Defocusing NLS equation, integrable PDEs, normal forms, action and angle variables, Zakharov–Shabat operators The theme of this monograph is the nonlinear Schrödinger equation. This equation models slowly varying wave envelopes in dispersive media and arises in various physical systems such as water waves, plasma physics, solid state physics and nonlinear optics. More specifically, this book treats the defocusing nonlinear Schrödinger (dNLS) equation on the circle with a dynamical systems viewpoint. By developing the normal form theory it is shown that this equation is an integrable partial differential equation in the strongest possible sense. In particular, all solutions of the dNLS equation on the circle are periodic, quasi-periodic or almost-periodic in time and Hamiltonian perturbations of this equation can be studied near solutions far away from the equilibrium. The book is not only intended for specialists working at the intersection of integrable PDEs and dynamical systems, but also for researchers farther away from these fields as well as for graduate students. It is written in a modular fashion, each of its chapters and appendices can be read independently of each other. 3 15 2014 978-3-03719-131-6 978-3-03719-631-1 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/131 http://www.ems-ph.org/doi/10.4171/131 Robert Marsh University of Leeds, UK Commutative rings and algebras Combinatorics Algebraic geometry Nonassociative rings and algebras 13F60; 05E40, 14M15, 17B22, 17B63, 18E30, 20F55, 51F15, 52B05, 52B11, 57Q15 Associahedron, cluster algebra, cluster complex, Dynkin diagram, finite mutation type, Grassmannian, Laurent phenomenon, reflection group, periodicity, polytope, quiver mutation, root system, Somos sequence, surface Cluster algebras are combinatorially defined commutative algebras which were introduced by S. Fomin and A. Zelevinsky as a tool for studying the dual canonical basis of a quantized enveloping algebra and totally positive matrices. The aim of these notes is to give an introduction to cluster algebras which is accessible to graduate students or researchers interested in learning more about the field, while giving a taste of the wide connections between cluster algebras and other areas of mathematics. The approach taken emphasizes combinatorial and geometric aspects of cluster algebras. Cluster algebras of finite type are classified by the Dynkin diagrams, so a short introduction to reflection groups is given in order to describe this and the corresponding generalized associahedra. A discussion of cluster algebra periodicity, which has a close relationship with discrete integrable systems, is included. The book ends with a description of the cluster algebras of finite mutation type and the cluster structure of the homogeneous coordinate ring of the Grassmannian, both of which have a beautiful description in terms of combinatorial geometry. 1 18 2014 978-3-03719-130-9 978-3-03719-630-4 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/130 http://www.ems-ph.org/doi/10.4171/130 Isabelle Gallagher Université Paris-Diderot France Laure Saint-Raymond Ecole Normale Superieure, France Benjamin Texier Université Paris Diderot - Paris 7, France Partial differential equations 35Q20; 35Q70 Boltzmann equation, particle systems, propagation of chaos, BBGKY hierarchy, hard spheres, clusters, collision trees The question addressed in this monograph is the relationship between the time-reversible Newton dynamics for a system of particles interacting via elastic collisions, and the irreversible Boltzmann dynamics which gives a statistical description of the collision mechanism. Two types of elastic collisions are considered: hard spheres, and compactly supported potentials.. Following the steps suggested by Lanford in 1974, we describe the transition from Newton to Boltzmann by proving a rigorous convergence result in short time, as the number of particles tends to infinity and their size simultaneously goes to zero, in the Boltzmann-Grad scaling. Boltzmann’s kinetic theory rests on the assumption that particle independence is propagated by the dynamics. This assumption is central to the issue of appearance of irreversibility. For finite numbers of particles, correlations are generated by collisions. The convergence proof establishes that for initially independent configurations, independence is statistically recovered in the limit. This book is intended for mathematicians working in the fields of partial differential equations and mathematical physics, and is accessible to graduate students with a background in analysis. 1 18 2014 978-3-03719-129-3 978-3-03719-629-8 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/129 http://www.ems-ph.org/doi/10.4171/129 Second edition Masoud Khalkhali The University of Western Ontario, London, Canada Global analysis, analysis on manifolds 58-02; 58B34 Noncommutative space, noncommutative quotient, groupoid, cyclic cohomology, Connes–Chern character, index formula This text provides an introduction to noncommutative geometry and some of its applications. It can be used either as a textbook for a graduate course or for self-study. It will be useful for graduate students and researchers in mathematics and theoretical physics and all those who are interested in gaining an understanding of the subject. One feature of this book is the wealth of examples and exercises that help the reader to navigate through the subject. While background material is provided in the text and in several appendices, some familiarity with basic notions of functional analysis, algebraic topology, differential geometry and homological algebra at a first year graduate level is helpful. Developed by Alain Connes since the late 1970s, noncommutative geometry has found many applications to long-standing conjectures in topology and geometry and has recently made headways in theoretical physics and number theory. The book starts with a detailed description of some of the most pertinent algebra-geometry correspondences by casting geometric notions in algebraic terms, then proceeds in the second chapter to the idea of a noncommutative space and how it is constructed. The last two chapters deal with homological tools: cyclic cohomology and Connes–Chern characters in K-theory and K-homology, culminating in one commutative diagram expressing the equality of topological and analytic index in a noncommutative setting. Applications to integrality of noncommutative topological invariants are given as well. Two new sections have been added to this second edition: one concerns the Gauss–Bonnet theorem and the definition and computation of the scalar curvature of the curved noncommutative two torus, and the second is a brief introduction to Hopf cyclic cohomology. The bibliography has been extended and some new examples are presented. 12 13 2013 978-3-03719-128-6 978-3-03719-628-1 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/128 http://www.ems-ph.org/doi/10.4171/128 François Labourie Université Paris Sud, Orsay, France Differential geometry Several complex variables and analytic spaces Global analysis, analysis on manifolds 53D30, 53C10; 58D27, 32G15, 58J28 Surface groups, symplectic geometry, Lie groups, character variety The subject of these notes is the character variety of representations of a surface group in a Lie group. We emphasize the various points of view (combinatorial, differential, algebraic) and are interested in the description of its smooth points, symplectic structure, volume and connected components. We also show how a three manifold bounded by the surface leaves a trace in this character variety. These notes were originally designed for students with only elementary knowledge of differential geometry and topology. In the first chapters, we do not insist in the details of the differential geometric constructions and refer to classical textbooks, while in the more advanced chapters proofs occasionally are provided only for special cases where they convey the flavor of the general arguments. These notes could also be used by researchers entering this fast expanding field as motivation for further studies proposed in a concluding paragraph of every chapter. 12 13 2013 978-3-03719-127-9 978-3-03719-627-4 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/127 http://www.ems-ph.org/doi/10.4171/127 Francis Buekenhout Université libre de Bruxelles, Belgium Bernhard Mühlherr Justus Liebig University Giessen, Germany Jean-Pierre Tignol Université catholique de Louvain, Belgium Hendrik Van Maldeghem Ghent University, Belgium General Combinatorics Nonassociative rings and algebras Group theory and generalizations 00B60, 05-XX, 17-XX, 20-XX, 22-XX, 51-XX Jacques Tits was awarded the Wolf Prize in 1993 and the Abel Prize (jointly with John Thompson) in 2008. The impact of his contributions in algebra, group theory and geometry made over a span of more than five decades is incalculable. Many fundamental developments in several fields of mathematics have their origin in ideas of Tits. A number of Tits’ papers mark the starting point of completely new directions of research. Outstanding examples are papers on quadratic forms, on Kac–Moody groups and on what subsequently became known as the Tits-alternative. These volumes contain an almost complete collection of Tits’ mathematical writings. They include, in particular, a number of published and unpublished manuscripts which have not been easily accessible until now. This collection of Tits’ contributions in one place makes the evolution of his mathematical thinking visible. The development of his theory of buildings and BN-pairs and its bearing on the theory of algebraic groups, for example, reveal a fascinating story. Along with Tits’ mathematical writings, these volumes contain biographical data, survey articles on aspects of Tits’ work and comments by the editors on the content of some of his papers. With the publication of these volumes, a major piece of 20th century mathematics is being made available to a wider audience. 11 15 2013 978-3-03719-126-2 978-3-03719-626-7 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/126 http://www.ems-ph.org/doi/10.4171/126 David Benson University of Aberdeen, UK Henning Krause University of Bielefeld, Germany Andrzej Skowroński Nicolaus Copernicus University, Toruń, Poland Associative rings and algebras 16Gxx; 13Dxx, 18Exx, 20Cxx Representation theory, finite dimensional algebras, Lie algebras, cluster algebras, homological algebra, derived categories, triangulated categories, rings and modules This volume presents a collection of articles devoted to representations of algebras and related topics. Dististinguished experts in this field presented their work at the International Conference on Representations of Algebras which took place 2012 in Bielefeld. Many of the expository surveys are included here. Researchers of representation theory will find in this volume interesting and stimulating contributions to the development of the subject. 1 3 2014 978-3-03719-125-5 978-3-03719-625-0 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/125 http://www.ems-ph.org/doi/10.4171/125

Lidia Angeleri Hügel Università degli Studi di Verona, VERONA, ITALY Associative rings and algebras Commutative rings and algebras General Infinite dimensional tilting modules are abundant in representation theory. They occur when studying torsion pairs in module categories, when looking for complements to partial tilting modules, or in connection with the Homological Conjectures. They share many properties with classical tilting modules, but they also give rise to interesting new phenomena as they are intimately related with localization, both at the level of module categories and of derived categories. In these notes, we review the main features of infinite dimensional tilting modules. We discuss the relationship with approximation theory and with localization. Finally, we focus on some classification results and we give a geometric interpretation of tilting. 1 37 1 10.4171/125-1/1 http://www.ems-ph.org/doi/10.4171/125-1/1

David Benson University of Aberdeen, ABERDEEN, UNITED KINGDOM Group theory and generalizations Algebraic geometry General This is a transcription of a series of four lectures given at the Seattle Summer $\pi$-School on Cohomology and Support in Representation Theory and Related Topics, 27–30 July 2012. These were repeated in a mildly compressed form as a series of three lectures at the Workshop on Representations of Algebras in Bielefeld, 8–11 August 2012. Most of the material here also appears in greatly expanded form in a set of notes available online [2], which I eventually hope to publish as a book. This document may serve as an introduction to that longer work. The sections here correspond to the lectures given in Seattle. The first lecture introduces the concept of modules of constant Jordan type, and develops some of their properties, especially the question of what Jordan types can occur. To get much further with this subject it is necessary to work with vector bundles on projective space. So the second lecture gives an algebraic introduction to this subject, and explains how vector bundles arise from modules of constant Jordan type. We give an outline of the realisation theorem, which states that (up to a Frobenius twist if $p$ is odd) every vector bundle comes from a module of constant Jordan type. The third lecture introduces the theory of Chern classes, and uses these to provide restrictions on the behaviour of modules of constant Jordan types. In particular, this explains the appearance of the Frobenius twist for $p$ odd in the realisation theorem. Finally in the fourth lecture we give a short algebraic proof of the Hirzebruch–Riemann–Roch theorem for projective space. Schwarzenberger’s conditions on Chern roots are usually regarded as a consequence of Hirzebruch–Riemann–Roch, but in our development the logical order is reversed. Hirzebruch–Riemann–Roch is then used to provide further restrictions on possible Jordan types of modules. 39 63 1 10.4171/125-1/2 http://www.ems-ph.org/doi/10.4171/125-1/2

Klaus Bongartz Bergische Universität Wuppertal, WUPPERTAL, GERMANY Associative rings and algebras General We give a survey on the theory of representation-finite and certain minimal representation-infinite algebras. The main goals are the existence of multiplicative bases and of coverings with good properties. Both are attained using ray-categories. As applications we include a proof of a sharper version of the second Brauer–Thrall conjecture and of the fact that there are no gaps in the lengths of the indecomposables. 65 101 1 10.4171/125-1/3 http://www.ems-ph.org/doi/10.4171/125-1/3

Jonathan Brundan University of Oregon, EUGENE, UNITED STATES Associative rings and algebras Nonassociative rings and algebras General This is a brief introduction to the quiver Hecke algebras of Khovanov, Lauda and Rouquier, emphasizing their application to the categorification of quantum groups. The text is based on lectures given by the author at the ICRA workshop in Bielefeld in August, 2012. 103 133 1 10.4171/125-1/4 http://www.ems-ph.org/doi/10.4171/125-1/4

Thomas Brüstle Bishop's University, SHERBROOKE, CANADA Dong Yang Nanjing University, NANJING, CHINA General The exchange graph of a cluster algebra encodes the combinatorics of mutations of clusters. Through the recent “categorifications” of cluster algebras using representation theory one obtains a whole variety of exchange graphs associated with objects such as a finite-dimensional algebra or a differential graded algebra concentrated in non-positive degrees. These constructions often come from variations of the concept of tilting, the vertices of the exchange graph being torsion pairs, t-structures, silting objects, support $\tau$-tilting modules and so on. All these exchange graphs stemming from representation theory have the additional feature that they are the Hasse quiver of a partial order which is naturally defined for the objects. In this sense, the exchange graphs studied in this article can be considered as a generalization or as a completion of the poset of tilting modules which has been studied by Happel and Unger. The goal of this article is to axiomatize the thus obtained structure of an ordered exchange graph, to present the various constructions of ordered exchange graphs and to relate them among each other. 135 193 1 10.4171/125-1/5 http://www.ems-ph.org/doi/10.4171/125-1/5

Sergey Mozgovoy Trinity College, DUBLIN, IRELAND Commutative rings and algebras Associative rings and algebras Category theory; homological algebra General These notes represent three lectures given at the International Conference on Representations of Algebras (ICRA) held at Bielefeld in August 2012. The goal of the lectures was to introduce refined Donaldson–Thomas invariants for the categories of modules over Jacobian algebras associated to quivers with potentials. We define refined Donaldson–Thomas invariants and compute them in some simple cases and then discuss their basic properties, including integrality, positivity, and wall-crossing phenomena. 195 210 1 10.4171/125-1/6 http://www.ems-ph.org/doi/10.4171/125-1/6

Hiraku Nakajima Kyoto University, KYOTO, JAPAN Commutative rings and algebras Nonassociative rings and algebras Partial differential equations General We propose an approach to Geiss–Leclerc–Schroer’s conjecture on the cluster algebra structure on the coordinate ring of a unipotent subgroup and the dual canonical base. It is based on singular supports of perverse sheaves on the space of representations of a quiver, which give the canonical base. 211 230 1 10.4171/125-1/7 http://www.ems-ph.org/doi/10.4171/125-1/7

Julia Pevtsova University of Washington, SEATTLE, UNITED STATES Group theory and generalizations Associative rings and algebras General This is a survey article covering developments in representation theory of finite group schemes over the last fifteen years. We start with the finite generation of cohomology of a finite group scheme and proceed to discuss various consequences and theories that ultimately grew out of that result. This includes the theory of one-parameter subgroups and rank varieties for infinitesimal group schemes; the $\pi$-points and $\Pi$-support spaces for finite group schemes, modules of constant rank and constant Jordan type, and construction of bundles on varieties closely related to $\operatorname{Proj}\operatorname{H}^{\bullet}(G,k)$ for an infinitesimal group scheme $G$. The material is mostly complementary to the article of D. Benson on elementary abelian $p$-groups in the same volume; we concentrate on the aspects of the theory which either hold generally for any finite group scheme or are specific to finite group schemes which are not finite groups. In the last section we discuss varieties of elementary subalgebras of modular Lie algebras, generalizations of modules of constant Jordan type, and new constructions of bundles on projective varieties associated to a modular Lie algebra. 231 261 1 10.4171/125-1/8 http://www.ems-ph.org/doi/10.4171/125-1/8

Mike Prest University of Manchester, MANCHESTER, UNITED KINGDOM Associative rings and algebras Mathematical logic and foundations General Existence of superdecomposable pure-injective modules reflects complexity in the category of finite-dimensional representations. We describe the relation in terms of pointed modules. We present methods for producing superdecomposable pure-injectives and give some details of recent work of Harland doing this in the context of tubular algebras. 263 296 1 10.4171/125-1/9 http://www.ems-ph.org/doi/10.4171/125-1/9

Jan Šťovíček Charles University, PRAHA 8, CZECH REPUBLIC Algebraic topology Category theory; homological algebra General Our aim is to give a fairly complete account on the construction of compatible model structures on exact categories and symmetric monoidal exact categories, in some cases generalizing previously known results. We describe the close connection of this theory to approximation theory and cotorsion pairs. We also discuss the motivating applications with the emphasis on constructing monoidal model structures for the derived category of quasi-coherent sheaves of modules over a scheme. 297 367 1 10.4171/125-1/10 http://www.ems-ph.org/doi/10.4171/125-1/10

Kaspar Nipp ETH Zürich, Switzerland Daniel Stoffer ETH Zürich, Switzerland Dynamical systems and ergodic theory Ordinary differential equations Numerical analysis 37-02; 37Cxx, 37Dxx, 34Cxx, 34Dxx, 65Lxx, 65P10 Discrete and continuous dynamical systems, invariant manifolds, foliation, singular perturbations, geometric numerical integration, differential algebraic equations In this book dynamical systems are investigated from a geometric viewpoint. Admitting an invariant manifold is a strong geometric property of a dynamical system. This text presents rigorous results on invariant manifolds and gives examples of possible applications. In the first part discrete dynamical systems in Banach spaces are considered. Results on the existence and smoothness of attractive and repulsive invariant manifolds are derived. In addition, perturbations and approximations of the manifolds and the foliation of the adjacent space are treated. In the second part analogous results for continuous dynamical systems in finite dimensions are established. In the third part the theory developed is applied to problems in numerical analysis and to singularly perturbed systems of ordinary differential equations. The mathematical approach is based on the so-called graph transform, already used by Hadamard in 1901. The aim is to establish invariant manifold results in a simple setting providing quantitative estimates. The book is targeted at researchers in the field of dynamical systems interested in precise theorems easy to apply. The application part might also serve as an underlying text for a student seminar in mathematics. 8 7 2013 978-3-03719-124-8 978-3-03719-624-3 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/124 http://www.ems-ph.org/doi/10.4171/124 21 Hans Triebel University of Jena, Germany Functional analysis Partial differential equations Fourier analysis 46-02, 46E35, 42B35, 42C40, 35K05, 35Q30, 76D03, 76D05 Function spaces, Morrey–Campanato spaces, Gagliardo–Nirenberg inequalities, heat equations, Navier–Stokes equations In this book a new approach is presented to exhibit relations between Sobolev spaces, Besov spaces, and Hölder–Zygmund spaces on the one hand and Morrey–Campanato spaces on the other. Morrey–Campanato spaces extend the notion of functions of bounded mean oscillation. These spaces play an important role in the theory of linear and nonlinear PDEs. Chapters 1–3 deal with local smoothness spaces in Euclidean n-space based on the Morrey–Campanato refinement of the Lebesgue spaces. The presented approach relies on wavelet decompositions. This is applied in Chapter 4 to Gagliardo–Nirenberg inequalities. Chapter 5 deals with linear and nonlinear heat equations in global and local function spaces. The obtained assertions about function spaces and nonlinear heat equations are used in Chapter 6 to study Navier–Stokes equations. The book is addressed to graduate students and mathematicians having a working knowledge of basic elements of (global) function spaces, and who are interested in applications to nonlinear PDEs with heat and Navier–Stokes equations as prototypes. 5 29 2013 978-3-03719-123-1 978-3-03719-623-6 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/123 http://www.ems-ph.org/doi/10.4171/123 20 Bogdan Bojarski IM PAN, Warsaw, Poland Vladimir Gutlyanskii National Academy of Science of Ukraine, Donetsk, Ukraine Olli Martio Finnish Academy of Science and Letters, Helsinki, Finland Vladimir Ryazanov National Academy of Science of Ukraine, Donetsk, Ukraine Functions of a complex variable Partial differential equations 30C65, 30C75, 35J46, 35J50, 35J56, 35J70, 35Q35, 35Q60, 37F30, 37F40, 37F45, 57R99 Quasiconformal mappings, bi-Lipschitz mappings, Beltrami equations, local and boundary behavior, infinitesimal space, convergence and compactness theory, asymptotic linearity, rotation problems, conformal differentiability This book is intended for researchers interested in new aspects of local behavior of plane mappings and their applications. The presentation is self-contained, but the reader is assumed to know basic complex and real analysis. The study of the local and boundary behavior of quasiconformal and bi-Lipschitz mappings in the plane forms the core of the book. The concept of the infinitesimal space is used to investigate the behavior of a mapping at points without differentiability. This concept, based on compactness properties, is applied to regularity problems of quasiconformal mappings and quasiconformal curves, boundary behavior, weak and asymptotic conformality, local winding properties, variation of quasiconformal mappings, and criteria of univalence. Quasiconformal and bi-Lipschitz mappings are instrumental for understanding elasticity, control theory and tomography and the book also offers a new look at the classical areas such as the boundary regularity of a conformal map. Complicated local behavior is illustrated by many examples. The text offers a detailed development of the background for graduate students and researchers. Starting with the classical methods to study quasiconformal mappings, this treatment advances to the concept of the infinitesimal space and then relates it to other regularity properties of mappings in Part II. The new unexpected connections between quasiconformal and bi-Lipschitz mappings are treated in Part III. There is an extensive bibliography. 5 29 2013 978-3-03719-122-4 978-3-03719-622-9 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/122 http://www.ems-ph.org/doi/10.4171/122 19 Wolfgang Reiter University of Vienna, Austria Jakob Yngvason University of Vienna, Austria History and biography Quantum theory 01-02, 81-02, 81-03, 81P05, 81P15 Erwin Schrödinger (1887–1961) was an Austrian physicist famous for the equation named after him and which earned him the Nobel Prize in 1933. This book contains lectures presented at the international symposium Erwin Schrödinger – 50 Years After held at the Erwin Schrödinger International Institute for Mathematical Physics in January 2011 to commemorate the 50th anniversary of Schrödinger’s death. The text covers a broad spectrum of topics ranging from personal reminiscences to foundational questions of quantum mechanics and historical accounts of Schrödinger’s work. Besides the lectures presented at the symposium the volume also contains articles specially written for this occasion. The contributions give an overview of Schrödinger’s legacy to the sciences from the standpoint of some of present day’s leading scholars in the field. The book addresses students and researchers in mathematics, physics and the history of science. 4 11 2013 978-3-03719-121-7 978-3-03719-621-2 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/121 http://www.ems-ph.org/doi/10.4171/121

Walter Thirring , WIEN, AUSTRIA History and biography Quantum theory General In this article the author recollects some memories of his encounters with Schrödinger, beginning in 1937 when he first met him in Vienna up to the final years after Schrödinger’s return to Vienna. 1 7 1 10.4171/121-1/1 http://www.ems-ph.org/doi/10.4171/121-1/1

Jürgen Renn , BERLIN, GERMANY History and biography Quantum theory General In the context of a new analysis of the notebooks of Erwin Schrödinger, the paper deals with the question of the relation between Schrödinger's creation of wave mechanics and the contemporary efforts by Werner Heisenberg and his colleagues to establish a new quantum mechanics. How can one explain, from a broader historical and epistemological perspective, the astonishing simultaneity and complementarity of these discoveries? The paper argues that neither the physical problems with which both approaches deal nor what ultimately turned out to be their common mathematical ground are sufficient to explain their complementarity. Instead, their closeness is explained by analyzing their common roots in classical mechanics and its transformation in the light of the most fundamental new quantum law, the relation between energy and frequency found by Planck. It is shown, in particular, that for both approaches a bridge between quantum and classical aspects involving this relation was crucial. In the case of Heisenberg, this bridge was given by Bohr’s correspondence principle. In the case of Schrödinger it was constituted by Hamilton’s optical-mechanical analogy. 9 36 1 10.4171/121-1/2 http://www.ems-ph.org/doi/10.4171/121-1/2

Jürg Fröhlich ETH Zürich, ZÜRICH, SWITZERLAND Baptiste Schubnel ETH Zürich, ZÜRICH, SWITZERLAND Quantum theory General This paper reviews some of our understanding of general quantum mechanics. It starts with the exposition of an abstract algebraic formalism useful to formulate classical and quantum-mechanical models of physical systems. It then highlights the essential differences between classical models (commutative algebra) and quantum-mechanical models (non-commutative algebra) of physical systems. It is explained in which sense classical models are “realistic” and deterministic, while quantum-mechanical models are intrinsically probabilistic – in spite of the fact that the Heisenberg time-evolution of operators representing physical quantities is “deterministic”. The quantum theory of time-ordered sequences of measurements is developed in some detail, and the crucial role of “decoherence” in the emergence of facts – or “(almost) consistent histories” – is explained. Some technical matters (Bell inequalities, quantum marginal problem) are discussed in appendices. 37 84 1 10.4171/121-1/3 http://www.ems-ph.org/doi/10.4171/121-1/3

Anthony Leggett University of Illinois at Urbana-Champaign, URBANA, UNITED STATES Quantum theory General In a famous 1935 paper, Erwin Schroedinger pointed out the apparently bizarre consequences which flow from the assumption that the formalism of quantum mechanics can be extrapolated from the world of electrons and atoms to the level of our own direct consciousness (the “Cat” paradox). In this paper I first discuss some alleged resolutions of the paradox, then point out that the premise is not self-evident, and define a class of alternative (“macrorealistic”) scenarios. I then review the extent to which the experimental predictions of quantum mechanics are currently tested in the direction of the everyday world, and explain a crucial test, yet to be performed, of these predictions vis-a-vis those of macrorealism. 87 108 1 10.4171/121-1/4 http://www.ems-ph.org/doi/10.4171/121-1/4

Markus Müller Universität Innsbruck, INNSBRUCK, AUSTRIA Peter Zoller Universität Innsbruck, INNSBRUCK, AUSTRIA Quantum theory General We summarize some of the recent developments in building a quantum simulator with quantum optical systems, in particular laser cooled trapped ions. Our discussion focuses on digital and open system quantum simulation as realized recently for the first time. 109 121 1 10.4171/121-1/5 http://www.ems-ph.org/doi/10.4171/121-1/5

Rainer Kaltenbaek Universität Wien, WIEN, AUSTRIA Markus Aspelmeyer Universität Wien, WIEN, AUSTRIA Quantum theory General Quantum optomechanics exploits radiation pressure effects inside optical cavities. It can be used to generate quantum states of the center-of-mass motion of massive mechanical objects, thereby opening up a new parameter regime for macroscopic quantum experiments. The challenging experimental conditions to maintain and observe quantum coherence for increasingly large objects may require a space environment rather than an earth-bound laboratory. We introduce a possible space experiment to study the wave-packet expansion of massive objects. This forms the basis for Schrödinger cat states of unprecedented size and mass. 123 132 1 10.4171/121-1/6 http://www.ems-ph.org/doi/10.4171/121-1/6

Helge Kragh University of Aarhus, AARHUS, DENMARK Quantum theory General Parts of Erwin Schrödinger’s career and views of science can be illuminated through his interaction with another pioneer of quantum theory, Niels Bohr. Both physicists endeavoured to understand nature on a basic level, where physics cannot be easily separated from natural philosophy in a broad sense of the term. In spite of having much in common, they fundamentally disagreed about the meaning and aim of quantum mechanics. The essay describes the Schrödinger–Bohr dialogue as it unfolded from their first contact in 1921 to their last exchange of letters in 1952. 135 151 1 10.4171/121-1/7 http://www.ems-ph.org/doi/10.4171/121-1/7

Anne Kox University of Amsterdam, AMSTERDAM, NETHERLANDS Quantum theory General In 1923 the Austrian Erwin Schrödinger, professor of physics in Zurich, and his Dutch colleague Hendrik Lorentz, grand old man of classical physics, engaged in a fascinating correspondence on the foundations of quantum physics. In this lecture I will go back to that year, give a sketch of the personalities of Lorentz and Schrödinger and of their respective careers, and explain the importance for the history of physics of their remarkable exchange of ideas. 153 162 1 10.4171/121-1/8 http://www.ems-ph.org/doi/10.4171/121-1/8

Olivier Darrigol Université Paris 7 Denis Diderot, PARIS, FRANCE Quantum theory General In 1923 Louis de Broglie arrived at a shocking notion of matter waves through reasoning so forceful that Albert Einstein and a few quantum experts took it seriously. The purpose of this paper is to analyze de Broglie's discovery and to show how some of his heuristics prevented him to introduce the equation that now bears Schrödinger’s name. 165 174 1 10.4171/121-1/9 http://www.ems-ph.org/doi/10.4171/121-1/9

Rafał Latała University of Warsaw, Poland Andrzej Ruciński A. Mickiewicz University, Poznań, Poland Paweł Strzelecki University of Warsaw, Poland Jacek Świątkowski University of Wrocław, Poland Dariusz Wrzosek University of Warsaw, Poland Piotr Zakrzewski University of Warsaw, Poland General 00Bxx The European Congress of Mathematics, held every four years, has become a well-established major international mathematical event. Following those in Paris (1992), Budapest (1996), Barcelona (2000), Stockholm (2004) and Amsterdam (2008), the Sixth European Congress of Mathematics (6ECM) took place in Kraków, Poland, July 2–7, 2012, with about 1000 participants from all over the world. Ten plenary, thirty-three invited lectures and three special lectures formed the core of the program. As at all the previous EMS congresses, ten outstanding young mathematicians received the EMS prizes in recognition of their research achievements. In addition, two more prizes were awarded: the Felix Klein Prize for a remarkable solution of an industrial problem, and – for the first time – the Otto Neugebauer Prize for a highly original and influential piece of work in the history of mathematics. The program was complemented by twenty-four minisymposia with nearly 100 talks, spread over all areas of mathematics. Six panel discussions were organized, covering a variety of issues ranging from the financing of mathematical research to gender imbalance in mathematics. These proceedings present extended versions of most of the invited talks which were delivered during the congress, providing a permanent record of the best what mathematics offers today. 1 3 2014 978-3-03719-120-0 978-3-03719-620-5 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/120 http://www.ems-ph.org/doi/10.4171/120

Adrian Constantin King's College London, LONDON, UNITED KINGDOM Mathematical logic and foundations Combinatorics Functional analysis General We discuss some recent mathematical investigations that o er insight and open up promising perspectives on the fundamental aspect of uid mechanics concerned with the motion beneath a surface water wave. 1 12 1 10.4171/120-1/1 http://www.ems-ph.org/doi/10.4171/120-1/1

Camillo De Lellis Universität Zürich, ZÜRICH, SWITZERLAND László Székelyhidi Jr. Universität Leipzig, LEIPZIG, GERMANY Partial differential equations Ordinary differential equations Differential geometry Fluid mechanics It is known since the pioneering works of Sche er and Shnirelman that there are nontrivial distributional solutions to the Euler equations which are compactly supported in space and time. Obviously these solutions do not respect the classical conservation law for the total kinetic energy and they are therefore very irregular. In recent joint works we have proved the existence of continuous and even Holder continuous solutions which dissipate the kinetic energy. Our theorem might be regarded as a rst step towards a conjecture of Lars Onsager, which in 1949 asserted the existence of dissipative Hölder solutions for any Hölder exponent smaller than 1/3. 13 29 1 10.4171/120-1/2 http://www.ems-ph.org/doi/10.4171/120-1/2

Herbert Edelsbrunner Institute of Science and Technology Austria, KLOSTERNEUBURG, AUSTRIA Dmitriy Morozov Lawrence National Laboratory, BERKELEY, UNITED STATES Algebraic topology Computer science General Persistent homology is a recent grandchild of homology that has found use in science and engineering as well as in mathematics. This paper surveys the method as well as the applications, neglecting completeness in favor of highlighting ideas and directions. 31 50 1 10.4171/120-1/3 http://www.ems-ph.org/doi/10.4171/120-1/3

Misha Gromov Le Bois-Marie, BURES-SUR-YVETTE, FRANCE Measure and integration Dynamical systems and ergodic theory Statistical mechanics, structure of matter General Mathematics is about \interesting structures". What make a structure interesting is an abundance of interesting problems; we study a structure by solving these problems. The worlds of science, as well as of mathematics itself, is abundant with gems (germs?) of simple beautiful ideas. When and how may such an idea direct you toward beautiful mathematics? I try to present in this talk a few suggestive examples. 51 78 1 10.4171/120-1/4 http://www.ems-ph.org/doi/10.4171/120-1/4

Christopher Hacon University of Utah, SALT LAKE CITY, UNITED STATES General We discuss recent results in the Minimal Model Program that have lead to several breakthroughs in the classi cation of algebraic varieties 79 90 1 10.4171/120-1/5 http://www.ems-ph.org/doi/10.4171/120-1/5

Alexander Braverman Brown University, PROVIDENCE, UNITED STATES David Kazhdan The Hebrew University of Jerusalem, JERUSALEM, ISRAEL General 91 117 1 10.4171/120-1/6 http://www.ems-ph.org/doi/10.4171/120-1/6

Sylvia Serfaty Université Pierre et Marie Curie (Paris VI), PARIS CEDEX 05, FRANCE Partial differential equations Convex and discrete geometry Statistical mechanics, structure of matter General We consider three diff erent models coming from physics and involving the Coulomb interaction of points in the plane. The first is the classical Coulomb gas in an external potential, the second is the Ginzburg–Landau model of superconductivity, the third is the Ohta–Kawasaki model of polymers. In superconductivity, one observes in certain regimes the emergence of densely packed point vortices forming perfect triangular lattices named "Abrikosov lattices" in physics. We show how these Abrikosov lattices are expected to emerge in the three systems, via the minimization of a "Coulombian renormalized energy" that we defi ned with Etienne Sandier. We also present applications to the statistical mechanics of Coulomb gases, which are related to some random matrix models, and show how the previous results lead to expecting crystallisation in the low temperature limit. 119 135 1 10.4171/120-1/7 http://www.ems-ph.org/doi/10.4171/120-1/7

Saharon Shelah The Hebrew University of Jerusalem, JERUSALEM, ISRAEL Mathematical logic and foundations General Model theory deals with general classes of structures = models, so the class of rings, the class of algebraically closed elds and other such speci c classes. Counting of so called complete types over models in the class has an important role in the development of model theory in general and so called stability theory in particular. In particular understanding the stable classes: those with relatively few complete types over structures from the class, has been central in model theory and its applications. Lately, we have a parallel recounting theorem under reasonable restrictions, counting types up to conjugacy. The classes for which we have few complete types up to conjugacy are proved to be so called dependent. This is a strong indication that there is much to be said on the classes whose models in the relevant cases has few complete types over them, so called dependent classes. 137 157 1 10.4171/120-1/8 http://www.ems-ph.org/doi/10.4171/120-1/8

Michel Talagrand Université Pierre et Marie Curie-Paris 6, PARIS CEDEX 6, FRANCE Probability theory and stochastic processes Algebraic geometry General Kolmogorov introduced a very powerful method called chaining in order to provide bounds for the supremum of a stochastic process. Even in the case of Gaussian processes, this method has reached its seemingly nal form only about 10 years ago. When used optimally in the case of Gaussian processes, it then yields optimal bounds. We explain why we believe that this is also true in many more cases. 159 178 1 10.4171/120-1/9 http://www.ems-ph.org/doi/10.4171/120-1/9

Anton Alekseev Université de Genève, GENÈVE, SWITZERLAND Nonassociative rings and algebras General 179 193 1 10.4171/120-1/10 http://www.ems-ph.org/doi/10.4171/120-1/10

Jean Bertoin Universität Zürich, ZÜRICH, SWITZERLAND Statistical mechanics, structure of matter Probability theory and stochastic processes General Smoluchowski's coagulation equations can be used as elementary mathematical models for the formation of polymers. We review here some recent contributions on a variation of this model in which the number of aggregations for each atom is a priori limited. Macroscopic results in the deterministic setting can be explained at the microscopic level by considering a version of stochastic coalescence with limited aggregations, which can be related to the so-called random con guration model of random graph theory 195 209 1 10.4171/120-1/11 http://www.ems-ph.org/doi/10.4171/120-1/11

Serge Cantat Ecole Normale Supérieure, PARIS CEDEX 05, FRANCE Algebraic geometry Group theory and generalizations Several complex variables and analytic spaces Dynamical systems and ergodic theory We survey a few results concerning the Cremona group in two variables. 211 225 1 10.4171/120-1/12 http://www.ems-ph.org/doi/10.4171/120-1/12

Vicent Caselles Universitat Pompeu-Fabra, BARCELONA, SPAIN Computer science Partial differential equations Numerical analysis General In this paper we study some variational models for exemplar-based image inpainting, also referred to as nonlocal methods. Nonlocal methods for denoising and inpainting have gained considerable attention due to their good performance on textured images, a known weakness of classical local methods which are performant in recovering the geometric structure of the image. We rst review a general variational framework for the problem of nonlocal inpainting that exploits the self-similarity of natural images to copy information in a consistent way from the known parts of the image. We single out some particular methods and we review the main properties of the corresponding energies and their minima. We discuss the basic algorithms to minimize the energies and we display some numerical experiments illustrating the main properties of the proposed models. 227 242 1 10.4171/120-1/13 http://www.ems-ph.org/doi/10.4171/120-1/13

Alessandra Celletti Università di Roma 'Tor Vergata', ROMA, ITALY Mechanics of particles and systems General We present a review of KAM theory for conservative and dissipative systems with applications to several models. The theory can be developed under general assumptions, namely a Diophantine condition on the frequency and a non-degeneracy requirement. In particular, we present a KAM theory for conformally symplectic systems, which provides also a very e cient numerical technique to determine the breakdown value of invariant attractors. Applications to model problems, most notably in Celestial Mechanics, are considered. Explicit estimates have been developed in the conservative case, where a computer-assisted implementation of KAM theory gives results in good agreement with the experimental values. Some KAM results for the dissipative standard map and for the dissipative spin-orbit problem in Celestial Mechanics are also discussed. 243 257 1 10.4171/120-1/14 http://www.ems-ph.org/doi/10.4171/120-1/14

Pierre Colmez Université Pierre et Marie Curie, PARIS, FRANCE Number theory General We give a loose introduction to the circle of ideas around the $p$-adic local Langlands correspondence for GL$_2$(Q$_p$). 259 284 1 10.4171/120-1/15 http://www.ems-ph.org/doi/10.4171/120-1/15

Tom Coates Imperial College London, LONDON, UNITED KINGDOM Alessio Corti Imperial College London, LONDON, UNITED KINGDOM Sergey Galkin The University of Tokyo, CHIBA, JAPAN Vasily Golyshev Russian Academy of Sciences, MOSCOW, RUSSIAN FEDERATION Alexander Kasprzyk Imperial College London, LONDON, UNITED KINGDOM Algebraic geometry General 285 300 1 10.4171/120-1/16 http://www.ems-ph.org/doi/10.4171/120-1/16

Hélène Esnault FU Berlin, BERLIN, GERMANY Algebraic geometry General We discuss analogies between the fundamental groups of flat bundles in characteristic 0 and $p > 0$. 301 313 1 10.4171/120-1/17 http://www.ems-ph.org/doi/10.4171/120-1/17

Alexander Gaifullin Steklov Mathematical Institute, MOSCOW, RUSSIAN FEDERATION Manifolds and cell complexes Several complex variables and analytic spaces Convex and discrete geometry Differential geometry In 1940s Steenrod asked if every homology class $z \in H_n(X; \mathbb Z)$ of every topological space $X$ can be realised by an image of the fundamental class of an oriented closed smooth manifold. Thom found a non-realisable 7-dimensional class and proved that for every $n$, there is a positive integer $k(n)$ such that the class $k(n)z$ is always realisable. The proof was by methods of algebraic topology and gave no information on the topology of the manifold which realises the homology class. We give a purely combinatorial construction of a manifold that realises a multiple of a given homology class. For every $n$, this construction yields a manifold $M^n_0$ with the following universality property: For any $X$ and $z \in H_n(X; \mathbb Z)$, a multiple of $z$ can be realised by an image of a (non-rami ed) finite-sheeted covering of $M^n_0$ . Manifolds satisfying this property are called URC-manifolds. The manifold $M^n_0$ is a so-called small cover of the permutahedron, i. e., a manifold glued in a special way out of $2n$ permutahedra. (The permutahedron is a special convex polytope with $(n+1)$! vertices.) Among small covers over other simple polytopes, we find a broad class of examples of URC-manifolds. In particular, in dimension 4, we fi nd a hyperbolic URC-manifold. Thus we prove a conjecture of Kotschick and Loh claiming that a multiple of every homology class can be realised by an image of a hyperbolic manifold. We also obtain some new results on the relationship of URC-manifolds with theory of simplicial volume. 315 330 1 10.4171/120-1/18 http://www.ems-ph.org/doi/10.4171/120-1/18

Isabelle Gallagher Université Paris 7, PARIS CEDEX 05, FRANCE Partial differential equations Fourier analysis General We review some recent results concerning the Cauchy problem for the three dimensional, homogeneous, incompressible Navier{Stokes equations. We show in particular that the set of initial data generating a global smooth solution is open and connected, and we discuss under which conditions it is open for weak topology. 331 345 1 10.4171/120-1/19 http://www.ems-ph.org/doi/10.4171/120-1/19

Olle Häggström Chalmers University of Technology, GÖTEBORG, SWEDEN Statistics General Science can be described as a systematic attempt to extract reliable information about the world. The cognitive capacities of homo sapiens come with various biases, such as our tendencies (a) to detect patterns in what is actually just noise, and (b) to be overly con dent in our conclusions. Thus, the scienti c method needs to involve safeguards against drawing incorrect conclusions due to such biases. A crucial part of the necessary toolbox is the theory of statistical inference. There exists a large and well-developed (but of course incomplete) body of such theory, which, however, researchers across practically all of the empirical sciences do not have su cient access to. The lack of statistical knowledge therefore forms a serious bottleneck in the quest for reliable scienti c advances. As has been observed by several authors in recent years, statistical malpractice is widespread across a broad spectrum of disciplines, including (but not limited to) medicine, cognitive sciences, Earth sciences and social sciences. Here I will rst try to describe the overall situation and provide some concrete examples, and then move on to discuss the more di cult issue of what can and needs to be done. 347 360 1 10.4171/120-1/20 http://www.ems-ph.org/doi/10.4171/120-1/20

Arieh Iserles University of Cambridge, CAMBRIDGE, UNITED KINGDOM Numerical analysis Nonassociative rings and algebras General The discretisation of a linear Schrodinger equation is di cult due to the presence of a small parameter which induces high oscillations. A standard approach consists of a spectral semidiscretisation, followed by an exponential splitting. This, however, is sub-optimal, because, unless we are willing to compute an exceedingly large number of matrix exponentials per step, the very high precision of the space discretisation is marred by low order of the time solver. The underlying reason is that no commutators are allowed in the arguments of the exponentials, to avoid costly computations. It turns out, however, that once we split rst, discretise later, commutators are benign. Thus, we consider formally objects in the free Lie algebra spanned by the Laplacian and by a multiplication with the interaction potential. We demonstrate that it is embedded in a larger Lie algebra where all commutators can be resolved easily. This is the starting point for our symmetric Zassenhaus splitting, expressing the solution operator as a product of simple exponentials of asymptotically decreasing arguments. Once the underlying di erential operator is discretised and the exponentials approximated by suitable means, the outcome is an exceedingly accurate scheme bearing the price tag of $\mathcal O(M$ log$M$) operations per step, where $M$ is the number of degrees of freedom in the discretisation. This is joint work with Philipp Bader (Valencia), Karolina Kropielnicka (Cambridge and Gda nsk) and Pranav Singh (Cambridge). 361 374 1 10.4171/120-1/21 http://www.ems-ph.org/doi/10.4171/120-1/21

Alexander Kechris California Institute of Technology, PASADENA, UNITED STATES Mathematical logic and foundations Topological groups, Lie groups Dynamical systems and ergodic theory General topology A topological group $G$ is Polish if its topology admits a compatible separable complete metric. Such a group is non-archimedean if it has a basis at the identity that consists of open subgroups. This class of Polish groups includes the pro finite groups and ($\mathbb Q_p; +$) but our main interest here will be on non-locally compact groups. In recent years there has been considerable activity in the study of the dynamics of Polish non-archimedean groups and this has led to interesting interactions between logic, fi nite combinatorics, group theory, topological dynamics, ergodic theory and representation theory. In this paper I will give a survey of some of the main directions in this area of research. 375 397 1 10.4171/120-1/22 http://www.ems-ph.org/doi/10.4171/120-1/22

Bernhard Keller Université Paris Diderot - Paris 7, PARIS CEDEX 13, FRANCE Commutative rings and algebras Nonassociative rings and algebras General Cluster algebras were invented by Sergey Fomin and Andrei Zelevinsky at the beginning of the year 2000. Their motivations came from Lie theory and more precisely from the study of the so-called canonical bases in quantum groups and that of total positivity in algebraic groups. Since then, cluster algebras have been linked to many other subjects ranging from higher Teichmuller theory through discrete dynamical systems to combinatorics, algebraic geometry and representation theory. According to Fomin–Zelevinsky's philosophy, each cluster algebra should admit a `canonical' basis, which should contain the cluster monomials. This led them to formulate, about ten years ago, the conjecture on the linear independence of the cluster monomials. In these notes, we give a concise introduction to cluster algebras and sketch the ingredients of a proof of the conjecture. The proof is valid for all cluster algebras associated with quivers and was obtained in recent joint work with G. Cerulli Irelli, D. Labardini-Fragoso and P.-G. Plamondon. 399 413 1 10.4171/120-1/23 http://www.ems-ph.org/doi/10.4171/120-1/23

Sławomir Kołodziej Jagellonian University, KRAKOW, POLAND Several complex variables and analytic spaces General Canonical Kahler metrics, such as Ricci-flat or Käahler–Einstein, are obtained via solving the complex Monge-Ampère equation. The famous Calabi–Yau theorem asserts the existence and regularity of solutions to this equation on compact Käahler manifolds for smooth data. In this note we shall present methods, based on pluripotential theory, which yield the results on the existence and stability of the weak solutions of the Monge–Amp ère equation for possibly degenerate, non-smooth right hand side. Those weak solutions have also interesting applications in geometry. They lead to canonical metrics with singularities, which may occur as the limits of the Kähler–Ricci flow or the limits of families of Calabi–Yau metrics when the Kähler class hits the boundary of the Kähler cone. 415 428 1 10.4171/120-1/24 http://www.ems-ph.org/doi/10.4171/120-1/24

Gady Kozma Weizmann Institute of Science, REHOVOT, ISRAEL Probability theory and stochastic processes General We show that linearly reinforced random walk has a recurrent phase in every graph. The proof does not use the magic formula. 429 443 1 10.4171/120-1/25 http://www.ems-ph.org/doi/10.4171/120-1/25

Frank Merle Université de Cergy-Pontoise, CERGY-PONTOISE CEDEX, FRANCE Partial differential equations General In a joint work with H. Zaag, we consider the semilinear wave equation with focusing power nonlinearity in one space dimension. Blow-up solutions are known to exist, and the solution can be de ned on some domain of de nition under the blow-up curve {$t = T(x)$}. Considering an arbitrary blow-up solution, our goal is to describe its behavior near the blow-up curve, and the geometry of the blow-up curve itself. Such properties are linked to the notion of non-characteristic points on the curve. First, we nd criteria on initial data to ensure the existence or the non-existence of characteristic points. Then, we prove the regularity of the blow-up curve away from characteristic points, and show a surprising isolatedness property for characteristic points, together with the classi cation of the behavior of the solution near them. In order to do this, we introduce for this problem a notion of critical space. Furthermore, we link the geometrical properties of the blow-up curve with the problem of decomposing a general solution into a sum of solitons. 445 457 1 10.4171/120-1/26 http://www.ems-ph.org/doi/10.4171/120-1/26

Andrey Mironov Siberian Branch of the Russian Academy of Sciences, NOVOSIBIRSK, RUSSIAN FEDERATION Dynamical systems and ergodic theory General The theory of commuting ordinary di erential operators was developed from the beginning of the XX century on. The problem of nding commuting di fferential operators is solved for the case of operators of rank one. For operators of rank greater than one it is still open. In this paper we discuss some results related to operators of rank greater than one. 459 473 1 10.4171/120-1/27 http://www.ems-ph.org/doi/10.4171/120-1/27

David Nualart University of Kansas, LAWRENCE, UNITED STATES Probability theory and stochastic processes General In this note we survey some recent results on the stochastic calculus with respect to a fractional Brownian motion using Riemann sums approximations. 475 488 1 10.4171/120-1/28 http://www.ems-ph.org/doi/10.4171/120-1/28

Alexander Olevskii Tel Aviv University, RAMAT AVIV, TEL AVIV, ISRAEL Fourier analysis Functions of a complex variable Information and communication, circuits General The concepts in the title are closely related and belong to the central objects in Fourier Analysis, partly inspired by the signal theory. I’ll present some recent results in these topics. Sec. 2–6 are devoted to the sampling and interpolation problems, sec. 7–8 to the completeness of translates. 489 502 1 10.4171/120-1/29 http://www.ems-ph.org/doi/10.4171/120-1/29

Leonid Parnovski University College London, LONDON, UNITED KINGDOM Partial differential equations Operator theory Quantum theory General I discuss various recent results on the spectral properties of multidimensional periodic and, to a smaller extent, almost-periodic operators, including proving the Bethe–Sommerfeld conjecture and obtaining the asymptotic expansion of the integrated density of states for large energies. 503 513 1 10.4171/120-1/30 http://www.ems-ph.org/doi/10.4171/120-1/30

Benjamin Schlein Universität Bonn, BONN, GERMANY Quantum theory Probability theory and stochastic processes Statistical mechanics, structure of matter General We report on recent results concerning the derivation of e ective evolution equations starting from many body quantum dynamics. In particular, we obtain rigorous derivations of nonlinear Hartree equations in the bosonic mean eld limit, with precise bounds on the rate of convergence. Moreover, we present a central limit theorem for the fluctuations around the Hartree dynamics. 515 529 1 10.4171/120-1/31 http://www.ems-ph.org/doi/10.4171/120-1/31

Piotr Śniady Polish Academy of Sciences, WARSAW 10, POLAND Group theory and generalizations Combinatorics Functional analysis General The representation theory of the symmetric groups S(n) is intimately related to combinatorics: combinatorial objects such as Young tableaux and combinatorial algorithms such as Murnaghan–Nakayama rule. In the limit as n tends to infinity, the structure of these combinatorial objects and algorithms becomes complicated and it is hard to extract from them some meaningful answers to asymptotic questions. In order to overcome these difficulties, a kind of dual combinatorics of the representation theory of the symmetric groups was initiated in 1990s. We will concentrate on one of its highlights: Kerov polynomials which express characters in terms of, so called, free cumulants. 531 545 1 10.4171/120-1/32 http://www.ems-ph.org/doi/10.4171/120-1/32

Hao Jia University of Minnesota, MINNEAPOLIS, UNITED STATES Vladimír Šverák University of Minnesota, MINNEAPOLIS, UNITED STATES Partial differential equations Fluid mechanics General We discuss the forward self-similar solutions of the Navier{Stokes equations. It appears these solutions may provide an interesting window into non-perturbative regimes of the solutions of the equations. 547 553 1 10.4171/120-1/33 http://www.ems-ph.org/doi/10.4171/120-1/33

Stevo Todorčević University of Toronto, TORONTO, ONTARIO, CANADA Mathematical logic and foundations Combinatorics Functional analysis General Understanding the possible conditional structure in a given weakly-null sequence $(x_i)$ in some normed space $X$ lies in the heart of several classical problems of this area of mathematics. We will expose the set-theoretic and Ramsey-theoretic methods relevant to both the lack and the existence of this conditional structure. We will concentrate on more recent results and will point out problems for further study. 555 572 1 10.4171/120-1/34 http://www.ems-ph.org/doi/10.4171/120-1/34

Simon Brendle Stanford University, STANFORD, UNITED STATES Differential geometry Relativity and gravitational theory General A classical theorem due to Alexandrov asserts that the only embedded surfaces in Euclidean space with constant mean curvature are the round spheres. Alexandrov's theorem is closely related to the classical isoperimetric inequality, which asserts that round spheres have minimal surface area among all surfaces that enclose a given amount of volume. It is an interesting question whether Alexandrov's theorem can be generalized to surfaces of constant mean curvatures in a Riemannian manifold. In particular, if the ambient Riemannian manifold is rotationally symmetric, can one conclude that every embedded surface of constant mean curvature is a sphere of symmetry? This question turns out to have interesting links with general relativity. We also discuss minimal surfaces in the sphere $S^3$ and our recent solution of Lawson's conjecture. 573 583 1 10.4171/120-1/35 http://www.ems-ph.org/doi/10.4171/120-1/35

Alessio Figalli University of Texas at Austin, AUSTIN, UNITED STATES Calculus of variations and optimal control; optimization Partial differential equations General The aim of this note is to review recent stability results for some geometric and functional inequalities, and to describe applications to the long-time asymptotic of evolution equations 585 599 1 10.4171/120-1/36 http://www.ems-ph.org/doi/10.4171/120-1/36

Adrian Ioana University of California, San Diego, LA JOLLA, UNITED STATES Functional analysis Measure and integration Dynamical systems and ergodic theory General We survey some recent progress on the classi cation of von Neumann algebras arising from countable groups and their measure preserving actions on probability spaces. In particular, we present results which provide classes of (W -superrigid) groups and actions that can be entirely reconstructed from their von Neumann algebras. We also discuss the recent finding of several large families of II$_1$ factors that have a unique group measure space decomposition. 601 625 1 10.4171/120-1/37 http://www.ems-ph.org/doi/10.4171/120-1/37

Mathieu Lewin Université de Cergy-Pontoise, CERGY-PONTOISE, FRANCE Partial differential equations Quantum theory General We describe several recent results obtained in collaboration with P. Gravejat, C. Hainzl, E. S er e and J.P. Solovej, concerning a nonlinear model for the relativistic quantum vacuum in interaction with a classical electromagnetic field. 627 641 1 10.4171/120-1/38 http://www.ems-ph.org/doi/10.4171/120-1/38

Ciprian Manolescu UCLA, LOS ANGELES, UNITED STATES Manifolds and cell complexes General We review the use of grid diagrams in the development of Heegaard Floer theory. We describe the construction of the combinatorial link Floer complex, and the resulting algorithm for unknot detection. We also explain how grid diagrams can be used to show that the Heegaard Floer invariants of 3-manifolds and 4-manifolds are algorithmically computable (mod 2). 643 657 1 10.4171/120-1/39 http://www.ems-ph.org/doi/10.4171/120-1/39

Grégory Miermont Université Paris-Sud 11, ORSAY CEDEX, FRANCE Probability theory and stochastic processes Combinatorics Statistical mechanics, structure of matter General In the recent years, much progress has been made in the mathematical understanding of the scaling limit of random maps, making precise the sense in which random embedded graphs approach a model of continuum surface. In particular, it is now known that many natural models of random plane maps, for which the faces degrees remain small, admit a universal scaling limit, the Brownian map. Other models, favoring large faces, also admit a one-parameter family of scaling limits, called stable maps. The latter are believed to describe the asymptotic geometry of random maps carrying statistical physics models, as has now been established in some important cases (including the so-called rigid $O(n)$ model on quadrangulations). 659 673 1 10.4171/120-1/40 http://www.ems-ph.org/doi/10.4171/120-1/40

Tom Sanders Oxford University, OXFORD, UNITED KINGDOM Group theory and generalizations Number theory General Our aim is to discuss the structure of subsets of Abelian groups which behave 'a bit like' cosets (of subgroups). One version of 'a bit like' can be arrived at by relaxing the usual characterisation of cosets: a subset $S$ of an Abelian group is a coset if for every three elements $x, y, z \in S$ we have $x+y–z \in S$. What happens if this is not true 100% of the time but is true, say, 1% of the time? It turns out that this is a situation which comes up quite a lot, and one possible answer is called Frei man's theorem. We shall discuss it and some recent related quantitative advances. 675 689 1 10.4171/120-1/41 http://www.ems-ph.org/doi/10.4171/120-1/41

Corinna Ulcigrai University of Bristol, BRISTOL, UNITED KINGDOM Dynamical systems and ergodic theory General Parabolic flows are dynamical systems in which nearby trajectories diverge with polynomial speed. A classical example is the horocycle flow on a surface of constant negative curvature. Other important classes of examples are smooth area-preserving flows on surfaces, whose study is connected with Teichmueller dynamics, and Heisenberg nilflows. We survey some of the chaotic properties of these flows and some recent results on time changes of the above mentioned classes of examples. We focus in particular on mixing and we explain the shearing mechanism which is responsible for mixing in parabolic dynamics. 691 705 1 10.4171/120-1/42 http://www.ems-ph.org/doi/10.4171/120-1/42

Emmanuel Trélat Université Pierre et Marie Curie (Paris 6), PARIS CEDEX 05, FRANCE Calculus of variations and optimal control; optimization Dynamical systems and ergodic theory Numerical analysis Systems theory; control In this proceedings article we rst shortly report on some classical techniques of nonlinear optimal control such as the Pontryagin Maximum Principle and the conjugate point theory, and on their numerical implementation. We illustrate these issues with problems coming from aerospace applications such as the orbit transfer problem which is taken as a motivating example. Such problems are encountered in a longstanding collaboration with the european space industry EADS Astrium. On this kind of nonacademic problem it is shown that the knowledge resulting from the maximum principle is insu fficient for solving adequately the problem, in particular due to the di culty of initializing the shooting method, which is an approach for solving the boundary value problem resulting from the application of the maximum principle. On the orbit transfer problem we show how the shooting method can be successfully combined with a numerical continuation method in order to improve signi cantly its performances. We comment on assumptions ensuring the feasibility of continuation or homotopy methods, which consist of deforming continuously a problem towards a simpler one, and then of solving a series of parametrized problems to end up with the solution of the initial problem. Finally, in view of designing low cost interplanetary space missions, we show how optimal control can be also combined with dynamical system theory, which allows to put in evidence nice properties of the celestial dynamics around Lagrange points that are of great interest for mission design. 707 726 1 10.4171/120-1/43 http://www.ems-ph.org/doi/10.4171/120-1/43

Jan Hogendijk University of Utrecht, UTRECHT, NETHERLANDS History and biography Geometry General We discuss medieval Arabic and Persian sources on the design and construction of geometric ornaments in Islamic civilization. 727 741 1 10.4171/120-1/44 http://www.ems-ph.org/doi/10.4171/120-1/44

José Francisco Rodrigues FC Universidade de Lisboa, LISBOA, PORTUGAL General The Planet Earth System is composed of several sub-systems: the atmosphere, the liquid oceans, the internal structure and the icecaps and the biosphere. In all of them Mathematics, enhanced by the supercomputers, has currently a key role through the \universal method" for their study, which consists of mathematical modeling, analysis, simulation and control, as it was re-stated by Jacques-Louis Lions in [41]. Much before the advent of computers, the representation of the Earth, navigation and cartography have contributed in a decisive form to the mathematical sciences. Nowadays the International Geosphere-Biosphere Program, sponsored by the International Council of Scienti c Unions, may contribute to stimulate several mathematical research topics. In this article, we present a brief historical introduction to some of the essential mathematics for understanding the Planet Earth, stressing the importance of Mathematical Geography and its role in the Scienti c Revolution(s), the modeling e orts of Winds, Heating, Earthquakes, Climate and their in uence on basic aspects of the theory of Partial Di erential Equations. As a special topic to illustrate the wide scope of these (Geo)physical problems we describe brie y some examples from History and from current research and advances in Free Boundary Problems arising in the Planet Earth. Finally we conclude by referring the potential impact of the international initiative Mathematics of Planet Earth (http://www.mpe2013.org) in Raising Public Awareness of Mathematics, in Research and in the Communication of the Mathematical Sciences to the new generations. 743 762 1 10.4171/120-1/45 http://www.ems-ph.org/doi/10.4171/120-1/45

P.D. Welch University of Bristol, BRISTOL, UNITED KINGDOM History and biography Mathematical logic and foundations General We sketch a brief outline of the mathematical, and in particular the logical, achievements of Turing in this, his centenary year. 763 777 1 10.4171/120-1/46 http://www.ems-ph.org/doi/10.4171/120-1/46

Artur Siemaszko University of Warmia and Mazury in Olsztyn, OLSZTYN, POLAND Maciej Wojtkowski University of Warmia and Mazury in Olsztyn, OLSZTYN, POLAND Dynamical systems and ergodic theory General We develop the connection of Berg partitions with special substitution tilings of two tiles. We obtain a new proof that the number of Berg partitions with a fi xed connectivity matrix is equal to half of the sum of its entries, [12]. This approach together with the formula of S e ebold [10], for the number of substitutions preserving a given Sturmian sequence, shows that all of the combinatorial substitutions can be realized geometrically as Berg partitions. We treat Sturmian tilings as intersection tilings of bi-partitions. Using the symmetries of bi-partitions we obtain geometrically the palindromic properties of Sturmian sequences (Theorem 3) established combinatorially by de Luca and Mignosi, [6]. 779 790 1 10.4171/120-1/47 http://www.ems-ph.org/doi/10.4171/120-1/47

Carel Faber Royal Institute of Technology, Stockholm, Sweden Gavril Farkas Humboldt-Universität zu Berlin, Berlin, Germany Robin de Jong University of Leiden, The Netherlands Algebraic geometry Number theory 14; 11 Algebraic geometry, arithmetic geometry This volume contains 21 articles written by leading experts in the fields of algebraic and arithmetic geometry. The treated topics range over a variety of themes, including moduli spaces of curves and abelian varieties, algebraic cycles, vector bundles and coherent sheaves, curves over finite fields, and algebraic surfaces, among others. The volume originates from the conference “Geometry and Arithmetic”, which was held on the island of Schiermonnikoog in The Netherlands in September 2010. 10 19 2012 978-3-03719-119-4 978-3-03719-619-9 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/119 http://www.ems-ph.org/doi/10.4171/119

Valery Alexeev University of Georgia, ATHENS, UNITED STATES David Swinarski Fordham University, BRONX, UNITED STATES Algebraic geometry General We introduce and study the GIT cone of $\overline{M}_{0,n}$, which is generated by the pullbacks of the natural ample line bundles on the GIT quotients $(\mathbb P^1)^n/\!/SL(2)$. We give an explicit formula for these line bundles and prove a number of basic results about the GIT cone. As one application, we prove unconditionally that the log canonical models of $\overline{M}_{0,n}$ with a symmetric boundary divisor coincide with the moduli spaces of weighted curves or with the symmetric GIT quotient, extending the result of Matt Simpson. 1 21 1 10.4171/119-1/1 http://www.ems-ph.org/doi/10.4171/119-1/1

Ingrid Bauer Universität Bayreuth, BAYREUTH, GERMANY Fabrizio Catanese Universität Bayreuth, BAYREUTH, GERMANY Algebraic geometry General We show that a family of minimal surfaces of general type with $p_g = 0, K^2=7$, constructed by Inoue in 1994, is indeed a connected component of the moduli space: indeed that any surface which is homotopically equivalent to an Inoue surface belongs to the Inoue family. The ideas used in order to show this result motivate us to give a new definition of varieties, which we propose to call Inoue type manifolds: these are obtained as quotients $ \hat{X} / G$, where $ \hat{X} $ is an ample divisor in a $K(\Gamma, 1)$ projective manifold $Z$, and $G$ is a finite group acting freely on $ \hat{X} $. For these types of manifolds we prove a similar theorem to the above, even if weaker, that manifolds homotopically equivalent to Inoue type manifolds are again Inoue type manifolds. 23 56 1 10.4171/119-1/2 http://www.ems-ph.org/doi/10.4171/119-1/2

Arnaud Beauville Université de Nice, NICE CEDEX 2, FRANCE Algebraic geometry General We prove that the symmetric sextic Fano threefold, defined by the equations $\sum X_i=\sum X_i^2=\sum X_i^3=0$ in $\mathbb{P}^6$, is not rational. In view of the work of Prokhorov [P], our result implies that the alternating group $\mathfrak{A}_7$ admits only one embedding into the Cremona group $\mathrm{Cr}_3$ up to conjugacy. 57 60 1 10.4171/119-1/3 http://www.ems-ph.org/doi/10.4171/119-1/3

Ciro Ciliberto Università di Roma, ROMA, ITALY Flaminio Flamini Università di Roma Tor Vergata, ROMA, ITALY Algebraic geometry General In this note we give an easy proof of the existence of generically smooth components of the expected dimension of certain Brill–Noether loci of stable rank 2 vector bundles on a curve with general moduli, with related applications to the Hilbert scheme of scrolls. 61 74 1 10.4171/119-1/4 http://www.ems-ph.org/doi/10.4171/119-1/4

José Ignacio Cogolludo-Agustín Universidad de Zaragoza, ZARAGOZA, SPAIN Remke Kloosterman Humboldt-Universität zu Berlin, BERLIN, GERMANY Algebraic geometry Number theory General We prove the existence of three irreducible curves $C_{12,m}$ of degree 12 with the same number of cusps and different Alexander polynomials. This exhibits a Zariski triple. Moreover we provide a set of generators for the elliptic threefold with constant $j$-invariant 0 and discriminant curve $C_{12,m}$. Finally we consider a general degree $d$ base change of $C_{12d,m}$ and calculate the dimension of the equisingular deformation space. 75 89 1 10.4171/119-1/5 http://www.ems-ph.org/doi/10.4171/119-1/5

Jean-Marc Couveignes Université Bordeaux 1, TALENCE CEDEX, FRANCE Bas Edixhoven Universiteit Leiden, LEIDEN, NETHERLANDS Algebraic geometry Number theory General This article gives an introduction for mathematicians interested in numerical computations in algebraic geometry and number theory to some recent progress in algorithmic number theory, emphasising the key role of approximate computations with modular curves and their Jacobians. These approximations are done in polynomial time in the dimension and the required number of significant digits. We explain the main ideas of how the approximations are done, illustrating them with examples, and we sketch some applications in number theory. 91 112 1 10.4171/119-1/6 http://www.ems-ph.org/doi/10.4171/119-1/6

Friedrich Eusen , DÜSSELDORF, GERMANY Frank-Olaf Schreyer Universität des Saarlandes, SAARBRÜCKEN, GERMANY Algebraic geometry Number theory General In this note, we show that the spaces of global sections of exterior powers of a globally generated line bundle on a curve are not necessarily spanned by locally decomposable sections. The examples are based on the study of generic syzygy varieties. An application of these varieties is a short proof of Mukai's theorem that every smooth curve of genus 7 and Clifford index 3 arises as the intersection of the spinor variety $S \subset \mathbb P^{15}$ with a transversal $\mathbb P^6$. 113 123 1 10.4171/119-1/7 http://www.ems-ph.org/doi/10.4171/119-1/7

Angela Gibney University of Georgia, ATHENS, UNITED STATES Algebraic geometry Number theory General The divisors on $\overline{\operatorname {M}}_g$ that arise as the pullbacks of ample divisors along any extension of the Torelli map to any toroidal compactification of $\operatorname{A}_g$ form a 2-dimensional extremal face of the nef cone of $\overline{\operatorname {M}}_g$, which is explicitly described. 125 136 1 10.4171/119-1/8 http://www.ems-ph.org/doi/10.4171/119-1/8

Benedict Gross Harvard University, CAMBRIDGE, UNITED STATES Algebraic geometry Number theory General We reinterpret some results of D. Zagier on the traces of singular moduli, in terms of the generalized Jacobian of the modular curve of level 1, with respect to the divisor 2$(\infty)$. 137 141 1 10.4171/119-1/9 http://www.ems-ph.org/doi/10.4171/119-1/9

Günter Harder , BONN, GERMANY Algebraic geometry Number theory General In his paper [4], Gerard van der Geer discusses the Eisenstein cohomology with coefficients in a sheaf $\tilde M$, which is obtained from a representation for the group $\tilde\Gamma=\GSp_g(\mathbb Z)$. Since we have an arithmetic interpretation of this sheaf, we can endow these cohomology groups with the structure of a mixed motive. A certain part of this cohomology is the compactly supported Eisenstein cohomology and van der Geer determines the structure of this compactly supported Eisenstein motive in the case $g=2$ and a regular coefficient system [4], Cor.~10.2). At the end of this note we compute this part of the cohomology for an arbitrary coefficient system, again in the case $g=2$. 143 164 1 10.4171/119-1/10 http://www.ems-ph.org/doi/10.4171/119-1/10

Jochen Heinloth Universität Duisburg-Essen, ESSEN, GERMANY Algebraic geometry Number theory General We compute the cohomology of the moduli stack of coherent sheaves on a curve and find that it is a free graded algebra on infinitely many generators. 165 171 1 10.4171/119-1/11 http://www.ems-ph.org/doi/10.4171/119-1/11

Everett Howe Center for Communications Research, SAN DIEGO, UNITED STATES Kristin Lauter Theory Group, REDMOND, UNITED STATES Algebraic geometry Number theory General We provide new upper bounds on $N_q(g)$, the maximum number of rational points on a smooth absolutely irreducible genus-$g$ curve over $\mathbb {F}_q$, for many values of $q$ and $g$. Among other results, we find that $N_4(7) = 21$ and $N_8(5) = 29$, and we show that a genus-12 curve over $\mathbb {F}_2$ having 15 rational points must have characteristic polynomial of Frobenius equal to one of three explicitly given possibilities. We also provide sharp upper bounds for the lengths of the shortest vectors in Hermitian lattices of small rank and determinant over the maximal orders of small imaginary quadratic fields of class number 1. Some of our intermediate results can be interpreted in terms of Mordell–Weil lattices of constant elliptic curves over one-dimensional function fields over finite fields. Using the Birch and Swinnerton-Dyer conjecture for such elliptic curves, we deduce lower bounds on the orders of certain Shafarevich–Tate groups. 173 212 1 10.4171/119-1/12 http://www.ems-ph.org/doi/10.4171/119-1/12

Hiroyuki Ito Tokyo University of Science, CHIBA-KEN, JAPAN Stefan Schröer Heinrich-Heine-Universität, DÜSSELDORF, GERMANY Algebraic geometry Number theory General We analyse the diagonal quotient for the product of certain Artin--Schreier curves. The smooth models are almost always surfaces of general type, with Chern slopes tending asymptotically to 1. The calculation of numerical invariants relies on a close examination of the relevant wild quotient singularity in characteristic $p$. It turns out that the canonical model has $q-1$ rational double points of type $A_{q-1}$, and embeds as a divisor of degree $q$ in $\mathbb P^3$, which is in some sense reminiscent of the classical Kummer quartic. 213 241 1 10.4171/119-1/13 http://www.ems-ph.org/doi/10.4171/119-1/13

Toshiyuki Katsura Hosei University, TOKYO, JAPAN Shigeyuki Kondō Nagoya University, NAGOYA, JAPAN Algebraic geometry Number theory General We construct, on a supersingular K3 surface with Artin invariant 1 in characteristic 2, a set of 21 disjoint smooth rational curves and another set of 21 disjoint smooth rational curves such that each curve in one set intersects exactly 5 curves from the other set with multiplicity 1 by using the structure of a generalized Kummer surface. As a corollary we have a concrete construction of a K3 surface with 21 rational double points of type A1 in characteristic 2. 243 255 1 10.4171/119-1/14 http://www.ems-ph.org/doi/10.4171/119-1/14

Sándor Kovács University of Washington, SEATTLE, UNITED STATES Algebraic geometry Number theory General It is proved that for projective varieties having Du Bois singularities is equivalent to the condition that the coherent cohomology groups of the structure sheaf coincide with the appropriate Hodge components of the singular cohomology groups 257 266 1 10.4171/119-1/15 http://www.ems-ph.org/doi/10.4171/119-1/15

H. Lange Universität Erlangen-Nünberg, ERLANGEN, GERMANY Peter Newstead University of Liverpool, LIVERPOOL, UNITED KINGDOM Algebraic geometry Number theory General In this paper, we construct stable bundles $E$ of rank 2 on suitably chosen curves of any genus $g\ge12$ with maximal Clifford index such that the Clifford index of $E$ takes the minimum possible value for curves with this property. 267 281 1 10.4171/119-1/16 http://www.ems-ph.org/doi/10.4171/119-1/16

Rahul Pandharipande ETH Zürich, ZÜRICH, SWITZERLAND A. Pixton Princeton University, PRINCETON, UNITED STATES Algebraic geometry Number theory General The stable pairs theory of local curves in 3-folds (equivariant with respect to the scaling 2-torus) is studied with stationary descendent insertions. Reduction rules are found to lower descendents when higher than the degree. Factorization then yields a simple proof of rationality in the stationary case and a proof of the functional equation related to inverting $q$. The method yields an effective determination of stationary descendent integrals. The series $\mathsf{Z}^{\mathsf{cap}}_{d,(d)}( \tau_d(\mathsf{p}))$ plays a special role and is calculated exactly using the stable pairs vertex and an analysis of the solution of the quantum differential equation for the Hilbert scheme of points of the plane. 283 307 1 10.4171/119-1/17 http://www.ems-ph.org/doi/10.4171/119-1/17

Dan Petersen Royal Institute of Technology, STOCKHOLM, SWEDEN Algebraic geometry Number theory General Ezra Getzler notes in the proof of the main theorem of [3] that “A proof of the theorem could no doubt be given using [a combinatorial interpretation in terms of a sum over necklaces]; however, we prefer to derive it directly from Theorem 2.2”. In this note we give such a direct combinatorial proof using wreath product symmetric functions. 309 316 1 10.4171/119-1/18 http://www.ems-ph.org/doi/10.4171/119-1/18

René Schoof Università di Roma Tor Vergata, ROMA, ITALY Algebraic geometry Number theory General The Jacobian $J_0(23)$ of the modular curve $X_0(23)$ is a semi-stable abelian variety over $\mathbb Q$ with good reduction outside 23. It is simple. We prove that every simple semi-stable abelian variety over $\mathbb Q$ with good reduction outside 23 is isogenous over $\mathbb Q$ to $J_0(23)$. 317 345 1 10.4171/119-1/19 http://www.ems-ph.org/doi/10.4171/119-1/19

Claire Voisin Institut de Mathématiques de Jussieu, PARIS, FRANCE Algebraic geometry Number theory General We study in this paper degree $4$ unramified cohomology with torsion coefficients of a smooth projective variety $X$. We show that if $CH_0(X)$ is small, it is isomorphic to the group of torsion codimension $3$ cycles with trivial Deligne cycle class, modulo algebraic equivalence. 347 368 1 10.4171/119-1/20 http://www.ems-ph.org/doi/10.4171/119-1/20

Yuri Zarhin Pennsylvania State University, UNIVERSITY PARK, UNITED STATES Algebraic geometry Number theory General It is well known that the cup-product pairing on the complementary integral cohomology groups (modulo torsion) of a compact oriented manifold is unimodular. We prove a similar result for the $\ell$-adic cohomology groups of smooth algebraic varieties. 369 376 1 10.4171/119-1/21 http://www.ems-ph.org/doi/10.4171/119-1/21

Strasbourg 2009 Vincent Blanlœil IRMA, Strasbourg, France Toru Ohmoto Hokkaido University, Sapporo, Japan Algebraic geometry Several complex variables and analytic spaces Manifolds and cell complexes Global analysis, analysis on manifolds 13A35, 14A22, 14B05, 14B07, 14B15, 14C17, 14D06, 14E15, 14E18, 14F99, 14H25, 14J17, 14M25, 14N99, 18F30, 19K10, 32A27, 32C37, 32F75, 32G15, 32SXX, 35Q75, 52C35, 53C05, 55N35, 55R40, 57N05, 57R18, 57R20, 58A30, 58K10, 58K40, 58K60, 60D05, 83C57 singularity theory, singularities, characteristic classes, Milnor fiber, jet schemes, equisingularity, intersection homology, knot theory, Hodge theory, Fulton–MacPherson bivariant theory, mixed weighted homogeneous, nearby cycles, vanishing cycles This volume arises from 5th Franco-Japanese Symposium on Singularities, held in Strasbourg in August 2009. The conference brought together an international group of researchers working on singularities in algebraic geometry, analytic geometry and topology, mainly from France and Japan. Besides, it also organized a special session, JSPS Forum on Singularities and Applications, which was aimed to introduce some recent applications of singularity theory to physics and statistics. This book comprises research papers and short lecture notes on advanced topics on singularities. Some surveys on applications that were presented in the Forum are also added. Topics covered include splice surface singularities, b-functions, equisingularity, degenerating families of Riemann surfaces, hyperplane arrangements, mixed singularities, jet schemes, noncommutative blow-ups, characteristic classes of singular spaces, and applications to geometric optics, cosmology and learning theory. Graduate students who wish to learn about various approaches to singularities, as well as experts in the field and researchers in other areas of mathematics and science will find the contributions to this volume a rich source for further study and research. 12 18 2012 978-3-03719-118-7 978-3-03719-618-2 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/118 http://www.ems-ph.org/doi/10.4171/118 20

Alain Joets Université Paris Sud-XI, ORSAY CEDEX, FRANCE General Optical caustics are bright patterns, formed by the local focalization of light rays. They are caused, for instance, by the reflection or the refraction of the sun rays through a wavy water surface. In the absence of an appropriate mathematical frame, their main characteristics have remained unrecognized for a long time and the caustics appeared in the literature under different names: evolutes, envelopes, focals, etc. The creation of the singularity theory in the middle of the 20th century radically changed the situation. Caustics are now understood as physical realizations of Lagrangian singularities. In this modelling, one predicts their local classification into five stable types (R. Thom, V. Arnold): folds, cusps, swallowtails, elliptic umbilics and hyperbolic umbilics. This local classification is indeed observed in experiments. However the global properties of the caustics are only partially taken into account by the Lagrangian model. In fact, it has been proved by Yu. Chekanov that the special form of the eikonal equation governing the propagation of the optical wave fronts imposes the existence of a topological constraint on the singular set (representing the caustic in the phase space) and restricts the number of possible bifurcations. Our experiments on caustics produced by bi-periodic structures in liquid crystals confirm the existence of the topological constraint, and validate the modelling of the caustics as special types of Lagrangian singularities. 1 17 1 10.4171/118-1/1 http://www.ems-ph.org/doi/10.4171/118-1/1

Helmut Hamm Universität Münster, MÜNSTER, GERMANY Several complex variables and analytic spaces Algebraic geometry Global analysis, analysis on manifolds General We will prove some generalization of the theorems of Lê and Ramanujam resp. of Timourian for the case where the ambient space is no longer $\mathbb{C}^m$. Furthermore we will derive some weaker result in the case of a family of non-isolated singularities. 19 37 1 10.4171/118-1/2 http://www.ems-ph.org/doi/10.4171/118-1/2

Shihoko Ishii University of Tokyo, TOKYO, JAPAN Akiyoshi Sannai Nagoya University, NAGOYA, JAPAN Kei-ichi Watanabe College of Humanities and Sciences, TOKYO, JAPAN Algebraic geometry General This paper studies the singularities of jet schemes of homogeneous hypersurfaces of general type. We obtain the condition of the degree and the dimension for the singularities of the jet schemes to be of dense $F$-regular type. This provides us with examples of singular varieties whose $m$-jet schemes have rational singularities for every $m$. 39 49 1 10.4171/118-1/3 http://www.ems-ph.org/doi/10.4171/118-1/3

Tatsuhiko Koike Keio University, YOKOHAMA, JAPAN Relativity and gravitational theory Partial differential equations Global analysis, analysis on manifolds General Many phenomena of importance in general relativity theory are related to singularities in mathematics. For a simple example, the spacetime regions of extreme gravitational field such as the beginning of the Universe and the fate of a massive star are described by singularities in the differential-geometric sense, i.e., curvature singularities of pseudo-Riemannian manifolds. This type of singularity is one of the main objects of interest in general relativity. A less trivial example is that the formation of a black hole horizon can be described as a blow-up solution of some partial differential equations in a certain coordinate system, which is a singularity in the analytic sense. Another is that the “shape” of the black hole horizon is fully characterised by the set of its nondifferential points which are singularities in the sense of singularity theory. I will explain these connections between singularity and relativity with some comments on my related works. 51 69 1 10.4171/118-1/4 http://www.ems-ph.org/doi/10.4171/118-1/4

Yukio Matsumoto Gakushuin University, TOKYO, JAPAN Algebraic geometry Several complex variables and analytic spaces Manifolds and cell complexes General Let $\Sigma_g$ be a closed oriented (topological) surface of genus $g$ ($\geqq 2$). Over the Teichmüller space $T(\Sigma_g)$ of $\Sigma_g$, Bers constructed a universal family $V(\Sigma_g)$ of curves of genus $g$, which would be well called “the tautological family of Riemann surfaces”. The mapping class group $\Gamma_g$ of $\Sigma_g$ acts on $V(\Sigma_g) \to T(\Sigma_g)$ in a fibration preserving manner. Dividing the fiber space by this action, we obtain an “orbifold fiber space” $Y(\Sigma_g) \to M(\Sigma_g)$, where $Y(\Sigma_g)$ and $M(\Sigma_g)$ denote $V(\Sigma_g)/\Gamma_g$ and $T(\Sigma_g)/\Gamma_g$, respectively. The latter quotient $M(\Sigma_g)$ is called the moduli space of $\Sigma_g$. The fiber space $Y(\Sigma_g) \to M(\Sigma_g)$ can be naturally compactified to another orbifold fiber space $\overline{Y(\Sigma_g)} \to \overline{M(\Sigma_g)}$. The base space $\overline{M(\Sigma_g)}$ is called the Deligne–Mumford compactification. Since this compactification is constructed by adding “stable curves” at infinity, it is usually accepted that the compactified moduli space $\overline{M(\Sigma_g)}$ is the coarse moduli space of stable curves of genus $g$. In this paper, we will sketch our argument which leads to a conclusion, somewhat contradictory to the above general acceptance, that the compactified family $\overline{Y(\Sigma_g)} \to \overline{M(\Sigma_g)}$ is the universal degenerating family of Riemann surfaces, i.e., it virtually parametrizes not only stable curves but also all types of degenerate and non-degenerate curves. Our argument is a combination of the Bers–Kra theory and Ashikaga’s precise stable reduction theorem. 71 102 1 10.4171/118-1/5 http://www.ems-ph.org/doi/10.4171/118-1/5

Yayoi Nakamura Kinki University, OSAKA, JAPAN Shinichi Tajima University of Tsukuba, IBARAKI, JAPAN Algebraic geometry Several complex variables and analytic spaces General The role of the weighted degree of algebraic local cohomology classes in the computation of $b$-function is discussed. The result which is similar to quasihomogeneous cases is observed for semiquasihomogeneous isolated singularities with $L(f)=2$. 103 115 1 10.4171/118-1/6 http://www.ems-ph.org/doi/10.4171/118-1/6

Toru Ohmoto Hokkaido University, SAPPORO, JAPAN Algebraic geometry General This is a note about the Chern–Schwartz–MacPherson class for certain algebraic stacks which has been introduced in [17]. We also discuss other singular Riemann–Roch type formulas in the same manner. 117 131 1 10.4171/118-1/7 http://www.ems-ph.org/doi/10.4171/118-1/7

Mutsuo Oka Tokyo University of Science, TOKYO, JAPAN Algebraic geometry General Let $f(\mathbf{z},\bar{\mathbf{z}})$ be a strongly polar homogeneous polynomial of $n$ variables $\mathbf{z} =(z_1,\dots, z_n)$. This polynomial defines a projective real algebraic variety $V = \{[\mathbf{z}] \in \mathbf{CP}^{n-1}\,|\,f(\mathbf{z},\bar{\mathbf{z}})=0 \}$ in the projective space $ \mathbf{CP}^{n-1}$. The behavior is different from that of the projective hypersurface. The topology is not uniquely determined by the degree of the variety even if $V$ is non-singular. We study a basic property of such a variety. 133 147 1 10.4171/118-1/8 http://www.ems-ph.org/doi/10.4171/118-1/8

Tomohiro Okuma Yamagata University, YAMAGATA, JAPAN Algebraic geometry Several complex variables and analytic spaces General This article is a survey of results on analytic invariants of splice quotient singularities induced by Neumann and Wahl. These singularities are natural and broad generalization of quasihomogeneous surface singularities with rational homology sphere links. The “leading terms” of the equations are constructed from the resolution graph. Some analytic invariants of splice quotients can explicitly be computed from their graph. 149 159 1 10.4171/118-1/9 http://www.ems-ph.org/doi/10.4171/118-1/9

Oswald Riemenschneider Universität Hamburg, HAMBURG, GERMANY Algebraic geometry General In my lecture at the Franco-Japanese Symposium on Singularities I gave an introduction to the work of Martin Hamm [3] concerning the explicit construction of the versal deformation of cyclic surface singularities. Since that part of his dissertation is already documented in a survey article (cf. [8]), I concentrate in the present note on some other aspect of [3]: the toric duality of the total spaces of the deformations over the monodromy coverings of the Artin components for the singularities $A_{n,q}$ and $A_{n,n - q}$ which themselves are toric duals of each other. Our exhibition is based – as in Hamm’s dissertation – on the algebraic aspects, i.e., the algebras and their generators of these total spaces. We prove Hamm’s remarkable duality result in this note first in detail for the hypersurface case $q = n - 1$ in which the interplay between algebra and geometry of the underlying polyhedral cones is rather obvious, especially when bringing also the “complementarity” of $A_{n,q}$ and $A_{n,n - q}$ into the game. We then treat the dual case $q = 1$ of cones over the rational normal curves once more in order to develop the necessary ideas for transforming the generators in such a way that it becomes transparent how to compute the dual, even in the general situation (which we explain in the last section by an example). 161 179 1 10.4171/118-1/10 http://www.ems-ph.org/doi/10.4171/118-1/10

Jörg Schürmann Universität Münster, MÜNSTER, GERMANY Algebraic geometry Several complex variables and analytic spaces Global analysis, analysis on manifolds General In this paper we give an introduction to our recent work on characteristic classes of complex hypersurfaces based on some talks given at conferences in Strasbourg, Oberwolfach, and Kagoshima. We explain the relation between nearby cycles for constructible functions or sheaves as well as for (relative) Grothendieck groups of algebraic varieties and mixed Hodge modules, and the specialization of characteristic classes of singular spaces like the Chern-, Todd-, Hirzebruch-, and motivic Chern-classes. As an application we get a description of the differences between the corresponding virtual and functorial characteristic classes of complex hypersurfaces in terms of vanishing cycles related to the singularities of the hypersurface. 181 205 1 10.4171/118-1/11 http://www.ems-ph.org/doi/10.4171/118-1/11

Tatsuo Suwa Hokkaido University, SAPPORO, JAPAN Algebraic geometry Several complex variables and analytic spaces Manifolds and cell complexes Global analysis, analysis on manifolds We present two types of residue theories for singular holomorphic distributions. The first one is for certain Chern polynomials of the normal sheaf of a distribution and the residues arise from the vanishing, by rank reason, of the relevant characteristic classes on the non-singular part. The second one is for certain Atiyah polynomials of vector bundles admitting an action of a distribution and the residues arise from the Bott type vanishing theorem on the non-singular part. 207 247 1 10.4171/118-1/12 http://www.ems-ph.org/doi/10.4171/118-1/12

Sumio Watanabe Tokyo Institute of Technology, YOKOHAMA, JAPAN Probability theory and stochastic processes General This paper introduces a recent advance in the research between algebraic geometry and statistical learning theory. A lot of statistical models used in information science contain singularities in their parameter spaces, to which the conventional theory can not be applied. The statistical foundation of singular models was been left unknown, because no mathematical base could be found. However, recently new theory was constructed based on algebraic geometry and algebraic analysis. In this paper, we show that statistical estimation process is determined by two birational invariants, the real log canonical threshold and the singular fluctuation. As a result, a new formula is derived, which enables us to estimate the generalization error without any knowledge of the information source. In the discussion, a relation between mathematics and the real world is introduced to pure mathematicians. 249 268 1 10.4171/118-1/13 http://www.ems-ph.org/doi/10.4171/118-1/13

Takehiko Yasuda Osaka University, OSAKA, JAPAN Algebraic geometry Commutative rings and algebras General We define the Frobenius morphism of certain class of noncommutative blowups in positive characteristic. Thanks to a nice property of the class, the defined morphism is flat. Therefore we say that the noncommutative blowups in this class are Kunz regular. One of such blowups is the one associated to a regular Galois alteration. As a consequence of de Jong's theorem, we see that for every variety over an algebraically closed field of positive characteristic, there exists a noncommutative blowup which is Kunz regular. We also see that a variety with F-pure and FFRT (finite F-representation type) singularities has a Kunz regular noncommutative blowup which is associated to an iteration of the Frobenius morphism of the variety. 269 283 1 10.4171/118-1/14 http://www.ems-ph.org/doi/10.4171/118-1/14

Shoji Yokura Kagoshima University, KAGOSHIMA, JAPAN Algebraic geometry Several complex variables and analytic spaces Algebraic topology Manifolds and cell complexes The Euler–Poincaré characteristic is a generalization of the cardinality (or counting) and its higher homological extension for singular varieties as a natural transformation (what could be put in as its “categorification”) is MacPherson’s Chern class transformation. This transformation furthermore has two main developments: a bivariant-theoretic analogue and a generating series of it, i.e. a zeta function. The motivic Hirzebruch class is a unified theory of the three well-known characteristic classes of singular varieties, i.e. the above MacPherson’s Chern class transformation, Baum–Fulton–MacPherson’s Riemann–Roch and Cappell–Shaneson’s $L$-class transformation, which extends Goresky–MacPherson’s $L$-class. In this paper we discuss a bivariant-theoretic analogue and a zeta function of the motivic Hirzebruch class. 285 343 1 10.4171/118-1/15 http://www.ems-ph.org/doi/10.4171/118-1/15

Masahiko Yoshinaga Kyoto University, KYOTO, JAPAN Algebraic geometry Several complex variables and analytic spaces Convex and discrete geometry General The purpose of this paper is applying minimality of hyperplane arrangements to local system cohomology groups. It is well known that twisted cohomology groups with coefficients in a generic rank one local system vanish except in the top degree, and bounded chambers form a basis of the remaining cohomology group. We determine precisely when this phenomenon happens for two-dimensional arrangements. 345 362 1 10.4171/118-1/16 http://www.ems-ph.org/doi/10.4171/118-1/16

Athanase Papadopoulos IRMA, Strasbourg, France Functions of a complex variable Several complex variables and analytic spaces Primary 30-00, 32-00, 57-00, 32G13, 32G15, 30F60; Secondary 11F06, 11F75, 14D20, 14H15, 14H60, 14H55, 14J60, 20F14, 20F28, 20F38, 20F65, 20F67, 20H10, 22E46, 30-03, 30C62, 30F20, 30F25, 30F10, 30F15, 30F30, 30F35, 30F40, 30F45, 32-03, 32S30, 37-99, 53A35, 53B35, 53C35, 53C50, 53C80, 53D55, 53Z05, 57M07, 57M20, 57M27, 57M50, 57M60, 57N16 Teichmüller theory is, since several decades, one of the most active research areas in mathematics, with a very wide range of points of view, including Riemann surface theory, hyperbolic geometry, low-dimensional topology, several complex variables, algebraic geometry, arithmetic, partial differential equations, dynamical systems, representation theory, symplectic geometry, geometric group theory, and mathematical physics. The present book is the fourth volume in a Handbook of Teichmüller Theory project that started as an attempt to present, in a most comprehensive and systematic way, the various aspects of this theory with its relations to all the fields mentioned. The handbook is addressed to researchers as well as graduate students. The present volume is divided into five parts: Part A: The metric and the analytic theory. Part B: Representation theory and generalized structures. Part C: Dynamics. Part D: The quantum theory. Part E: Sources. Parts A, B and D are sequels of parts on the same theme in previous volumes. Part E has a new character in the series; it contains the translation together with a commentary of an important paper by Teichmüller which is almost unknown even to specialists. Making clear the original ideas of and motivations for a theory is crucial for many reasons, and rendering available this translation together with the commentary that follows will give a new impulse and will contribute in putting the theory into a broader perspective. The various volumes in this collection are written by experts who have a broad view on the subject. In general, the chapters have an expository character, which is the original purpose of this handbook, while some of them contain new and important results. 5 30 2014 978-3-03719-117-0 978-3-03719-617-5 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/117 http://www.ems-ph.org/doi/10.4171/117 19

Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Several complex variables and analytic spaces General This is an extended introduction to the volume. It contains a detailed account of its content, with motivation and background material. 1 39 1 10.4171/117-1/1 http://www.ems-ph.org/doi/10.4171/117-1/1

Sumio Yamada Gakushuin University, TOKYO, JAPAN Global analysis, analysis on manifolds Partial differential equations General This is a survey on the topic of Weil–Petersson geometry of Teichmüller spaces. Although historically the subject has been developed as a branch of complex analysis, the treatment here is from the view-point of differential geometry, much influenced by the works of Eells, Earle, Fischer, Tromba and Wolpert over the several decades. The Weil–Petersson geometry has negative sectional curvature, and the main theme of this article is to exploit the convexity of the Weil–Petersson distance function induced by the negativity of the curvature, from Riemannian geometric approaches as well as those of the CAT(0) geometry and the Coxeter theory. Also we discuss the Weil–Petersson geometry of the universal Teichmüller space. In particular, the linear deformation theory of the hyperbolic metrics is treated synthetically in the sense that the universal Teichmüller space and the finite dimensional Teichmüller space are contrasted with each other. 43 111 1 10.4171/117-1/2 http://www.ems-ph.org/doi/10.4171/117-1/2

Hugo Parlier Université de Fribourg, FRIBOURG, SWITZERLAND Global analysis, analysis on manifolds Partial differential equations General The goal of the chapter is to present certain aspects of the relationship between the study of simple closed geodesics and Teichmüller spaces. The set of simple closed geodesics is more than a mere curiosity and has been central in the study of surfaces for quite some time. There are two main themes to the chapter. The first one is the study of the set of simple closed geodesics in contrast with the set of closed geodesics and the second is on systoles, their lengths and related quantities such as the lengths of minimal pants decompositions. 113 134 1 10.4171/117-1/3 http://www.ems-ph.org/doi/10.4171/117-1/3

Lizhen Ji University of Michigan, ANN ARBOR, UNITED STATES Differential geometry Group theory and generalizations Topological groups, Lie groups Functions of a complex variable Tits buildings $\Delta_\mathbb Q(\mathbf G)$ of linear algebraic groups $\mathbf G$ defined over the field of rational numbers $\mathbb Q$ have played an important role in understanding partial compactifications of symmetric spaces and compactifications of locally symmetric spaces, cohomological properties of arithmetic subgroups and S-arithmetic subgroups of $\mathbf G(\mathbb Q)$. Curve complexes $\mathcal C(S_{g,n})$ of surfaces $S_{g,n}$ were introduced to parametrize boundary components of partial compactifications of Teichmüller spaces and were later applied to understand properties of mapping class groups of surfaces and the geometry and topology of 3-dimensional manifolds. Tits buildings are spherical building. Another important class of buildings consists of Euclidean buildings, for example, the Bruhat 135 196 1 10.4171/117-1/4 http://www.ems-ph.org/doi/10.4171/117-1/4

Hideki Miyachi Osaka University, OSAKA, JAPAN General This is a survey of the development of the extremal length geometry of Teichmüller space after S. Kerckhoff, F. Gardiner and H. Masur’s works. We shall discuss intersection number in extremal length geometry, and we shall view a canonical compactification in extremal length geometry with respect to intersection number. This is an analogy with the Thurston compactification in hyperbolic geometry. We will also obtain a hyperboloid model of Teichmüller space in extremal length geometry. This is an analogy with a realization of Teichmüller space into the space of the geodesic currents observed by F. Bonahon. 197 234 1 10.4171/117-1/5 http://www.ems-ph.org/doi/10.4171/117-1/5

Ken’ichi Ohshika Osaka University Graduate School of Science, OSAKA, JAPAN Functions of a complex variable Manifolds and cell complexes General We explain three ways of compactifying Teichmüller spaces: the Thurston compactification, the Teichmüller compactification, and the Bers compactification. The mapping class group acts on the Thurston compactification continuously, but neither on the Teichmüller compactification nor on the Bers compactification. We introduce quotient spaces of the Teichmüller boundary and the Bers boundary to which the action of the mapping class group extends continuously, and show that any self-homeomorphism on these quotient spaces is induced from a unique extended mapping class. 235 254 1 10.4171/117-1/6 http://www.ems-ph.org/doi/10.4171/117-1/6

Ralph Kaufmann Purdue University, WEST LAFAYETTE, UNITED STATES Several complex variables and analytic spaces Category theory; homological algebra Algebraic topology Manifolds and cell complexes By defining natural gluing operations, we combine the classical approach to Teichmüller theory via foliations with the operadic approach to field theory. The result is a universal structure, which governs surprisingly many algebraic and geometric constructions in and around moduli spaces of curves, string (field) theory and string topology. Applications include solutions to Deligne’s conjecture, its cyclic version and the definition of string topology operations. For the algebraic operations, such as the Gerstenhaber bracket and the BV operator, we use a discretized version of the foliations. Interpreted as chain level operations, these give natural definitions of up to homotopy algebras, which have the benefit of being concrete and small enough to be handleable, but large enough to contain all the necessary homotopies, such as those giving rise to brace operations. There are other natural modifications, which we discuss, that lead to moduli space actions, actions of certain compactifications, an open/closed generalization, and the possibility of stabilization in the case of semi-simple algebras. 255 325 1 10.4171/117-1/7 http://www.ems-ph.org/doi/10.4171/117-1/7

Cormac Walsh Ecole Polytechnique, PALAISEAU CEDEX, FRANCE General We show that the horofunction boundary of Teichmüller space with Thurston’s Lipschitz metric is the same as the Thurston boundary. We use this to determine the isometry group of the Lipschitz metric, apart from in some exceptional cases. We also show that the Teichmüller spaces of different surfaces, when endowed with this metric, are not isometric, again with some possible exceptions of low genus. 327 353 1 10.4171/117-1/8 http://www.ems-ph.org/doi/10.4171/117-1/8

Lixin Liu Sun Yat-sen (Zhongshan) University, GUANGZHOU, GUANGDONG, CHINA Weixu Su Fudan University, SHANGHAI, CHINA Several complex variables and analytic spaces Functions of a complex variable Manifolds and cell complexes General We show that the horofunction compactification of Teichmüller space, endowed with the Teichmüller metric, is homeomorphic to the Gardiner–Masur compactification. 355 374 1 10.4171/117-1/9 http://www.ems-ph.org/doi/10.4171/117-1/9

Hideki Miyachi Osaka University, OSAKA, JAPAN Functions of a complex variable Real functions Functional analysis General In this chapter, we develop the function theory on Teichmüller space with the aim of studying the extremal length geometry on that space. We focus on the Lipschitz algebra on Teichmüller space in terms of the Teichmüller distance. We first study the algebraic structure of the Lipschitz algebra by showing a version of the Stone–Weierstrass theorem. Next, we discuss a geometric aspect. Indeed, we show that there is a canonical continuous surjection from the Lipschitz compactification to the Gardiner–Masur compactification of Teichmüller space, and we realize the Gardiner-Masur compactification as a $Q$-compactification, where $Q$ is a subset in the Lipschitz algebra. As a corollary, we conclude that any Lipschitz mapping from a metric space $M$ to Teichmüller space extends continuously from the Lipschitz compactification of $M$ to the Gardiner–Masur compactification. We also define a subalgebra $\mathcal{GM}$ of the Lipschitz algebra on Teichmüller space such that the $\mathcal{GM}$-compactification coincides with the Gardiner–Masur compactification. 375 413 1 10.4171/117-1/10 http://www.ems-ph.org/doi/10.4171/117-1/10

Zhong Li Peking University, BEIJING, CHINA Functions of a complex variable Several complex variables and analytic spaces A finite-dimensional Teichmüller space is a geodesic straight space in the sense of Busemann. However, any infinite-dimensional Teichmüller space is not a geodesic straight space. At some kind of points in infinite-dimensional Teichmüller spaces, the geometric behavior of these points with respect to the base point is very “bad”. Such phenomena were sufficiently investigated. The study led to a complete characterization of such points and promoted a better understanding of the geometry of Teichmüller spaces. This topic is closely connected with the extremal problem of quasi-conformal mappings. 415 437 1 10.4171/117-1/11 http://www.ems-ph.org/doi/10.4171/117-1/11

Hiroshige Shiga Tokyo Institute of Technology, TOKYO, JAPAN General Holomorphic families of Riemann surfaces appear in various fields of mathematics. In this chapter, we shall give a comprehensive view of the role of holomorphic families of Riemann surfaces and their monodromies in related fields. 439 458 1 10.4171/117-1/12 http://www.ems-ph.org/doi/10.4171/117-1/12

Oliver Baues Georg-August Universität Göttingen, GÖTTINGEN, GERMANY Differential geometry Number theory Group theory and generalizations Manifolds and cell complexes The group action which defines the moduli problem for the deformation space of flat affine structures on the two-torus is the action of the affine group Aff(2) on $\mathbb{R}^2$. Since this action has non-compact stabiliser GL(2,$\mathbb{R}$), the underlying locally homogeneous geometry is highly non-Riemannian. In this chapter, we describe the deformation space of all flat affine structures on the two-torus. In this context interesting phenomena arise in the topology of the deformation space, which, for example, is not a Hausdorff space. This contrasts with the case of constant curvature metrics, or conformal structures on surfaces, which are encountered in classical Teichmüller theory. As our main result on the space of deformations of flat affine structures on the two-torus we prove that the holonomy map from the deformation space to the variety of conjugacy classes of homomorphisms from the fundamental group of the two-torus to the affine group is a local homeomorphism. 461 537 1 10.4171/117-1/13 http://www.ems-ph.org/doi/10.4171/117-1/13

Marc Burger ETH Zentrum, ZÜRICH, SWITZERLAND Alessandra Iozzi ETH Zentrum, ZÜRICH, SWITZERLAND Anna Wienhard Universität Heidelberg, HEIDELBERG, GERMANY Algebraic geometry Topological groups, Lie groups Several complex variables and analytic spaces We review the Fricke–Klein space of a surface with puncture from the point of view of the bounded Euler class and Goldman’s theorem and we extend this point of view to the study of representations of fundamental groups of punctures surfaces into Lie groups of Hermitian type. 539 618 1 10.4171/117-1/14 http://www.ems-ph.org/doi/10.4171/117-1/14

Gaven Martin Massey University, AUCKLAND, NEW ZEALAND Functions of a complex variable General This chapter presents a survey of the many and various elements of the modern higher-dimensional theory of quasiconformal mappings and their wide and varied application. It is unified (and limited) by the theme of the author’s interests. Thus we will discuss the basic theory as it developed in the 1960s in the early work of F. W. Gehring and Yu G. Reshetnyak and subsequently explore the connections with geometric function theory, nonlinear partial differential equations, differential and geometric topology and dynamics as they ensued over the following decades. We give few proofs as we try to outline the major results of the area and current research themes. We do not strive to present these results in maximal generality, as to achieve this considerable technical knowledge would be necessary of the reader. We have tried to give a feel of where the area is, what are the central ideas and problems and where are the major current interactions with researchers in other areas. We have also added a bit of history here and there. We have not been able to cover the many recent advances generalising the theory to mappings of finite distortion and to degenerate elliptic Beltrami systems which connects the theory closely with the calculus of variations and nonlinear elasticity, nonlinear Hodge theory and related areas, although the reader may see shadows of this aspect in parts. In the sequel (with Bruce Palka) we will give a more detailed account of the basic techniques and how they are used with a view to providing tools for researchers who may come in contact with higher-dimensional quasiconformal mappings from their own area. 619 677 1 10.4171/117-1/15 http://www.ems-ph.org/doi/10.4171/117-1/15

Katsuhiko Matsuzaki Waseda University, TOKYO, JAPAN Functions of a complex variable Dynamical systems and ergodic theory General In this chapter, we give a synthetic treatment concerning recent theories on the dynamics of Teichmüller modular groups for Riemann surfaces of infinite topological type. We consider quotient spaces of the Teichmüller space by subgroups of the modular group in order to find a moduli space in an appropriate sense. Among a glossary of fundamental results on this subject matter, we emphasize the construction of the stable moduli space and the enlarged moduli space, both of which are regarded as generalizations of the concept of moduli spaces from finite type Riemann surfaces to infinite type surfaces. 681 716 1 10.4171/117-1/16 http://www.ems-ph.org/doi/10.4171/117-1/16

Xavier Buff Université Paul Sabatier, TOULOUSE CEDEX, FRANCE Guizhen Cui Chinese Academy of Sciences, BEIJING, CHINA Lei Tan Université d'Angers, ANGERS CEDEX, FRANCE General One fundamental theorem in the theory of holomorphic dynamics is Thurston’s topological characterization of postcritically finite rational maps. Its proof is a beautiful application of Teichmüller theory. In this chapter we provide a self-contained proof of a slightly generalized version of Thurston’s theorem (the marked Thurston’s theorem). We also mention some applications and related results, as well as the notion of deformation spaces of rational maps introduced by A. Epstein. 717 756 1 10.4171/117-1/17 http://www.ems-ph.org/doi/10.4171/117-1/17

Ren Guo Oregon State University, CORVALLIS, UNITED STATES Manifolds and cell complexes Group theory and generalizations General In this chapter, we survey the algebraic aspects of quantum Teichmüller space, generalized Kashaev algebra and a natural relationship between the two algebras. 759 784 1 10.4171/117-1/18 http://www.ems-ph.org/doi/10.4171/117-1/18

Oswald Teichmüller , Berlin, GERMANY General This is the English translation of Teichmüller’s article “Veränderliche Riemannsche Flächen”, Deutsche Math. 7 (1944), 344–359. 787 803 1 10.4171/117-1/19 http://www.ems-ph.org/doi/10.4171/117-1/19

Annette A’Campo-Neuen Universität Basel, BASEL, SWITZERLAND Norbert A’Campo Universität Basel, BASEL, SWITZERLAND Lizhen Ji University of Michigan, ANN ARBOR, UNITED STATES Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Global analysis, analysis on manifolds Partial differential equations General This is a commentary on Teichmüller’s article “Veränderliche Riemannsche Flächen”, Deutsche Math. 7 (1944), 344–359. 787 814 1 10.4171/117-1/20 http://www.ems-ph.org/doi/10.4171/117-1/20

Volume III: Standard Information for Operators Erich Novak University of Jena, Germany Henryk Woźniakowski Columbia University, New York, USA, and University of Warsaw, Poland Numerical analysis 65-02; 65Y20, 68Q17, 68Q25, 41A63, 65-02, 46E22, 28C20, 46E30, 65N99, 65R20 Multivariate approximation, linear problems, quasilinear problems, Poisson equation, Fredholm equation, power of function values, worst case setting, average case setting, randomized setting, high-dimensional numerical problems, Smolyak and weighted tensor product algorithms, weighted spaces, tractability, curse of dimension This three-volume set is a comprehensive study of the tractability of multivariate problems. Volume I covers algorithms using linear information consisting of arbitrary continuous linear functionals. Volumes II and III are devoted to algorithms using standard information consisting of function values. Approximation of linear and selected nonlinear functionals is dealt with in volume II, and linear and selected nonlinear operators are studied in volume III. To a large extent, volume III can be read independently of volumes I and II. The most important example studied in volume III is the approximation of multivariate functions. It turns out that many other linear and some nonlinear problems are closely related to the approximation of multivariate functions. While the lower bounds obtained in volume I for the class of linear information also yield lower bounds for the standard class of function values, new techniques for upper bounds are presented in volume III. One of the main issues here is to verify when the power of standard information is nearly the same as the power of linear information. In particular, for the approximation problem defined over Hilbert spaces, the power of standard and linear information is the same in the randomized and average case (with Gaussian measures) settings, whereas in the worst case setting this is not true. The book is of interest to researchers working in computational mathematics, especially in approximation of high-dimensiona problems. It may be well suited for graduate courses and seminars. The text contains 58 open problems for future research in tractability. 10 29 2012 978-3-03719-116-3 978-3-03719-616-8 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/116 http://www.ems-ph.org/doi/10.4171/116 18 Tokyo 2011 Yujiro Kawamata University of Tokyo, Japan Algebraic geometry Commutative rings and algebras Associative rings and algebras Primary: 13D09, 14-02, 14-06, 14F05, 16E35, 18E30; Secondary: 13D02, 13D10, 14A22, 14E08, 14E16, 14E30, 14J32, 14J33, 14J45, 14K05, 14L24, 14N35, 16E05, 16E40, 16F60, 18G10, 20G05, 20G15 Algebraic variety, derived category, triangulated category, Fourier–Mukai transform, cluster algebra, birational geometry, semi-orthogonal decomposition, exceptional collection, minimal model, flop, McKay correspondence, categorical action, abelian variety, non-commutative algebraic geometry, mirror symmetry, Donaldson–Thomas theory The study of derived categories is a subject that attracts increasingly many young mathematicians from various fields of mathematics, including abstract algebra, algebraic geometry, representation theory and mathematical physics. The concept of the derived category of sheaves was invented by Grothendieck and Verdier in the 1960s as a tool to express important results in algebraic geometry such as the duality theorem. In the 1970s, Beilinson, Gelfand and Gelfand discovered that a derived category of an algebraic variety may be equivalent to that of a finite dimensional non-commutative algebra, and Mukai found that there are non-isomorphic algebraic varieties that have equivalent derived categories. In this way the derived category provides a new concept that has many incarnations. In the 1990s, Bondal and Orlov uncovered an unexpected parallelism between derived categories and birational geometry. Kontsevich’s homological mirror symmetry provided further motivation for the study of derived categories. This book is the proceedings of a conference held at the University of Tokyo in January 2011 on the current status of the research on derived categories related to algebraic geometry. Most articles are survey papers on this rapidly developing field. The book is suitable for young mathematicians who want to enter this exciting field. Some basic knowledge of algebraic geometry is assumed. 1 7 2013 978-3-03719-115-6 978-3-03719-615-1 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/115 http://www.ems-ph.org/doi/10.4171/115

Marcello Bernardara Université Paul Sabatier, TOULOUSE CEDEX 9, FRANCE Michele Bolognesi Université de Montpellier, MONTPELLIER, FRANCE Algebraic geometry Category theory; homological algebra General We define, basing upon semiorthogonal decompositions of ${\rm D}^{\rm b}(X)$, categorical representability of a projective variety $X$ and describe its relation with classical representabilities of the Chow ring. For complex threefolds satisfying both classical and categorical representability assumptions, we reconstruct the intermediate Jacobian from the semiorthogonal decomposition. We discuss finally how categorical representability can give useful information on the birational properties of $X$ by providing examples and stating open questions. 1 25 1 10.4171/115-1/1 http://www.ems-ph.org/doi/10.4171/115-1/1

Alberto Canonaco Università degli Studi di Pavia, PAVIA, ITALY Paolo Stellari Università degli Studi di Milano, MILANO, ITALY Algebraic geometry Category theory; homological algebra General In this article some recent results about Fourier–Mukai functors are reported. In particular, given an exact functor between the bounded derived categories of coherent sheaves on two smooth projective varieties, we deal with the question whether this functor is of Fourier–Mukai type. Several related questions are answered and many open problems are stated. 27 60 1 10.4171/115-1/2 http://www.ems-ph.org/doi/10.4171/115-1/2

Sabin Cautis University of Southern California, LOS ANGELES, UNITED STATES Algebraic geometry Nonassociative rings and algebras General Stratified flops show up in the birational geometry of symplectic varieties such as resolutions of nilpotent orbits and moduli spaces of sheaves. Constructing derived equivalences between varieties related by such flops is, strangely enough, related to areas in representation theory and knot homology. In this paper we discuss how to construct such equivalences, explain the main tool for doing this (categorical Lie algebra actions) and comment on various related topics. 61 101 1 10.4171/115-1/3 http://www.ems-ph.org/doi/10.4171/115-1/3

Akira Ishii Hiroshima University, HIROSHIMA, JAPAN Kazushi Ueda Osaka University, OSAKA, JAPAN Category theory; homological algebra Associative rings and algebras General We show that the quotient stack of a Fermat variety with respect to a natural abelian group action has a full strong exceptional collection consisting of invertible sheaves. We also discuss a description of the derived category of the Fermat variety in terms of the coherent action of the group of characters on the derived category of the quotient stack. 103 110 1 10.4171/115-1/4 http://www.ems-ph.org/doi/10.4171/115-1/4

Dmitry Kaledin Steklov Mathematical Institute, MOSCOW, RUSSIAN FEDERATION Algebraic topology General We prove a simple homological expression for the homology of a connected spectrum represented by an infinite loop space via the Segal machine. The expression is essentially due to Pirashvili but not stated explicitly in his work; we give an independent proof. 111 121 1 10.4171/115-1/5 http://www.ems-ph.org/doi/10.4171/115-1/5

Bernhard Keller Université Paris Diderot - Paris 7, PARIS CEDEX 13, FRANCE Commutative rings and algebras Associative rings and algebras General This is an introductory survey on cluster algebras and their (additive) categorification using derived categories of Ginzburg algebras. After a gentle introduction to cluster combinatorics, we review important examples of coordinate rings admitting a cluster algebra structure. We then present the general definition of a cluster algebra and describe the interplay between cluster variables, coefficients, $c$-vectors and $g$-vectors. We show how $c$-vectors appear in the study of quantum cluster algebras and their links to the quantum dilogarithm. We then present the framework of additive categorification of cluster algebras based on the notion of quiver with potential and on the derived category of the associated Ginzburg algebra. We show how the combinatorics introduced previously lift to the categorical level and how this leads to proofs, for cluster algebras associated with quivers, of some of Fomin–Zelevinsky’s fundamental conjectures. 123 183 1 10.4171/115-1/6 http://www.ems-ph.org/doi/10.4171/115-1/6

Izuru Mori Shizuoka University, SHIZUOKA, JAPAN Associative rings and algebras General It is interesting to find a (non-affine) smooth algebraic variety which is derived equivalent to a noncommutative algebra. A full strong exceptional sequence, a tilting generator, and McKay correspondence provide such examples. In this survey paper, we will show that similar derived equivalences exist between noncommutative schemes and noncommutative algebras. 185 196 1 10.4171/115-1/7 http://www.ems-ph.org/doi/10.4171/115-1/7

Alexander Polishchuk University of Oregon, EUGENE, UNITED STATES Algebraic geometry General We partially generalize the theory of semihomogeneous bundles on an abelian variety $A$ developed by Mukai, Semi-homogeneous vector bundles on an abelian variety, J. Math. Kyoto Univ. 18 (1978), 239–272. This involves considering abelian subvarieties $Y\subset X_A=A\times\hat{A}$ and studying coherent sheaves on $A$ invariant under the action of $Y$. The natural condition to impose on $Y$ is that of being Lagrangian with respect to a certain skew-symmetric biextension $\mathcal{E}$ of $X_A\times X_A$ by $\mathbb{G}_m$. We prove that in this case any $Y$-invariant sheaf is a direct sum of several copies of a single coherent sheaf. We call such sheaves Lagrangian-invariant (or LI-sheaves). We also study LI-functors $D^b(A)\to D^b(B)$ associated with kernels in $D^b(A\times B)$ that are invariant with respect to some Lagrangian subvariety in $X_A\times X_B$. We calculate their composition and prove that in characteristic zero it can be decomposed into a direct sum of LI-functors. In the case $B=A$ this leads to an interesting central extension of the group of symplectic automorphisms of $X_A$ in the category of abelian varieties up to isogeny. 197 250 1 10.4171/115-1/8 http://www.ems-ph.org/doi/10.4171/115-1/8

Mihnea Popa University of Illinois at Chicago, CHICAGO, UNITED STATES Algebraic geometry Commutative rings and algebras General The paper establishes a natural relationship between generic vanishing in the context of vanishing theorems and birational geometry, and perverse coherent sheaves in the context of derived categories. It includes criteria for checking these conditions, and applications. 251 278 1 10.4171/115-1/9 http://www.ems-ph.org/doi/10.4171/115-1/9

Christian Schnell Stony Brook University, STONY BROOK, UNITED STATES Algebraic geometry General We show that the fundamental group is not invariant under derived equivalence of smooth projective varieties. 279 285 1 10.4171/115-1/10 http://www.ems-ph.org/doi/10.4171/115-1/10

Yukinobu Toda The University of Tokyo, Kashiwa, JAPAN Algebraic geometry Category theory; homological algebra General In this article, we give an introduction to Donaldson–Thomas theory and a survey of its recent developments. We also discuss some open problems related to Donaldson–Thomas theory. 287 318 1 10.4171/115-1/11 http://www.ems-ph.org/doi/10.4171/115-1/11

Michel Van den Bergh Hasselt University, HASSELT, BELGIUM Commutative rings and algebras Algebraic geometry Category theory; homological algebra General In these notes we provide the foundation for the deformation-theoretic parts of arXiv:0807.3753 and arXiv:math/0102005. 319 344 1 10.4171/115-1/12 http://www.ems-ph.org/doi/10.4171/115-1/12

Impanga Lecture Notes Piotr Pragacz IM PAN, Warsaw, Poland Algebraic geometry Number theory Several complex variables and analytic spaces Differential geometry 11S15, 13D10, 14-02, 14B05, 14B12, 14C17, 14C20, 14C35, 14D06, 14D15, 14D20, 14D23, 14E15, 14E30, 14F43, 14H10, 14H40, 14H42, 14J17, 14J28, 14J30, 14J32, 14J50, 14J70, 14K10, 14K25, 14L30, 14M15, 14M17, 14M25, 14N10, 14N15, 32G10, 32Q45, 34A30, 53D05, 55N91; 01-02, 01A70, 05E05, 11S85, 13A35, 13D10, 14C30, 14F18, 14J26, 14J32, 14N20, 19D55, 32S25, 53D20, 57R45 K3 surface, Enriques surface, Calabi–Yau threefold, linear system, Seshadri constant, multiplier ideal, differential form, log canonical treshold, Mori theory, canonical ring, deformation of morphism, Hodge numbers, logarithmic differential form, Prym variety, moduli space, syzygy, Wro´nski determinant, linear ODE, ramification locus, Schubert calculus, Grassmannian, Schubert variety, Schur function, singularity, Thom polynomial, P-ideal, equivariant localization, toric variety, symplectic manifold, toric stack, symplectic quotient, equivariant cohomology, Bloch group, field extension, norm The articles in this volume are the outcome of the Impanga Conference on Algebraic Geometry in 2010 at the Banach Center in Będlewo. The following spectrum of topics is covered: K3 surfaces and Enriques surfaces; Prym varieties and their moduli; invariants of singularities in birational geometry; differential forms on singular spaces; Minimal Model Program; linear systems; toric varieties; Seshadri and packing constants; equivariant cohomology; Thom polynomials; arithmetic questions. The main purpose of the volume is to give comprehensive introductions to the above topics through texts starting from an elementary level and ending with the discussion of current research. The first four topics are represented by the notes from the minicourses held during the conference. In the articles the reader will find classical results and methods, as well as modern ones. The book is addressed to researchers and graduate students in algebraic geometry, singularity theory and algebraic topology. Most of the material exposed in the volume has not yet appeared in book form. 8 15 2012 978-3-03719-114-9 978-3-03719-614-4 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/114 http://www.ems-ph.org/doi/10.4171/114

Piotr Blass Ulam University, Boynton Beach, UNITED STATES Algebraic geometry History and biography General We describe some aspects of the life and mathematical work of Oscar Zariski. In particular, we discuss the influence of his work and the work of his students on modern algebraic geometry. 1 15 1 10.4171/114-1/1 http://www.ems-ph.org/doi/10.4171/114-1/1

Klaus Altmann Freie Universität Berlin, BERLIN, GERMANY Nathan Ilten , BONN, GERMANY Lars Petersen Freie Universität Berlin, BERLIN, GERMANY Hendrik Süß Brandenburgische Technische Universität Cottbus, COTTBUS, GERMANY Robert Vollmert Freie Universität Berlin, BERLIN, GERMANY Algebraic geometry General This is a survey of the language of polyhedral divisors describing T-varieties. This language is explained in parallel to the well established theory of toric varieties. In addition to basic constructions, subjects touched on include singularities, separatedness and properness, divisors and intersection theory, cohomology, Cox rings, polarizations, and equivariant deformations, among others. 17 69 1 10.4171/114-1/2 http://www.ems-ph.org/doi/10.4171/114-1/2

Dave Anderson University of Washington, SEATTLE, UNITED STATES Algebraic geometry Combinatorics General Introduced by Borel in the late 1950s, equivariant cohomology encodes information about how the topology of a space interacts with a group action. Quite some time passed before algebraic geometers picked up on these ideas, but in the last twenty years, equivariant techniques have found many applications in enumerative geometry, Gromov–Witten theory, and the study of toric varieties and homogeneous spaces. In fact, many classical algebro-geometric notions, going back to the degeneracy locus formulas of Giambelli, are naturally statements about certain equivariant cohomology classes. These lectures survey some of the main features of equivariant cohomology at an introductory level. The first part is an overview, including basic definitions and examples. In the second lecture, I discuss one of the most useful aspects of the theory: the possibility of localizing at fixed points without losing information. The third lecture focuses on Grassmannians, and describes some recent positivity results about their equivariant cohomology rings. 71 92 1 10.4171/114-1/3 http://www.ems-ph.org/doi/10.4171/114-1/3

Thomas Bauer Philipps-Universität, MARBURG, GERMANY Cristiano Bocci Università di Siena, SIENA, ITALY Susan Cooper University of Nebraska-Lincoln, LINCOLN, UNITED STATES Sandra Di Rocco Royal Institute of Technology, STOCKHOLM, SWEDEN Marcin Dumnicki Jagiellonian University, KRAK&#211;W, POLAND Brian Harbourne University of Nebraska-Lincoln, LINCOLN, UNITED STATES Anders Lindquist Royal Institute of Technology, STOCKHOLM, SWEDEN Hans Munthe-Kaas University of Bergen, BERGEN, NORWAY Alex Küronya Budapest University of Technology & Economics, BUDAPEST, HUNGARY Rick Miranda Colorado State University, FORT COLLINS, UNITED STATES Joaquim Roé Universitat Autònoma de Barcelona, BELLATERRA, SPAIN Henry Schenck University of Illinois, URBANA, UNITED STATES Tomasz Szemberg Krakow Pedagogical Academy, KRAK&#211;W, POLAND Zach Teitler Boise State University, BOISE, UNITED STATES Algebraic geometry General This article presents problems discussed during the Mini-Workshop Linear Series on Algebraic Varieties held in the week of October 3–9, 2010, at the Mathematisches Forschungsinstitut at Oberwolfach, as well as results obtained there and shortly after the Workshop. The discussion revolves mainly around the speciality and the postulation problems as well as the containment problems for various powers of ideals. The main motivations originated in the Harbourne-Hirschowitz, Nagata and the Bounded Negativity Conjectures. 93 140 1 10.4171/114-1/4 http://www.ems-ph.org/doi/10.4171/114-1/4

Gergely Bérczi Oxford University, OXFORD, UNITED KINGDOM Algebraic topology Algebraic geometry Several complex variables and analytic spaces General This survey paper – based on my IMPANGA lectures given in the Banach Center, Warsaw in January 2011 – studies the moduli of holomorphic map germs from the complex line into complex compact manifolds with applications in global singularity theory and the theory of hyperbolic algebraic varieties. 141 167 1 10.4171/114-1/5 http://www.ems-ph.org/doi/10.4171/114-1/5

Paolo Cascini Imperial College London, LONDON, UNITED KINGDOM Vladimir Lazić Universität Bayreuth, BAYREUTH, GERMANY General We give a light introduction to some recent developments in Mori theory, and to our recent direct proof of the finite generation of the canonical ring. 169 187 1 10.4171/114-1/6 http://www.ems-ph.org/doi/10.4171/114-1/6

Sławomir Cynk Jagiellonian University, KRAK&#211;W, POLAND Sławomir Rams Jagiellonian University, KRAK&#211;W, POLAND Algebraic geometry Several complex variables and analytic spaces General In this survey paper we discuss various results on invariants of a resolution of a singular projective hypersurface. In particular, we prove certain defect-type formulae. In our proofs we make essential use of the sheaf of logarithmic differential forms. 189 213 1 10.4171/114-1/7 http://www.ems-ph.org/doi/10.4171/114-1/7

Gavril Farkas Humboldt-Universität zu Berlin, BERLIN, GERMANY Algebraic geometry Ordinary differential equations General This survey discusses the geometry of the moduli space of Prym varieties. Several applications of Prym in algebraic geometry are presented. The paper begins with with a historical discussion of the life and achievements of Friedrich Prym. Topics treated in subsequent sections include singularities and Kodaira dimension of the moduli space, syzygies of Prym-canonical embedding and the geometry of the moduli space $\mathcal{R}_g$ in small genus. 215 255 1 10.4171/114-1/8 http://www.ems-ph.org/doi/10.4171/114-1/8

Letterio Gatto Politecnico di Torino, TORINO, ITALY Inna Scherbak Tel Aviv University, TEL AVIV, ISRAEL Algebraic geometry Ordinary differential equations General The Wroński determinant (Wrońskian), usually introduced in standard courses in Ordinary Differential Equations (ODE), is a very useful tool in algebraic geometry to detect ramification loci of linear systems. The present survey aims to describe some “materializations” of the Wrońskian and of its close relatives, the generalized Wrońskians, in algebraic geometry. Emphasis will be put on the relationships between Schubert Calculus and ODE. 257 295 1 10.4171/114-1/9 http://www.ems-ph.org/doi/10.4171/114-1/9

Kevin Hutchinson University College Dublin, DUBLIN, IRELAND Masha Vlasenko Trinity College, DUBLIN, IRELAND Algebraic geometry $K$-theory General According to B. Totaro (Milnor K-theory is the simplest part of algebraic K-theory, K-theory 6 (1992), 177–189), there is a hope that the Chow groups of a field $k$ can be computed using a very small class of affine algebraic varieties (linear spaces in the right coordinates), whereas the current definition uses essentially all algebraic cycles in affine space. In this note we consider a simple modification of $\mathrm{CH}^2(\operatorname{Spec} (k),3)$ using only linear subvarieties in affine spaces and show that it maps surjectively to the Bloch group $B(k)$ for any infinite field $k$. We also describe the kernel of this map. 297 303 1 10.4171/114-1/10 http://www.ems-ph.org/doi/10.4171/114-1/10

Andreas Hochenegger Freie Universität Berlin, BERLIN, GERMANY Frederik Witt Universität Münster, MÜNSTER, GERMANY Algebraic geometry Differential geometry General Toric varieties play an important role both in symplectic and complex geometry. In symplectic geometry, the construction of a symplectic toric manifold from a smooth polytope is due to Delzant [D]. In algebraic geometry, there is a more general construction using fans rather than polytopes. However, in case the fan is induced by a smooth polytope Audin [Au] showed both constructions to give isomorphic projective varieties. For rational but not necessarily smooth polytopes the Delzant construction was refined by Lerman and Tolman [LT], leading to symplectic toric orbifolds or more generally, symplectic toric DM stacks [LM]. We show that the stacks resulting from the Lerman–Tolman construction are isomorphic to the stacks obtained by Borisov et al. [BCS] in case the stacky fan is induced by a polytope. No originality is claimed (cf. also the article by Sakai [S]). Rather we hope that this text serves as an example driven introduction to symplectic toric geometry for the algebraically minded reader. 305 331 1 10.4171/114-1/11 http://www.ems-ph.org/doi/10.4171/114-1/11

Clemens Jörder Universität Freiburg, FREIBURG, GERMANY Stefan Kebekus Universität Freiburg, FREIBURG, GERMANY Algebraic geometry Commutative rings and algebras Several complex variables and analytic spaces General Let $f \colon Y \to X$ be the inclusion map of a compact reduced subspace of a complex manifold, and let $\mathcal{F} \subseteq T_X$ be a subsheaf of the tangent bundle which is closed under the Lie bracket, but not necessarily a sheaf of $\mathcal{O}_X$-algebras. This paper discusses criteria to guarantee that infinitesimal deformations of $f$ which are induced by $\mathcal{F}$ lift to positive-dimensional deformations of $f$, where $f$ is deformed “along the sheaf $\mathcal{F}$”. In case where $X$ is complex-symplectic and $\mathcal{F}$ the sheaf of locally Hamiltonian vector fields, this partially reproduces known results on unobstructedness of deformations of Lagrangian submanifolds. The proof is rather elementary and geometric, constructing higher-order liftings of a given infinitesimal deformation using flow maps of carefully crafted time-dependent vector fields. 333 357 1 10.4171/114-1/12 http://www.ems-ph.org/doi/10.4171/114-1/12

Michał Kapustka Jagiellonian University, KRAK&#211;W, POLAND Algebraic geometry General We introduce a variety $\hat{G}_2$ parameterizing isotropic five-spaces of a general degenerate four-form in a seven dimensional vector space. It is in a natural way a degeneration of the variety $G_2$, the adjoint variety of the simple Lie group $\mathbb{G}_2$. It occurs that it is also the image of $\mathbb{P}^5$ by a system of quadrics containing a twisted cubic. Degenerations of this twisted cubic to three lines give rise to degenerations of $G_2$ which are toric Gorenstein Fano fivefolds. We use these two degenerations to construct geometric transitions between Calabi–Yau threefolds. We prove moreover that every polarized K3 surface of Picard number 2, genus 10, and admitting a $g^1_5$ appears as linear sections of the variety $\hat{G}_2$. 359 373 1 10.4171/114-1/13 http://www.ems-ph.org/doi/10.4171/114-1/13

Mateusz Michałek Jagiellonian University, KRAK&#211;W, POLAND Algebraic geometry General The present paper contains the notes taken by Mateusz Michałek from a series of lectures by Stefan Kebekus during IMPANGA Summer School 2010. The aim of this series of lectures is to give an exposition on the extension of well known results concerning differential forms on manifolds to the case of normal varieties. 375 388 1 10.4171/114-1/14 http://www.ems-ph.org/doi/10.4171/114-1/14

Shigeru Mukai Kyoto University, KYOTO, JAPAN Algebraic geometry General The main aim of these lectures is to study the connection between symplectic symmetries of $K3$ surfaces and the Mathieu group $M_{24}$, and its Enriques analogy, that is, a conjectural connection between semi-symplectic symmetries of Enriques surfaces and another Mathieu group $M_{12}$. 389 405 1 10.4171/114-1/15 http://www.ems-ph.org/doi/10.4171/114-1/15

Mircea Mustaţă University of Michigan, ANN ARBOR, UNITED STATES Algebraic geometry Commutative rings and algebras General We give an introduction to log canonical thresholds, and discuss some open problems and recent progress. 407 442 1 10.4171/114-1/16 http://www.ems-ph.org/doi/10.4171/114-1/16

Özer Öztürk Mimar Sinan Fine Arts University, BEŞIKTAŞ/ISTANBUL, TURKEY Piotr Pragacz Polish Academy of Sciences, WARSAW 10, POLAND Combinatorics Algebraic geometry Manifolds and cell complexes General We discuss computations of the Thom polynomials of singularity classes of maps in the basis of Schur functions. We survey the known results about the bound on the length and a rectangle containment for partitions appearing in such Schur function expansions. We describe several recursions for the coefficients. For some singularities, we give old and new computations of their Thom polynomials. 443 479 1 10.4171/114-1/17 http://www.ems-ph.org/doi/10.4171/114-1/17

Marek Szyjewski , KATOWICE, POLAND Number theory Field theory and polynomials General We investigate kernel of the norm map on power classes for cyclic field extensions. 481 487 1 10.4171/114-1/18 http://www.ems-ph.org/doi/10.4171/114-1/18

Halszka Tutaj-Gasińska Jagiellonian University, KRAK&#211;W, POLAND Algebraic geometry Differential geometry General This note is about a certain connection between Seshadri constants and symplectic packing. 489 499 1 10.4171/114-1/19 http://www.ems-ph.org/doi/10.4171/114-1/19

Peter Roquette University of Heidelberg, Germany History and biography 01-02, 03-03, 11-03, 12-03 , 16-03, 20-03; 01A60, 01A70, 01A75, 11E04, 11E88, 11R18 11R37, 11U10 The 20th century was a time of great upheaval and great progress, mathematics not excluded. In order to get the overall picture of trends, developments and results it is illuminating to look at their manifestations locally, in the personal life and work of people living at the time. The university archives of Göttingen harbor a wealth of papers, letters and manuscripts from several generations of mathematicians – documents which tell us the story of the historic developments from a local point of view. The present book offers a number of essays based on documents from Göttingen and elsewhere – essays which are not yet contained in the author’s Collected Works. These little pieces, independent from each other, are meant as contributions to the imposing mosaic of history of number theory. They are written for mathematicians but with no special background requirements. Involved are the names of Abraham Adrian Albert, Cahit Arf, Emil Artin, Richard Brauer, Otto Grün, Helmut Hasse, Klaus Hoechsmann, Robert Langlands, Heinrich-Wolfgang Leopoldt, Emmy Noether, Abraham Robinson, Ernst Steinitz, Hermann Weyl and others. 1 24 2013 978-3-03719-113-2 978-3-03719-613-7 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/113 http://www.ems-ph.org/doi/10.4171/113 Piotr Nowak IM PAN, Warsaw, Poland Guoliang Yu Texas A&M University, College Station, USA Geometry Group theory and generalizations 51-01; 50F99, 20F69; 19K56, 57-01, 46L87, 58B34, 46L99, 53C20 Large scale geometry, quasi-isometry, coarse geometry, asymptotic dimension, finite decomposition complexity, amenability, Property A, coarse embedding, expanders, a-T-menability, coarse homology, uniformly finite homology, Baum–Connes conjecture, coarse Baum–Connes conjecture, Novikov conjecture, Borel conjecture Large scale geometry is the study of geometric objects viewed from a great distance. The idea of large scale geometry can be traced back to Mostow’s work on rigidity and the work of Švarc, Milnor and Wolf on growth of groups. In the last decades, large scale geometry has found important applications in group theory, topology, geometry, higher index theory, computer science, and large data analysis. This book provides a friendly approach to the basic theory of this exciting and fast growing subject and offers a glimpse of its applications to topology, geometry, and higher index theory. The authors have made a conscientious effort to make the book accessible to advanced undergraduate students, graduate students, and non-experts. 10 10 2012 978-3-03719-112-5 978-3-03719-612-0 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/112 http://www.ems-ph.org/doi/10.4171/112 Translated from the Catalan by Ignacio Monreal Joaquim Bruna Universitat Autònoma de Barcelona, Spain Julià Cufí Universitat Autònoma de Barcelona, Spain Functions of a complex variable Several complex variables and analytic spaces 30-01, 31-01 Power series, holomorphic function, line integral, differential form, analytic function, zeros and poles, residues, simply connected domain, harmonic function, Dirichlet problem, Poisson equation, conformal mapping, homographic transformation, meromorphic function, infinite product, entire function, interpolation, band-limited function The theory of functions of a complex variable is a central theme in mathematical analysis that has links to several branches of mathematics. Understanding the basics of the theory is necessary for anyone who wants to have a general mathematical training or for anyone who wants to use mathematics in applied sciences or technology. The book presents the basic theory of analytic functions of a complex variable and their points of contact with other parts of mathematical analysis. This results in some new approaches to a number of topics when compared to the current literature on the subject. Some issues covered are: a real version of the Cauchy–Goursat theorem, theorems of vector analysis with weak regularity assumptions, an approach to the concept of holomorphic functions of real variables, Green’s formula with multiplicities, Cauchy’s theorem for locally exact forms, a study in parallel of Poisson’s equation and the inhomogeneous Cauchy–Riemann equations, the relationship between Green’s function and conformal mapping, the connection between the solution of Poisson’s equation and zeros of holomorphic functions, and the Whittaker–Shannon theorem of information theory. The text can be used as a manual for complex variable courses of various levels and as a reference book. The only prerequisites for reading it is a working knowledge of the topology of the plane and the differential calculus for functions of several real variables. A detailed treatment of harmonic functions also makes the book useful as an introduction to potential theory. 5 6 2013 978-3-03719-111-8 978-3-03719-611-3 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/111 http://www.ems-ph.org/doi/10.4171/111 Koen Thas Ghent University, Belgium Combinatorics Group theory and generalizations Geometry 05-02, 20-02, 51-02; 05B25, 05E18, 20B25, 20D15, 20D20, 51B25, 51E12 Generalized quadrangle, elation group, Moufang condition, p-group The notion of elation generalized quadrangle is a natural generalization to the theory of generalized quadrangles of the important notion of translation planes in the theory of projective planes. Almost any known class of finite generalized quadrangles can be constructed from a suitable class of elation quadrangles. In this book the author considers several aspects of the theory of elation generalized quadrangles. Special attention is given to local Moufang conditions on the foundational level, exploring for instance a question of Knarr from the 1990s concerning the very notion of elation quadrangles. All the known results on Kantor’s prime power conjecture for finite elation quadrangles are gathered, some of them published here for the first time. The structural theory of elation quadrangles and their groups is heavily emphasized. Other related topics, such as p-modular cohomology, Heisenberg groups and existence problems for certain translation nets, are briefly touched. The text starts from scratch and is essentially self-contained. Many alternative proofs are given for known theorems. Containing dozens of exercises at various levels, from very easy to rather difficult, this course will stimulate undergraduate and graduate students to enter the fascinating and rich world of elation quadrangles. The more accomplished mathematician will especially find the final chapters challenging. 6 13 2012 978-3-03719-110-1 978-3-03719-610-6 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/110 http://www.ems-ph.org/doi/10.4171/110 Alain-Sol Sznitman ETH Zürich, Switzerland Probability theory and stochastic processes Statistical mechanics, structure of matter 60K35, 60J27, 60G15, 82B41 Occupation times, Gaussian free field, Markovian loop, random interlacements This book grew out of a graduate course at ETH Zurich during the Spring term 2011. It explores various links between such notions as occupation times of Markov chains, Gaussian free fields, Poisson point processes of Markovian loops, and random interlacements, which have been the object of intensive research over the last few years. These notions are developed in the convenient set-up of finite weighted graphs endowed with killing measures. The book first discusses elements of continuous-time Markov chains, Dirichlet forms, potential theory, together with some consequences for Gaussian free fields. Next, isomorphism theorems and generalized Ray-Knight theorems, which relate occupation times of Markov chains to Gaussian free fields, are pre- sented. Markovian loops are constructed and some of their key properties derived. The field of occupation times of Poisson point processes of Markovian loops is investigated. Of special interest are its connection to the Gaussian free field, and a formula of Symanzik. Finally, links between random interlacements and Markovian loops are discussed, and some further connections with Gaussian free fields are mentioned. 5 11 2012 978-3-03719-109-5 978-3-03719-609-0 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/109 http://www.ems-ph.org/doi/10.4171/109 Volodymyr Mazorchuk Uppsala University, Sweden Category theory; homological algebra Nonassociative rings and algebras Manifolds and cell complexes 18-01, 18D05; 17B10, 17B55, 18A40, 18E30, 57M27 Categorification, 2-category, Lie algebra, module, category O, functor, knot, Jones polynomial, Verma module, translation functor The term “categorification” was introduced by Louis Crane in 1995 and refers to the process of replacing set-theoretic notions by the corresponding category-theoretic analogues. This text mostly concentrates on algebraical aspects of the theory, presented in the historical perspective, but also contains several topological applications, in particular, an algebraic (or, more precisely, representation-theoretical) approach to categorification. It consists of fifteen sections corresponding to fifteen one-hour lectures given during a Master Class at Aarhus University, Denmark in October 2010. There are some exercises collected at the end of the text and a rather extensive list of references. Video recordings of all (but one) lectures are available from the Master Class website. The book provides an introductory overview of the subject rather than a fully detailed monograph. Emphasis is on definitions, examples and formulations of the results. Most proofs are either briefly outlined or omitted. However, complete proofs can be found by tracking references. It is assumed that the reader is familiar with the basics of category theory, representation theory, topology and Lie algebra. 3 16 2012 978-3-03719-108-8 978-3-03719-608-3 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/108 http://www.ems-ph.org/doi/10.4171/108 Hans Triebel University of Jena, Germany Functional analysis Approximations and expansions Fourier analysis Computer science 46-02, 46E35, 42C40, 42B35, 68Q17, 41A55 Function spaces, Haar bases, Faber bases, Faber frames, numerical integration, discrepancy This book deals first with Haar bases, Faber bases and Faber frames for weighted function spaces on the real line and the plane. It extends results in the author’s book Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration (EMS, 2010) from unweighted spaces (preferably in cubes) to weighted spaces. The obtained assertions are used to study sampling and numerical integration in weighted spaces on the real line and weighted spaces with dominating mixed smoothness in the plane. A short chapter deals with the discrepancy for spaces on intervals. The book is addressed to graduate students and mathematicians having a working knowledge of basic elements of function spaces and approximation theory. 3 16 2012 978-3-03719-107-1 978-3-03719-607-6 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/107 http://www.ems-ph.org/doi/10.4171/107 Joachim Krieger EPFL Lausanne, Switzerland Wilhelm Schlag University of Chicago, USA Partial differential equations Differential geometry 35L05, 35L52, 53Z05 Energy critical wave equations, wave maps, concentration compactness, Bahouri–Gérard decomposition, Kenig–Merle method, hyperbolic plane Wave maps are the simplest wave equations taking their values in a Riemannian manifold $(M,g)$. Their Lagrangian is the same as for the scalar equation, the only difference being that lengths are measured with respect to the metric $g$. By Noether's theorem, symmetries of the Lagrangian imply conservation laws for wave maps, such as conservation of energy. In coordinates, wave maps are given by a system of semilinear wave equations. Over the past 20 years important methods have emerged which address the problem of local and global wellposedness of this system. Due to weak dispersive effects, wave maps defined on Minkowski spaces of low dimensions, such as $\mathbb R^{2+1}_{t,x}$, present particular technical difficulties. This class of wave maps has the additional important feature of being energy critical, which refers to the fact that the energy scales exactly like the equation. Around 2000 Daniel Tataru and Terence Tao, building on earlier work of Klainerman–Machedon, proved that smooth data of small energy lead to global smooth solutions for wave maps from 2+1 dimensions into target manifolds satisfying some natural conditions. In contrast, for large data, singularities may occur in finite time for $M =\mathbb S^2$ as target. This monograph establishes that for $\mathbb H$ as target the wave map evolution of any smooth data exists globally as a smooth function. While we restrict ourselves to the hyperbolic plane as target the implementation of the concentration-compactness method, the most challenging piece of this exposition, yields more detailed information on the solution. This monograph will be of interest to experts in nonlinear dispersive equations, in particular to those working on geometric evolution equations. 2 9 2012 978-3-03719-106-4 978-3-03719-606-9 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/106 http://www.ems-ph.org/doi/10.4171/106 Athanase Papadopoulos IRMA, Strasbourg, France Geometry Manifolds and cell complexes Primary 51-01, 51-02, 57-01, 57-02; Secondary 14H30, 14H52, 20F67, 20F69, 22E40, 22D40, 30F10, 30F20, 30F45, 30F60, 32G15, 37E30, 51F99, 51M10, 51E24, 53C21, 53C22, 53C23, 53C35, 53C70, 54C20, 57M15, 57M20, 57M27, 57M50, 57N10 Hyperbolic geometry, hyperbolic space, neutral geometry, Euclid’s axioms, hyperbolic trigonometry, spherical geometry, Khayyam–Saccheri quadrilaterals, trirectangular quadrilaterals, parallelism, horocycle, Lobachevsky parallelism function, Beltrami–Klein This book contains carefully revised and expanded versions of eight courses that were presented at the University of Strasbourg, during two geometry master classes, in 2008 and 2009. The aim of the master classes was to give to fifth-year students and PhD students in mathematics the opportunity to learn new topics that lead directly to the current research in geometry and topology. The courses were held by leading experts. The subjects treated include hyperbolic geometry, three-manifold topology, representation theory of fundamental groups of surfaces and of three-manifolds, dynamics on the hyperbolic plane with applications to number theory, Riemann surfaces, Teichmüller theory, Lie groups and asymptotic geometry. The text is addressed to students and mathematicians who wish to learn the subject. It can also be used as a reference book and as a textbook for short courses on geometry. 1 18 2012 978-3-03719-105-7 978-3-03719-605-2 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/105 http://www.ems-ph.org/doi/10.4171/105 18

Norbert A’Campo Universität Basel, BASEL, SWITZERLAND Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Differential geometry Geometry General These are notes on plane hyperbolic geometry. The presentation is rather unusual since it is model-free. The methods can be used in the same way for spherical geometry. In particular, we derive the trigonometric formulae without using any Euclidean model. After a brief introduction to the axioms and the basic notions, we study in detail the geometry of triangles and of quadrilaterals. We discuss the notions of area and of dissection, and we derive the trigonometric formulae. Then we present two models of the hyperbolic plane, namely, a disk model whose automorphisms are projective maps, and a model that arises from algebra, whose points are the prime ideals of the ring of polynomials with one real variable. We study parallelism, horocycles, parabolic transformations and the Lobachevsky parallelism function. Finally, we describe a 3-dimensional space which contains the hyperbolic plane, the Euclidean plane and the elliptic space of spherical geometry. In this 3-dimensional space, we can make continuous transitions between objects in the three 2-dimensional geometries (hyperbolic, Euclidean and spherical) and we can watch the transformation of their geometrical properties. 1 182 1 10.4171/105-1/1 http://www.ems-ph.org/doi/10.4171/105-1/1

Françoise Dal’Bo Université de Rennes I, RENNES CEDEX, FRANCE Dynamical systems and ergodic theory Number theory Group theory and generalizations Geometry The motivation of this text is to explain through two elementary examples some links between hyperbolic geometry and another domains of the mathematics. In the first example, we establish relations between the linear orbits of the discrete subgroups of SL(2,R) and the horocyclic trajectories on the hyperbolic surfaces. In the second one, we use the topology of the geodesic trajectories on the modular surface to obtain results in the theory of Diophantine approximations. 183 232 1 10.4171/105-1/2 http://www.ems-ph.org/doi/10.4171/105-1/2

Frank Herrlich Karlsruhe Institute of Technology, KARLSRUHE, GERMANY Algebraic geometry Functions of a complex variable Several complex variables and analytic spaces Manifolds and cell complexes Origamis are translation surfaces that arise from certain coverings of elliptic curves. We give four different characterizations of them, in combinatorial as well as in algebro-geometric terms. By affine variation of the translation structure, an origami defines an embedding of the upper half plane into Teichmüller space; its image in moduli space is an algebraic curve. As a Riemann surface, this algebraic curve is uniformized by the Veech group of the origami, a group closely related to the affine homeomorphisms of the translation surface. We explain these concepts and their interrelations, and illustrate them by examples. In particular we discuss the very helpful example of the quaternion origami or “Eierlegende Wollmilchsau”. 233 253 1 10.4171/105-1/3 http://www.ems-ph.org/doi/10.4171/105-1/3

Philipp Korablev Chelyabinsk State University, CHELYABINSK, RUSSIAN FEDERATION Sergey Matveev Chelyabinsk State University, CHELYABINSK, RUSSIAN FEDERATION Manifolds and cell complexes General The article consists of five parts intended for graduate students of mathematics or researchers seeking to extend their mathematical knowledge in mathematics, in particular, in 3-manifold topology. The first section is devoted to describing the most popular classical ways for presenting 3-manifolds: triangulations, Heegaard splittings, surgery along framed links, and special spines. The latter is used in the next section for an elementary description of “quantum” or “state sum” invariants of 3-manifolds discovered by V. Turaev and O. Viro. This description includes an explicit geometric construction of the first nontrivial invariant of TV-type, which can be calculated by looking at a special spine of a given 3-manifold. The third section is short but informative. It contains a very clear explanation of the famous JSJ-decomposition theorem, including its relation to the Thurston geometrization conjecture solved by G. Perelman. The next section is an exposition of the Haken theory of normal surfaces, which is one of the main tools for investigating 3-manifolds. It is very important since it explains how to construct the JSJ-decompositions algorithmically. The last section is devoted to description of the main steps in the proof of the algorithmic classification theorem for sufficiently large 3-manifolds, which includes algorithmic classification of knots. 255 284 1 10.4171/105-1/4 http://www.ems-ph.org/doi/10.4171/105-1/4

Gabriele Link Karlsruhe Institute of Technology, KARLSRUHE, GERMANY Differential geometry Topological groups, Lie groups General These notes provide an introduction to globally symmetric spaces with an emphasis on those of non-compact type. We intend to give a guideline through a part of the landscape rather than proving every detail; however, where proofs are omitted, the reader gets a precise reference. The first part of the text gives an overview on the geometry and algebraic coding of arbitrary globally symmetric spaces. We then restrict ourselves to symmetric spaces of non-compact type and describe the Iwasawa and Cartan decomposition in the second part. The last part is devoted to the study of the geometry at infinity of globally symmetric spaces of non-compact type. We give descriptions of the geometric boundary, the Furstenberg boundary and the Bruhat decomposition, which help to identify the pairs of boundary points which can be joined by a geodesic or flat. Finally we show how invariant Finsler distances on the globally symmetric space can be constructed with the help of Busemann functions. 285 332 1 10.4171/105-1/5 http://www.ems-ph.org/doi/10.4171/105-1/5

Julien Marché École Polytechnique, PALAISEAU, FRANCE Manifolds and cell complexes Dynamical systems and ergodic theory Quantum theory General These notes of a course given at IRMA in April 2009 cover some aspects of the representation theory of fundamental groups of manifolds of dimension at most 3 in compact Lie groups, mainly SU(2). We give detailed examples, develop the techniques of twisted cohomology and gauge theory. We review Chern–Simons theory and describe an integrable system for the representation space of a surface. Finally, we explain some basic ideas on geometric quantization. We apply them to the case of representation spaces by computing Bohr–Sommerfeld orbits with metaplectic correction. 333 370 1 10.4171/105-1/6 http://www.ems-ph.org/doi/10.4171/105-1/6

Carlo Petronio Università di Pisa, PISA, ITALY Manifolds and cell complexes General This survey article describes the algorithmic approaches successfully used over the time to construct hyperbolic structures on various 3-dimensional topological “objects” (cusped manifolds, manifolds with boundary, links, graphs, orbifolds). It also exposes the classification techniques based on hyperbolic geometry and the experimental classification results of several classes of 3-dimensional topological “objects” obtained over the time. 371 404 1 10.4171/105-1/7 http://www.ems-ph.org/doi/10.4171/105-1/7

Viktor Schroeder University of Zürich, ZÜRICH, SWITZERLAND Geometry Differential geometry General This article gives an introduction to certain aspects of the asymptotic geometry of metric spaces. Thus the focus is on the large scale geometry of a space while the local structure is neglected. In particular hyperbolic spaces (in the sense of Gromov) are discussed, for which the asymptotic geometry is encoded in the boundary at infinity. In analogy with the classical hyperbolic space and its boundary, the relation between the metric geometry of a Gromov hyperbolic space and the Möbius geometry of its boundary is studied in detail. 405 454 1 10.4171/105-1/8 http://www.ems-ph.org/doi/10.4171/105-1/8

Corrected 2nd printing, September 2013 Peter Neumann University of Oxford, UK History and biography Group theory and generalizations 01-02; 01-55, 01A75, 00B55, 11-03, 11A55, 12-03, 12E12, 12E20, 12F10, 20-02, 20-03, 20B05, 20B15, 20D05, 33-03, 33E05 History of mathematics, Galois, Galois Theory, group, Galois group, equation, theory of equations, Galois field, finite field, elliptic function, modular equation, primitive equation, primitive group, solubility, simple group, soluble group Although Évariste Galois was only 20 years old when he died, shot in a mysterious early-morning duel in 1832, his ideas, when they were published 14 years later, changed the course of algebra. He invented what is now called Galois Theory, the modern form of what was classically the Theory of Equations. For that purpose, and in particular to formulate a precise condition for solubility of equations by radicals, he also invented groups and began investigating their theory. His main writings were published in French in 1846 and there have been a number of French editions culminating in the great work published by Bourgne & Azra in 1962 containing transcriptions of every page and fragment of the manuscripts that survive. Very few items have been available in English up to now. The present work contains English translations of almost all the Galois material. They are presented alongside a new transcription of the original French, and are enhanced by three levels of commentary. An introduction explains the context of Galois' work, the various publications in which it appears, and the vagaries of his manuscripts. Then there is a chapter in which the five mathematical articles published in his lifetime are reprinted. After that come the Testamentary Letter and the First Memoir (in which Galois expounded the ideas now called Galois Theory), which are the most famous of the manuscripts. There follow the less well known manuscripts, namely the Second Memoir and the many fragments. A short epilogue devoted to myths and mysteries concludes the text. The book is written as a contribution to the history of mathematics but with mathematicans as well as historians in mind. It makes available to a wide mathematical and historical readership some of the most exciting mathematics of the first half of the 19th century, presented in its original form. The primary aim is to establish a text of what Galois wrote. Exegesis would fill another book or books, and little of that is to be found here. This work will be a resource for research in the history of mathematics, especially algebra, as well as a sourcebook for those many mathematicians who enliven their student lectures with reliable historical background. 10 20 2011 978-3-03719-104-0 978-3-03719-604-5 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/104 http://www.ems-ph.org/doi/10.4171/104 Athanase Papadopoulos IRMA, Strasbourg, France Functions of a complex variable Several complex variables and analytic spaces Primary 30-00, 32-00, 57-00, 32G13, 32G15, 30F60; secondary 11F06, 11F75, 14D20, 14H15, 14H60, 14H55, 14J60, 20F14, 20F28, 20F38, 20F65, 20F67, 20H10, 30C62, 30F20, 30F25, 30F10, 30F15, 30F30, 30F35, 30F40, 30F45, 53A35, 53B35, 53C35, 53C50, 53C80, 53D55, 53Z05, 57M07, 57M20, 57M27, 57M50, 57M60, 57N16 The subject of this handbook is Teichmüller theory in a wide sense, namely the theory of geometric structures on surfaces and their moduli spaces. This includes the study of vector bundles on these moduli spaces, the study of mapping class groups, the relation with 3-manifolds, the relation with symmetric spaces and arithmetic groups, the representation theory of fundamental groups, and applications to physics. Thus the handbook is a place where several fields of mathematics interact: Riemann surfaces, hyperbolic geometry, partial differential equations, several complex variables, algebraic geometry, algebraic topology, combinatorial topology, low-dimensional topology, theoretical physics, and others. This confluence of ideas towards a unique subject is a manifestation of the unity and harmony of mathematics. The present volume contains surveys on the fundamental theory as well as surveys on applications to and relations with the fields mentioned above. It is written by leading experts in the fields. Some of the surveys contain classical material, while others present the latest developments of the theory as well as open problems. The metric and the analytic theory. The group theory. The algebraic topology of mapping class groups and moduli spaces. Teichmüller theory and mathematical physics. The handbook is addressed to graduate students and researchers in all the fields mentioned. 6 8 2012 978-3-03719-103-3 978-3-03719-603-8 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/103 http://www.ems-ph.org/doi/10.4171/103 17

Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Several complex variables and analytic spaces Functions of a complex variable Manifolds and cell complexes General This introduction provides an overview on the contents of this volume, and at the same time is a survey of several aspects of Teichmülller theory with its relation to the moduli spaces of curves, mapping class groups and their various actions, 3-manifold topology and theoretical physics. 1 34 1 10.4171/103-1/1 http://www.ems-ph.org/doi/10.4171/103-1/1

Jean-Pierre Otal Université Paul Sabatier, TOULOUSE CEDEX 9, FRANCE Functions of a complex variable General We expose a proof of the existence and uniqueness of homeomorphic solutions to the Beltrami equation $\bar \partial f = \mu \partial f$ a.e. in two cases. First, when the $L^\infty$ norm of $\mu$ is $ 37 70 1 10.4171/103-1/2 http://www.ems-ph.org/doi/10.4171/103-1/2

Jun Hu Brooklyn College of CUNY, BROOKLYN, UNITED STATES Functions of a complex variable Several complex variables and analytic spaces Dynamical systems and ergodic theory General Based on the recent works of several people, we present a self-contained exposition of the earthquake theory on the hyperbolic plane, initiated by Thurston. 71 122 1 10.4171/103-1/3 http://www.ems-ph.org/doi/10.4171/103-1/3

Caroline Series University of Warwick, COVENTRY, UNITED KINGDOM Functions of a complex variable Several complex variables and analytic spaces General We survey the known results about lines of minima introduced by Kerckhoff in [19], and also Rafi’s results about curves which are short on surfaces along Teichmüller geodesics, in particular their use in proving that lines of minima are Teichmüller quasi-geodesics. 123 153 1 10.4171/103-1/4 http://www.ems-ph.org/doi/10.4171/103-1/4

Lizhen Ji University of Michigan, ANN ARBOR, UNITED STATES Number theory Group theory and generalizations Topological groups, Lie groups Functions of a complex variable In this chapter, we discuss similarities, differences and interaction between two natural and important classes of groups: arithmetic subgroups $\Gamma$ of Lie groups $G$ and mapping class groups mod of surfaces of genus $g$ with $n$ punctures. We also mention similar properties and problems for related groups such as outer automorphism groups $\operatorname{Out}(F_n)$, Coxeter groups and hyperbolic groups. Since groups are often effectively studied by suitable spaces on which they act, we also discuss related properties of actions of arithmetic groups on symmetric spaces and actions of mapping class groups on Teichmüller spaces. Interaction between locally symmetric spaces and moduli spaces of Riemann surfaces through the example of the Jacobian map will also be discussed in the last part of this chapter. Since reduction theory, i.e., finding good fundamental domains for proper actions of discrete groups, is crucial to transformation group theory, i.e., to understand the algebraic structures of groups, properties of group actions and geometry, topology and compactifications of the quotient spaces, we discuss many different approaches to reduction theory of arithmetic groups acting on symmetric spaces. These results for arithmetic groups motivate some results on fundamental domains for the action of mapping class groups on Teichmüller spaces. For example, the Minkowski reduction theory of quadratic forms is generalized to the action of $\operatorname{Mod}_g=\operatorname{Mod}_{g, 0}$ on the Teichmüller space $\mathcal{T}_g$ to construct an intrinsic fundamental domain consisting of finitely many cells, solving a weaker version of a folklore conjecture in the theory of Teichmüller spaces. 157 295 1 10.4171/103-1/5 http://www.ems-ph.org/doi/10.4171/103-1/5

John McCarthy Michigan State University, EAST LANSING, UNITED STATES Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Several complex variables and analytic spaces Combinatorics Group theory and generalizations Functions of a complex variable In this chapter, we review the actions of the (extended) mapping class group of a surface of finite type by simplicial automorphisms on various abstract simplicial complexes (the curve complex, the arc complex, the cut system complex, the pants decomposition complex, the ideal triangulation complex, etc.). Each of these complexes has some special combinatorial features, and there are interesting questions that are particular to each one. We study in detail the actions on recently defined complexes, namely, the complex of domains and some of its subcomplexes. In most cases mentioned, the automorphism group of the simplicial automorphism coincides with the (injective) image of the mapping class group in that automorphism group. In the case of the complex of domains, this does not hold, as soon as the surface has at least two boundary components. We compute the automorphism group of the complex of domains, and we introduce a subcomplex of this complex, the truncated complex of domains, whose automorphism group is the extended mapping class group. The survey contains expository sections on surfaces and related objects and on simplicial complexes. 297 423 1 10.4171/103-1/6 http://www.ems-ph.org/doi/10.4171/103-1/6

Valentina Disarlo Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Several complex variables and analytic spaces Group theory and generalizations Functions of a complex variable Manifolds and cell complexes The complex of domains $D(S)$ is a geometric tool with a very rich simplicial structure, it contains the curve complex $C(S)$ as a simplicial subcomplex. In this chapter we shall regard it as a metric space, endowed with the metric which makes each simplex Euclidean with edges of length 1, and we shall discuss its coarse geometry. We prove that for every subcomplex $\Delta(S)$ of $D(S)$ which contains the curve complex $C(S)$, the natural simplicial inclusion $C(S) \to \Delta(S)$ is an isometric embedding and a quasi-isometry. We prove that, except a few cases, the arc complex $A(S)$ is quasi-isometric to the subcomplex $P_\partial(S)$ of $D(S)$ spanned by the vertices which are peripheral pair of pants, and we give necessary and sufficient conditions on $S$ for the simplicial inclusion $P_\partial(S) \to D(S)$ to be a quasi-isometric embedding. We then apply these results to the arc and curve complex $AC(S)$. We give a new proof of the fact that $AC(S)$ is quasi-isometric to $C(S)$, and we discuss the metric properties of the simplicial inclusion $A(S) \to AC(S)$. 425 439 1 10.4171/103-1/7 http://www.ems-ph.org/doi/10.4171/103-1/7

Mustafa Korkmaz Middle East Technical University, ANKARA, TURKEY Manifolds and cell complexes Group theory and generalizations Functions of a complex variable General The mapping class group of an orientable surface is the group of orientation-preserving self-diffeomorphisms of the surface modulo isotopy. This family of groups plays a central role in low dimensional topology. Therefore, its algebraic structure is of interest. This chapter focuses on the generating sets of the mapping class groups, specifically various minimal generating sets. After the necessary definitions are given, Dehn twists and relations among them are introduced. Next, the minimal generating sets of the mapping class group are considered: It turns out that mapping class group is generated by two elements, by two torsion elements, and by four involutions. Finally, known results about the minimal generating sets for the extended mapping class groups, the hyperelliptic mapping class groups, and the mapping class groups of nonorientable surface are considered. 441 463 1 10.4171/103-1/8 http://www.ems-ph.org/doi/10.4171/103-1/8

Kazuo Habiro Kyoto University, KYOTO, JAPAN Gwénaël Massuyeau Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Manifolds and cell complexes Group theory and generalizations General Let $\Sigma$ be a compact oriented surface. A homology cobordism of $\Sigma$ is a cobordism $C$ between two copies of $\Sigma$, such that both the “top” inclusion and the “bottom” inclusion $\Sigma \subset C$ induce isomorphisms in homology. Homology cobordisms of $\Sigma$ form a monoid, into which the mapping class group of $\Sigma$ embeds by the mapping cylinder construction. In this chapter, we survey recent works on the structure of the monoid of homology cobordisms, and we outline their relations with the study of the mapping class group. We are mainly interested in the cases where $\partial \Sigma$ is empty or connected. 465 529 1 10.4171/103-1/9 http://www.ems-ph.org/doi/10.4171/103-1/9

Manifolds and cell complexes Group theory and generalizations General This is a survey of Magnus representations with particular emphasis on their applications to mapping class groups and monoids (groups) of homology cobordisms of surfaces. In the first half, we begin by recalling the basics of the Fox calculus and overview Magnus representations for automorphism groups of free groups and mapping class groups of surfaces with related topics. In the latter half, we discuss in detail how the theory in the first half extends to homology cobordisms of surfaces and present a number of applications from recent researches. 531 594 1 10.4171/103-1/10 http://www.ems-ph.org/doi/10.4171/103-1/10

Louis Funar Université Grenoble I, SAINT MARTIN D'HERES CEDEX, FRANCE Christophe Kapoudjian Université Paul Sabatier – Toulouse 3, TOULOUSE CEDEX, FRANCE Vlad Sergiescu Université Grenoble I, SAINT MARTIN D'HERES CEDEX, FRANCE Manifolds and cell complexes Group theory and generalizations General We consider Thompson's groups from the perspective of mapping class groups of surfaces of infinite type. This point of view leads us to the braided Thompson groups, which are extensions of Thompson’s groups by infinite (spherical) braid groups. We will outline the main features of these groups and some applications to the quantization of Teichmüller spaces. The chapter provides an introduction to the subject with an emphasis on some of the authors results. 595 664 1 10.4171/103-1/11 http://www.ems-ph.org/doi/10.4171/103-1/11

Dimitri Zvonkine Université Pierre et Marie Curie – Paris 6, PARIS CEDEX 05, FRANCE General The material of this chapter is based on a series of three lectures for graduate students that the author gave at theJournées mathématiques de Glanon in July 2006. We introduce moduli spaces of smooth and stable curves, the tautological cohomology classes on these spaces, and explain how to compute all possible intersection numbers between these classes. 667 716 1 10.4171/103-1/12 http://www.ems-ph.org/doi/10.4171/103-1/12

Ib Madsen University of Copenhagen, COPENHAGEN, DENMARK Algebraic geometry Algebraic topology General This is a survey on the proof of a generalized version of the Mumford conjecture obtained in joint work with M. Weiss stating that a certain map between some classifying spaces which a priori have different natures induces an isomorphism at the level of integral homology. We also discuss our proof of the original Mumford conjecture stating that the stable rational cohomology of the moduli space of Riemann surfaces is a certain polynomial algebra generated by the Mumford–Morita–Miller cohomology classes of even degrees. 717 746 1 10.4171/103-1/13 http://www.ems-ph.org/doi/10.4171/103-1/13

Lizhen Ji University of Michigan, ANN ARBOR, UNITED STATES Steven Zucker The Johns Hopkins University, BALTIMORE, UNITED STATES Several complex variables and analytic spaces Algebraic geometry Algebraic topology Global analysis, analysis on manifolds Let ${\mathcal M}_{g,n}$ be the moduli space of algebraic curves of genus $g$ with $n$ punctures, which is a noncompact orbifold. Let $\overline{\mathcal M}^{DM}_{g,n}$ denote its Deligne–Mumford compactification. Then ${\mathcal M}_{g,n}$ admits a class of canonical Riemannian and Finsler metrics. We probe the analogy between ${\mathcal M}_{g,n}$ (resp. Teichmüller spaces) with these metrics and certain noncompact locally symmetric spaces (resp. symmetric spaces of noncompact type) with their natural metrics. In this chapter, we observe that for all $1 < p < \infty$, the $L^p$-cohomology of ${\mathcal M}_{g,n}$ with respect to these Riemannian metrics that are complete can be identified with the (ordinary) cohomology of $\overline{\mathcal M}^{DM}_{g,n}$, and hence the $L^p$-cohomology is the same for different values of $p$. This suggests a “rank-one nature” of the moduli space ${\mathcal M}_{g,n}$ from the point of view of $L^p$-cohomology. On the other hand, the $L^p$-cohomology of ${\mathcal M}_{g,n}$ with respect to the incomplete Weil Petersson metric is either the cohomology of $\overline{\mathcal M}^{DM}_{g,n}$ or that of ${\mathcal M}_{g,n}$ itself, depending on whether $p\leq \frac{4}{3}$ or not. At the end of the chapter, we pose several natural problems on the geometry and analysis of these complete Riemannian metrics. 747 775 1 10.4171/103-1/14 http://www.ems-ph.org/doi/10.4171/103-1/14

Kirill Krasnov University of Nottingham, NOTTINGHAM, UNITED KINGDOM Jean-Marc Schlenker Université Paul Sabatier, TOULOUSE CEDEX, FRANCE Several complex variables and analytic spaces Functions of a complex variable Differential geometry General We survey the renormalized volume of hyperbolic 3-manifolds, as a tool for Teichmüller theory, using simple differential geometry arguments to recover results sometimes first achieved by other means. One such application is McMullen’s quasifuchsian (or more generally Kleinian) reciprocity, for which different arguments are proposed. Another is the fact that the renormalized volume of quasifuchsian (or more generally geometrically finite) hyperbolic 3-manifolds provides a Kähler potential for the Weil–Petersson metric on Teichmüller space. Yet another is the fact that the grafting map is symplectic, which is proved using a variant of the renormalized volume defined for hyperbolic ends. 779 819 1 10.4171/103-1/15 http://www.ems-ph.org/doi/10.4171/103-1/15

Rinat Kashaev Université de Genève, GENÈVE, SWITZERLAND Manifolds and cell complexes Dynamical systems and ergodic theory Quantum theory General The discrete Liouville equation is reviewed both classically and quantum mechanically from the viewpoints of its integrable structure and its relationships to (quantum) Teichmüller theory. 821 851 1 10.4171/103-1/16 http://www.ems-ph.org/doi/10.4171/103-1/16

Basic Representation Theory Andrzej Skowroński Nicolaus Copernicus University, Toruń, Poland Kunio Yamagata Tokyo University of Agriculture and Technology, Japan Associative rings and algebras 16-01; 13E10, 15A63, 15A69, 16Dxx, 16E30, 16G10, 16G20, 16G70, 16K20, 16W30, 51F15 Algebra, module, representation, quiver, ideal, radical, simple module, semisimple module, uniserial module, projective module, injective module, simple algebra, semisimple algebra, separable algebra, hereditary algebra, Nakayama algebra, Frobenius algebra, symmetric algebra, selfinjective algebra, Brauer tree algebra, enveloping algebra, Coxeter group, Coxeter graph, Hecke algebra, coalgebra, comodule, Hopf algebra, Hopf module, syzygy module, periodic module, periodic algebra, irreducible homomorphism, almost split sequence, Auslander–Reiten translation, Auslander–Reiten quiver, extension spaces, projective dimension, injective dimension, category, functor, Nakayama functor, Nakayama automorphism, Morita equivalence, Morita–Azumaya duality This is the first of two volumes which will provide a comprehensive introduction to the modern representation theory of Frobenius algebras. The first part of the book serves as a general introduction to basic results and techniques of the modern representation theory of finite dimensional associative algebras over fields, including the Morita theory of equivalences and dualities and the Auslander–Reiten theory of irreducible morphisms and almost split sequences. The second part is devoted to fundamental classical and recent results concerning the Frobenius algebras and their module categories. Moreover, the prominent classes of Frobenius algebras, the Hecke algebras of Coxeter groups and the finite dimensional Hopf algebras over fields are exhibited. This volume is self-contained and the only prerequisite is a basic knowledge of linear algebra. It includes complete proofs of all results presented and provides a rich supply of examples and exercises. The text is primarily addressed to graduate students starting research in the representation theory of algebras as well mathematicians working in other fields. 12 13 2011 978-3-03719-102-6 978-3-03719-602-1 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/102 http://www.ems-ph.org/doi/10.4171/102 Andrzej Skowroński Nicolaus Copernicus University, Toruń, Poland Kunio Yamagata Tokyo University of Agriculture and Technology, Japan Associative rings and algebras 13Dxx, 13Fxx, 14Bxx, 14Hxx, 14Lxx, 14Mxx, 14Nxx, 15Axx, 16Dxx, 16Exx, 16Gxx, 16Sxx, 16Wxx, 17Bxx, 18Exx, 19Kxx, 20Cxx, 20Jxx This book is concerned with recent trends in the representation theory of algebras and its exciting interaction with geometry, topology, commutative algebra, Lie algebras, combinatorics, quantum algebras, and theoretical physics. The collection of articles, written by leading researchers in the field, is conceived as a sort of handbook providing easy access to the present state of knowledge and stimulating further development. The topics under discussion include quivers, quivers with potential, bound quiver algebras, Jacobian algebras, cluster algebras and categories, Calabi–Yau algebras and categories, triangulated and derived categories, quantum loop algebras, Nakajima quiver varieties, Yang–Baxter equations, T-systems and Y-systems, dilogarithm and quantum dilogarithm identities, stable module categories, localizing and colocalizing subcategories, cohomologies of groups, support varieties, fusion systems, Hochschild cohomologies, weighted projective lines, coherent sheaves, Kleinian and Fuchsian singularities, stable categories of vector bundles, nilpotent operators, Artin–Schelter regular algebras, Fano algebras, deformations of algebras, module varieties, degenerations of modules, singularities of orbit closures, coalgebras and comodules, representation types of algebras and coalgebras, Tits and Euler forms of algebras, Galois coverings of algebras, tilting and cluster tilting theory, algebras of small homological dimensions, Auslander–Reiten theory. The book consists of thirteen self-contained expository survey and research articles and is addressed to researchers and graduate students in algebra as well as a broader mathematical community. They contain a large number of examples and open problems and give new perspectives for research in the field. 9 24 2011 978-3-03719-101-9 978-3-03719-601-4 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/101 http://www.ems-ph.org/doi/10.4171/101

Claire Amiot Université de Strasbourg, STRASBOURG CEDEX, FRANCE Associative rings and algebras Commutative rings and algebras Category theory; homological algebra General Cluster categories have been introduced by Buan, Marsh, Reineke, Reiten and Todorov in order to categorify Fomin–Zelevinsky cluster algebras. This survey motivates and outlines the construction of a generalization of cluster categories, and explains different applications of these new categories in representation theory. 1 53 1 10.4171/101-1/1 http://www.ems-ph.org/doi/10.4171/101-1/1

David Benson University of Aberdeen, ABERDEEN, UNITED KINGDOM Srikanth Iyengar University of Nebraska-Lincoln, LINCOLN, UNITED STATES Henning Krause Universität Bielefeld, BIELEFELD, GERMANY Group theory and generalizations Commutative rings and algebras Associative rings and algebras Category theory; homological algebra This survey article is intended as an introduction to the recent categorical classification theorems of the three authors, restricting to the special case of the category of modules for a finite group. 55 83 1 10.4171/101-1/2 http://www.ems-ph.org/doi/10.4171/101-1/2

Bernhard Keller Université Paris Diderot - Paris 7, PARIS CEDEX 13, FRANCE Associative rings and algebras Algebraic geometry Nonassociative rings and algebras Category theory; homological algebra These are expanded notes from three survey lectures given at the 14th International Conference on Representations of Algebras (ICRA XIV) held in Tokyo in August 2010. We first study identities between products of quantum dilogarithm series associated with Dynkin quivers following Reineke. We then examine similar identities for quivers with potential and link them to Fomin–Zelevinsky’s theory of cluster algebras. Here we mainly follow ideas due to Bridgeland, Fock–Goncharov, Kontsevich–Soibelman and Nagao. 85 116 1 10.4171/101-1/3 http://www.ems-ph.org/doi/10.4171/101-1/3

Bernard Leclerc Université de Caen, CAEN CEDEX, FRANCE General These notes reflect the contents of three lectures given at the workshop of the 14th International Conference on Representations of Algebras (ICRA XIV), held in August 2010 in Tokyo. We first provide an introduction to quantum loop algebras and their finite-dimensional representations. We explain in particular Nakajima’s geometric description of the irreducible $q$-characters in terms of graded quiver varieties. We then present a recent attempt to understand the tensor structure of the category of finite-dimensional representations by means of cluster algebras. This takes the form of a general conjecture depending on a level $\ell\in \mathbb{N}$. The conjecture for $\ell = 1$ is now proved thanks to some joint work with Hernandez, and a subsequent paper of Nakajima. The general case is still open. 117 152 1 10.4171/101-1/4 http://www.ems-ph.org/doi/10.4171/101-1/4

Helmut Lenzing Universität Paderborn, PADERBORN, GERMANY Algebraic geometry Associative rings and algebras General This survey covers the topics of my mini-course on “Weighted projective lines” given at the Workshop of ICRA XIV (XIV International Conference at Workshop, Tokyo, August 6–15, 2010). Weighted projective lines, and their defining equations, have a long history going back to Klein and Poincaré. Accordingly their study has a high contact surface with many mathematical subjects, classical and modern. Among the many related subjects we mention representation theory of algebras and groups, invariant theory, function theory, orbifolds, 3-manifolds, singularities and the study of nilpotent operators. Since the formal definition of the category of coherent sheaves by W. Geigle and the author in 1987, substantial progress has been made by a number of authors. As a recent application, the analysis of the singularity category of triangle singularities, Kleinian and Fuchsian singularities is covered in some detail. In the center of this analysis is the structure of the corresponding stable categories of vector bundles. The study is in the spirit of Buchweitz (1987) and Orlov (2005) and concerns work in progress with Kussin and Meltzer, recent work by Kajiura, Saito, and Takahashi (2007, 2009), and joint work with J. A. de la Peña (2006). These methods are further applied to the study by C. M. Ringel and M. Schmidmeier (2008) on the invariant subspace problem for nilpotent operators. 153 187 1 10.4171/101-1/5 http://www.ems-ph.org/doi/10.4171/101-1/5

Markus Linckelmann City University, LONDON, UNITED KINGDOM $K$-theory General Block algebras are algebras which arise as indecomposable direct factors of finite group algebras, where we choose as a base ring an algebraically closed field k of prime characteristic p. The purpose of the present notes is to describe some connections between invariants of the module categories of block algebras and of their associated fusion systems, with a particular emphasis on cohomological invariants. 189 250 1 10.4171/101-1/6 http://www.ems-ph.org/doi/10.4171/101-1/6

Piotr Malicki Nicolaus Copernicus University, TORU&#323;, POLAND Andrzej Skowroński Nicolaus Copernicus University, TORU&#323;, POLAND Associative rings and algebras General We survey old and new results on the structure and homological properties of Artin algebras whose Auslander–Reiten quiver admits a separating family of connected components. 251 353 1 10.4171/101-1/7 http://www.ems-ph.org/doi/10.4171/101-1/7

Izuru Mori Shizuoka University, SHIZUOKA, JAPAN Associative rings and algebras General In noncommutative algebraic geometry, it is interesting to classify homologically nice classes of connected graded algebras. On the other hand, in representation theory of finite dimensional algebras, it is interesting to classify homologically nice classes of finite dimensional algebras. In this survey paper, we will show that there are strong interactions between these classification problems. 355 406 1 10.4171/101-1/8 http://www.ems-ph.org/doi/10.4171/101-1/8

Tomoki Nakanishi Nagoya University, NAGOYA, CHIKUSA-KU, JAPAN Commutative rings and algebras Nonassociative rings and algebras General We consider two kinds of periodicities of mutations in cluster algebras. For any sequence of mutations under which exchange matrices are periodic, we define the associated T- and Y-systems. When the sequence is ‘regular’, they are particularly natural generalizations of the known ‘classic’ T- and Y-systems. Furthermore, for any sequence of mutations under which seeds are periodic, we formulate the associated dilogarithm identity. We prove the identities when exchange matrices are skew symmetric. 407 443 1 10.4171/101-1/9 http://www.ems-ph.org/doi/10.4171/101-1/9

José Antonio Peña Universidad Nacional Autónoma de México, MÉXICO, D.F., MEXICO Andrzej Skowroński Nicolaus Copernicus University, TORU&#323;, POLAND Linear and multilinear algebra; matrix theory Algebraic geometry Associative rings and algebras General We survey some old and recent results concerning properties of the Tits quadratic forms of triangular finite dimensional algebras of finite and tame representation type over an algebraically closed field, and realization of their roots as dimension vectors of indecomposable modules. In particular, we discuss when we may recover the representation type of a triangular algebra from the combinatorial properties of its Tits quadratic form. 445 499 1 10.4171/101-1/10 http://www.ems-ph.org/doi/10.4171/101-1/10

Claus Michael Ringel Universität Bielefeld, BIELEFELD, GERMANY Associative rings and algebras General Let $k$ be a field. A finite dimensional k-algebra is said to be minimal representation-infinite provided it is representation-infinite and all its proper factor algebras are representation-finite. Our aim is to classify the special biserial algebras which are minimal representation-infinite. The second part describes the corresponding module categories. 501 560 1 10.4171/101-1/11 http://www.ems-ph.org/doi/10.4171/101-1/11

Daniel Simson Nicolaus Copernicus University, TORU&#323;, POLAND Associative rings and algebras General We study the structure of $K$-coalgebras over a field $K$ and comodule categories. In particular, we discuss the concepts of tame comodule type, of discrete comodule type, of polynomial growth, and of wild comodule type, for $K$-coalgebras $C$, introduced by the author in [83], [84], [93], and intensively studied during the last decade. Among other things, we show that, over an algebraicaly closed field $K$, the tame-wild dichotomy holds, for a wide class of coalgebras $C$ of infinite dimension, including the class of semiperfect coalgebras and the class of incidence coalgebras $K^{\Box} I$ of interval finite posets $I$. Tools and techniques applied in the study of $K$-coalgebras $C$, their comodules, and representation types, are presented. Characterisations of large classes of coalgebras of tame comodule type are presented, including path coalgebras of quivers, string coalgebras, and the incidence coalgebras of interval finite posets. 561 660 1 10.4171/101-1/12 http://www.ems-ph.org/doi/10.4171/101-1/12

Grzegorz Zwara Nicolaus Copernicus University, TORU&#323;, POLAND Associative rings and algebras Algebraic geometry General Finite dimensional modules (or representations of quivers) form affine varieties equipped with algebraic group actions, such that the orbits correspond bijectively to the isomorphism classes of modules. The Zariski-closures of orbits are the objects of our interest. We study their geometric properties and relate them to properties of the corresponding modules. 661 725 1 10.4171/101-1/13 http://www.ems-ph.org/doi/10.4171/101-1/13

Erwan Faou ENS Cachan Bretagne, France Numerical analysis Partial differential equations Dynamical systems and ergodic theory 65P10, 37M15, 35Q41 Geometric numerical integration, symplectic integrators, backward error analysis, Schrödinger equations, long time behavior The goal of geometric numerical integration is the simulation of evolution equations possessing geometric properties over long times. Of particular importance are Hamiltonian partial differential equations typically arising in application fields such as quantum mechanics or wave propagation phenomena. They exhibit many important dynamical features such as energy preservation and conservation of adiabatic invariants over long time. In this setting, a natural question is how and to which extent the reproduction of such long time qualitative behavior can be ensured by numerical schemes. Starting from numerical examples, these notes provide a detailed analysis of the Schrödinger equation in a simple setting (periodic boundary conditions, polynomial nonlinearities) approximated by symplectic splitting methods. Analysis of stability and instability phenomena induced by space and time discretization are given, and rigorous mathematical explanations for them. The book grew out of a graduate level course and is of interest to researchers and students seeking an introduction to the subject matter. 1 14 2012 978-3-03719-100-2 978-3-03719-600-7 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/100 http://www.ems-ph.org/doi/10.4171/100 Anders Björn Linköping University, Sweden Jana Björn Linköping University, Sweden Potential theory 31-02, 31E05; 28A12, 30L99, 31C05, 31C15, 31C40, 31C45, 35B45, 35B65, 35D30, 35J20, 35J25, 35J60, 35J67, 35J70, 35J92, 46E35, 47J20, 49J10, 49J27, 49J40, 49J52, 49N60, 49Q20, 58C99, 58J05, 58J32 Boundary regularity, capacity, Dirichlet problem, doubling measure, interior regularity, metric space, minimizer, Newtonian space, nonlinear, obstacle problem, Perron solution, p-harmonic function, Poincaré inequality, potential theory, Sobolev space, upper gradient The p-Laplace equation is the main prototype for nonlinear elliptic problems and forms a basis for various applications, such as injection moulding of plastics, nonlinear elasticity theory and image processing. Its solutions, called p-harmonic functions, have been studied in various contexts since the 1960s, first on Euclidean spaces and later on Riemannian manifolds, graphs and Heisenberg groups. Nonlinear potential theory of p-harmonic functions on metric spaces has been developing since the 1990s and generalizes and unites these earlier theories. This monograph gives a unified treatment of the subject and covers most of the available results in the field, so far scattered over a large number of research papers. The aim is to serve both as an introduction to the area for an interested reader and as a reference text for an active researcher. The presentation is rather self-contained, but the reader is assumed to know measure theory and functional analysis. The first half of the book deals with Sobolev type spaces, so-called Newtonian spaces, based on upper gradients on general metric spaces. In the second half, these spaces are used to study p-harmonic functions on metric spaces and a nonlinear potential theory is developed under some additional, but natural, assumptions on the underlying metric space. Each chapter contains historical notes with relevant references and an extensive index is provided at the end of the book. 11 5 2011 978-3-03719-099-9 978-3-03719-599-4 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/099 http://www.ems-ph.org/doi/10.4171/099 17 Marek Jarnicki Jagiellonian University, Kraków, Poland Peter Pflug University of Oldenburg, Germany Several complex variables and analytic spaces 32-02, 32D15, 32A10, 32A17, 32D05, 32D10, 32D26, 32U15 Separately holomorphic/meromorphic functions, Riemann domains, N-fold crosses, generalized crosses, relative extremal functions, holomorphic extension with singularities The story of separately holomorphic functions began about 100 years ago. During the second half of the 19th century, it became known that a separately continuous function is not necessarily continuous as a function of all variables. At the beginning of the 20th century, the study of separately holomorphic functions started due to the fundamental work of Osgood and Hartogs. This book provides the first self-contained and complete presentation of the study of separately holomorphic functions, starting from its birth up to current research. Most of the results presented have never been published before in book form. The text is divided into two parts. A more elementary one deals with separately holomorphic functions “without singularities”, another addresses the situation of existing singularities. A discussion of the classical results related to separately holomorphic functions leads to the most fundamental result, the classical cross theorem as well as various extensions and generalizations to more complicated “crosses”. Additionally, several applications for other classes of “separately regular” functions are given. A solid background in basic complex analysis is a prerequisite. In order to make the book self-contained, all the results needed for its understanding are collected in special introductory chapters and referred to at the beginning of each section. The book is addressed to students and researchers in several complex variables as well as to mathematicians and theoretical physicists who are interested in this area of mathematics. 8 11 2011 978-3-03719-098-2 978-3-03719-598-7 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/098 http://www.ems-ph.org/doi/10.4171/098 16 Claudio Carmeli University of Genova, Italy Lauren Caston RAND Corporation, Santa Monica, CA, USA Rita Fioresi University of Bologna, Italy Global analysis, analysis on manifolds Algebraic geometry Nonassociative rings and algebras Several complex variables and analytic spaces 58-02; 58A50, 58C50, 14M30, 17A70, 32C11, 81Q60 Supermanifolds, supersymmetry, superschemes, Lie superalgebras Supersymmetry is a highly active area of considerable interest among physicists and mathematicians. It is not only fascinating in its own right, but there is also indication that it plays a fundamental role in the physics of elementary particles and gravitation. The purpose of the book is to lay down the foundations of the subject, providing the reader with a comprehensive introduction to the language and techniques, with a special attention to giving detailed proofs and many clarifying examples. It is aimed ideally at a second year graduate student. After the first three introductory chapters, the text divides into two parts: the theory of smooth supermanifolds and Lie supergroups, including the Frobenius theorem, and the theory of algebraic superschemes and supergroups. There are three appendices, the first introducing Lie superalgebras and representations of classical Lie superalgebras, the second collecting some relevant facts on categories, sheafification of functors and commutative algebra, and the third explaining the notion of Fréchet space in the super context. 8 6 2011 978-3-03719-097-5 978-3-03719-597-0 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/097 http://www.ems-ph.org/doi/10.4171/097 Damien Calaque ETH Zurich, Switzerland Carlo Rossi Max Planck Institute for Mathematics, Bonn, Germany Commutative rings and algebras Algebraic geometry Nonassociative rings and algebras 13D03, 17B56, 14F43 Lie algebra, Hochschild cohomology, complex manifolds, deformation theory, Kontsevich’s graphical calculus, Atiyah class, Duflo isomorphism, Todd class Duflo isomorphism first appeared in Lie theory and representation theory. It is an isomorphism between invariant polynomials of a Lie algebra and the center of its universal enveloping algebra, generalizing the pioneering work of Harish-Chandra on semi-simple Lie algebras. Later on, Duflo’s result was refound by Kontsevich in the framework of deformation quantization, who also observed that there is a similar isomorphism between Dolbeault cohomology of holomorphic polyvector fields on a complex manifold and its Hochschild cohomology. The present book, which arose from a series of lectures by the first author at ETH, derives these two isomorphisms from a Duflo-type result for Q-manifolds. All notions mentioned above are introduced and explained in the book, the only prerequisites being basic linear algebra and differential geometry. In addition to standard notions such as Lie (super)algebras, complex manifolds, Hochschild and Chevalley–Eilenberg cohomologies, spectral sequences, Atiyah and Todd classes, the graphical calculus introduced by Kontsevich in his seminal work on deformation quantization is addressed in details. The book is well-suited for graduate students in mathematics and mathematical physics as well as for researchers working in Lie theory, algebraic geometry and deformation theory. 6 11 2011 978-3-03719-096-8 978-3-03719-596-3 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/096 http://www.ems-ph.org/doi/10.4171/096 Kenji Nakanishi Kyoto University, Japan Wilhelm Schlag University of Chicago, USA Partial differential equations 35L70, 35Q55, 37D10, 37K40, 37K45 Nonlinear dispersive equations, wave, Klein–Gordon, Schrödinger equations, scattering theory, stability theory, solitons, ground states, global existence, finite time blow up, soliton resolution conjecture, hyperbolic dynamics, stable, unstable, center-stable, invariant manifolds The notion of an invariant manifold arises naturally in the asymptotic stability analysis of stationary or standing wave solutions of unstable dispersive Hamiltonian evolution equations such as the focusing semilinear Klein–Gordon and Schrödinger equations. This is due to the fact that the linearized operators about such special solutions typically exhibit negative eigenvalues (a single one for the ground state), which lead to exponential instability of the linearized flow and allows for ideas from hyperbolic dynamics to enter. One of the main results proved here for energy subcritical equations is that the center-stable manifold associated with the ground state appears as a hyper-surface which separates a region of finite-time blowup in forward time from one which exhibits global existence and scattering to zero in forward time. Our entire analysis takes place in the energy topology, and the conserved energy can exceed the ground state energy only by a small amount. This monograph is based on recent research by the authors and the proofs rely on an interplay between the variational structure of the ground states on the one hand, and the nonlinear hyperbolic dynamics near these states on the other hand. A key element in the proof is a virial-type argument excluding almost homoclinic orbits originating near the ground states, and returning to them, possibly after a long excursion. These lectures are suitable for graduate students and researchers in partial differential equations and mathematical physics. For the cubic Klein–Gordon equation in three dimensions all details are provided, including the derivation of Strichartz estimates for the free equation and the concentration-compactness argument leading to scattering due to Kenig and Merle. 9 2 2011 978-3-03719-095-1 978-3-03719-595-6 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/095 http://www.ems-ph.org/doi/10.4171/095 An Algorithmic Theory Shmuel Onn Technion - Israel Institute of Technology, Haifa, Israel Operations research, mathematical programming Combinatorics Linear and multilinear algebra; matrix theory Computer science 05Axx, 05Cxx, 05Dxx, 05Exx, 11Dxx, 11Hxx, 11Pxx, 13Pxx, 14Qxx, 15Axx, 15Bxx, 51Mxx, 52Axx, 52Bxx, 52Cxx, 62Hxx, 62Kxx, 62Qxx, 65Cxx, 68Qxx, 68Rxx, 68Wxx, 90Bxx, 90Cxx Integer programming, combinatorial optimization, optimization, linear programming, stochastic programming, randomized algorithm, approximation algorithm, polynomial time, transportation problem, multi index transportation problem, transshipment problem, multicommodity flow, congestion game, spanning tree, matroid, submodular function, matching, partitioning, clustering, polytope, zonotope, edge direction, totally unimodular matrix, test set, Graver base, contingency table, statistical table, multiway table, disclosure control, data security, privacy, algebraic statistics, experimental design, Frobenius number, Grobner base, Hilbert scheme, zero-dimensional ideal This monograph develops an algorithmic theory of nonlinear discrete optimization. It introduces a simple and useful setup which enables the polynomial time solution of broad fundamental classes of nonlinear combinatorial optimization and integer programming problems in variable dimension. An important part of this theory is enhanced by recent developments in the algebra of Graver bases. The power of the theory is demonstrated by deriving the first polynomial time algorithms in a variety of application areas within operations research and statistics, including vector partitioning, matroid optimization, experimental design, multicommodity flows, multi-index transportation and privacy in statistical databases. The monograph is intended for graduate students and researchers. It is accessible to anyone with standard undergraduate knowledge and mathematical maturity. 9 7 2010 978-3-03719-093-7 978-3-03719-593-2 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/093 http://www.ems-ph.org/doi/10.4171/093 Jacqueline Stedall University of Oxford, UK History and biography 01-02; 01A40; 01A45; 01A50 Algebra, equations, renaissance, early modern This book is an exploration of a claim made by Lagrange in the autumn of 1771 as he embarked upon his lengthy ‘Réflexions sur la résolution algébrique des équations’: that there had been few advances in the algebraic solution of equations since the time of Cardano in the mid sixteenth century. That opinion has been shared by many later historians. The present study attempts to redress that view and to examine the intertwined developments in the theory of equations from Cardano to Lagrange. A similar historical exploration led Lagrange himself to insights that were to transform the entire nature and scope of algebra. Progress was not confined to any one country: at different times mathematicians in Italy, France, the Netherlands, England, Scotland, Russia, and Germany contributed to the discussion and to a gradual deepening of understanding. In particular, the national Academies of Berlin, St Petersburg, and Paris in the eighteenth century were crucial in supporting informed mathematical communities and encouraging the wider dissemination of key ideas. This study therefore truly highlights the existence of a European mathematical heritage. The book is written in three parts. Part I offers an overview of the period from Cardano to Newton (from 1545 to 1707) and is arranged chronologically. Part II covers the period from Newton to Lagrange (from 1707 to 1770) and treats the material according to key themes. Part III is a brief account of the aftermath of the discoveries made in the 1770s. The book attempts throughout to capture the reality of mathematical discovery by inviting the reader to follow in the footsteps of the authors themselves, with as few changes as possible to the original notation and style of presentation. 3 29 2011 978-3-03719-092-0 978-3-03719-592-5 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/092 http://www.ems-ph.org/doi/10.4171/092 ℋ2-Matrix Compression, Algorithms and Analysis Corrected 2nd printing, September 2013 Steffen Börm Kiel University, Germany Numerical analysis 65-02; 65F05, 65F30, 65N22, 65N38, 65R20 Hierarchical matrix, data-sparse approximation, boundary element method, preconditioner Hierarchical matrices present an efficient way of treating dense matrices that arise in the context of integral equations, elliptic partial differential equations, and control theory. While a dense n × n matrix in standard representation requires n2 units of storage, a hierarchical matrix can approximate the matrix in a compact representation requiring only O(nk log n) units of storage, where k is a parameter controlling the accuracy. Hierarchical matrices have been successfully applied to approximate matrices arising in the context of boundary integral methods, to construct preconditioners for partial differential equations, to evaluate matrix functions and to solve matrix equations used in control theory. ℋ2-matrices offer a refinement of hierarchical matrices: using a multilevel representation of submatrices, the efficiency can be significantly improved, particularly for large problems. This books gives an introduction to the basic concepts and presents a general framework that can be used to analyze the complexity and accuracy of ℋ2-matrix techniques. Starting from basic ideas of numerical linear algebra and numerical analysis, the theory is developed in a straightforward and systematic way, accessible to advanced students and researchers in numerical mathematics and scientific computing. Special techniques are only required in isolated sections, e.g., for certain classes of model problems. 12 1 2010 978-3-03719-091-3 978-3-03719-591-8 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/091 http://www.ems-ph.org/doi/10.4171/091 14 Koichiro Harada The Ohio State University, Columbus, OH, USA Group theory and generalizations Number theory 20B05, 11F03 Monster simple group, congruence groups, modular functions, eta function This is an almost verbatim reproduction of the author’s lecture notes written in 1983–84 at the Ohio State University, Columbus, Ohio, USA. A substantial update is given in the bibliography. Over the last 20 plus years, there has been an energetic activity in the field of finite simple group theory related to the monster simple group. Most notably, influential works have been produced in the theory of vertex operator algebras whose research was stimulated by the moonshine of the finite groups. Still, we can ask the same questions now just as we did some 30–40 years ago: What is the monster simple group? Is it really related to the theory of the universe as it was vaguely so envisioned? What lays behind the moonshine phenomena of the monster group? It may appear that we have only scratched the surface. These notes are primarily reproduced for the benefit of young readers who wish to start learning about modular functions used in moonshine. 9 29 2010 978-3-03719-090-6 978-3-03719-590-1 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/090 http://www.ems-ph.org/doi/10.4171/090 Schweizerische Mathematische Gesellschaft – Société Mathématique Suisse – Swiss Mathematical Society 1910 Bruno Colbois University of Neuchâtel, Switzerland Christine Riedtmann University of Bern, Switzerland Viktor Schroeder University of Zurich, Switzerland History and biography 00Bxx This book includes twenty-three essays to celebrate the 100th anniversary of the Swiss Mathematical Society. The life and work of outstanding mathematicians, extraordinary conferences held in Switzerland such as the three International Congresses of Mathematicians, the influence of women in Swiss mathematics are among the topics. The articles, including many photographs, old and recent, give a vivid picture of hundred years of mathematical life in Switzerland. Dieses Buch ist eine Festschrift zum 100-jährigen Bestehen der Schweizerischen Mathematischen Gesellschaft. Es enthält dreiundzwanzig Beiträge zur Mathematik in der Schweiz. Geschichtliches und Biographisches über herausragende Mathematiker an Schweizer Universitäten, grosse Tagungen wie etwa die drei Internationalen Mathematiker-Kongresse, die Rolle der Frauen in der Schweizer Mathematik sind nur einige Themen. Insgesamt vermitteln die verschiedenen Essays zusammen mit den zahlreichen Abbildungen ein höchst lebendiges und anschauliches Panorama eines Jahrhunderts Schweizer Mathematik. Cet ouvrage a été édité pour marquer le 100e anniversaire de la Société Mathématique Suisse. Il rassemble vingt-trois articles consacrés aux mathématiques en Suisse. Parmi beaucoup d'autres choses, les écrits évoquent la vie et l'œuvre de grands mathématiciens des universités suisses, les grands événements, dont les trois Congrès internationaux de mathématiques, ou encore la présence des femmes dans les mathématiques suisses. Agrémenté de nombreuses photos, anciennes et récentes, ce livre donne une image très vivante de cent années de vie mathématique en Suisse. 8 24 2010 978-3-03719-089-0 978-3-03719-589-5 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/089 http://www.ems-ph.org/doi/10.4171/089

Michel Plancherel (1885–1967) ETH Zürich, , SWITZERLAND History and biography General 1 21 1 10.4171/089-1/1 http://www.ems-ph.org/doi/10.4171/089-1/1

Erwin Neuenschwander Universität Zürich, ZÜRICH, SWITZERLAND History and biography General 23 105 1 10.4171/089-1/2 http://www.ems-ph.org/doi/10.4171/089-1/2

Christian Blatter ETH Zürich, ZÜRICH, SWITZERLAND History and biography General The author, born 1935, spent his youth and student years in Basel (Switzerland). His tale begins when he enters the “Humanistisches Gymnasium” there; it ends with the legendary Colloquium on Differential Geometry and Topology, held 1960 in Zurich on the occasion of the 50-th anniversary of the Swiss Mathematical Society, where our hero for the first time meets the international mathematical scene. In between we make the acquaintance with the relevant mathematical figures (and some other scientists) in Basel at the time, we hear about the author’s struggling with various minors (philosophy, physics, chemistry), about his part time job as proofreader for a local newspaper, about his year as famulus of the great Ostrowski, and about more leisure-oriented student activities. All this converges to serious mathematical work on a theme between complex analysis and differential geometry (a hint of which is explained in the text at hand). The lively account of the author shows that then and there studying meant working in very diverse environments, as it does today, but all in all had an essential quality of liberty – Bologna was still forty years ahead. 107 128 1 10.4171/089-1/3 http://www.ems-ph.org/doi/10.4171/089-1/3

Johann Jakob Burckhardt (1903–2006) Unversität Basel, , SWITZERLAND Adolf Schnyder , THERWIL, SWITZERLAND History and biography General Andreas Speiser is well known for his work in mathematics and philosophy. In mathematics, his main areas of interest were number theory and group theory. His book Die Theorie der Gruppen von endlicher Ordnung was published in 1923 (5th ed. 1980) as volume 5 of Springer's Grundlagen series. From 1928 to 1965 Speiser was “General Redactor” of the Euler edition, 37 volumes of which appeared in this period, among them 11 volumes that were edited by Speiser himself. As far as philosophy is concerned, Speiser published a commentary on Plato’s Parmenides (Ein Parmenideskommentar, 1937) and an account on his involvement with the philosophy of Fichte and Hegel (Elemente der Philosophie und der Mathematik, 1952). Speiser was an intellectual of the old school in the best sense of the word, his interests and his education spread over wide areas, from mathematics and philosophy to music, architecture, poetry and arts. With his article, it is the authors’ intention to explain Speiser’s work to a broader audience. 129 161 1 10.4171/089-1/4 http://www.ems-ph.org/doi/10.4171/089-1/4

Peter Buser Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND History and biography General This is a historical article about the discovery of the length spectrum of a compact Riemann surface. It begins with Euler’s product formula for the prime numbers from 1737, proceeds to Riemann’s zeta function and then focusses on the theorems about length and eigenvalue spectra that were discovered by Heinz Huber in the 1950s. Emphasis is given to the comparison of Wiener’s proof of the prime number theorem with Huber’s proof of the asymptotic formula for the lengths of closed geodesics. The history ends with a brief outlook to current research. 163 193 1 10.4171/089-1/5 http://www.ems-ph.org/doi/10.4171/089-1/5

Srishti Chatterji , LAUSANNE, SWITZERLAND Manuel Ojanguren Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND History and biography General The object of this essay is to give an impressionistic overview of the mathematical period during which Georges de Rham played an important role, both through his own mathematical creation as well as his influence on other mathematicians. Our understanding of the man and his work has been much influenced by a study of his far-flung correspondence and his unpublished papers to which we had access. Some general introductory material has been appended to clarify the position of Swiss mathematics in general in order to initiate foreign readers to the life and times of de Rham. 195 240 1 10.4171/089-1/6 http://www.ems-ph.org/doi/10.4171/089-1/6

Jean Descloux Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND Dominique de Werra Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND History and biography General At the request of the Director of the Federal Polytechnic School of Lausanne, Professor Blanc (1910–2006) initiated an ambitious programme in applied mathematics: new courses, the creation of an institute for research, and an orientation towards economics. With the acquisition of a computer in 1958, he promoted numerical analysis, computer science and operational research. When the school received a federal status in 1969, he took this opportunity to endow it with a department of mathematics dedicated to applications. 241 245 1 10.4171/089-1/7 http://www.ems-ph.org/doi/10.4171/089-1/7

Shalom Eliahou Université du Littoral Côte d'Opale, CALAIS, CEDEX, FRANCE Pierre de la Harpe Université de Genève, GENÈVE 4, SWITZERLAND Jean-Claude Hausmann Université de Genève, GENÈVE 4, SWITZERLAND Claude Weber Université de Genève, GENÈVE 4, SWITZERLAND General This obituary of Michel Kervaire provides some biographical data, and describes his activity in organising spring seminars in Les Plans-sur-Bex. Some of his mathematical results are briefly discussed: his discovery of a topological manifold without any smooth structure, his results with Milnor on spheres with several smooth structures, his subject-founding paper on high-dimensional knots, the Kervaire conjecture on group presentations, and his results from the period 1987–2007 in algebra, combinatorics and number theory. A short last section discusses dramatic recent progress on the Kervaire invariant problem by Hill, Hopkins and Ravenel in 2009. 247 255 1 10.4171/089-1/8 http://www.ems-ph.org/doi/10.4171/089-1/8

Walter Gautschi Purdue University, WEST LAFAYETTE, UNITED STATES History and biography General The life of A. M. Ostrowski, full professor of mathematics at the University of Basel from 1927 to 1958, is sketched, and his extensive and many-sided work is summarized and illustrated by a few selected contributions. A complete list of his Ph.D. students is assembled and the careers of some of them are traced. 257 278 1 10.4171/089-1/9 http://www.ems-ph.org/doi/10.4171/089-1/9

Martin Gutknecht Eidgenössische Technische Hochschule, ZÜRICH, SWITZERLAND History and biography Numerical analysis General The Institute for Applied Mathematics at ETH Zurich, founded by Eduard Stiefel in January 1948, made seminal early contributions to numerical analysis and computer science. In numerical analysis the most important ones were the conjugate gradient method and the qd and LR algorithms – which are the predecessors of the ubiquitous QR algorithm. Those in computer science were the construction of the Swiss computer ERMETH, the introduction and development of compilers, and the collaboration at the ground-breaking computer language ALGOL 60. An important side-effect was the education of excellent young numerical analysts, quite a few of which emigrated to the United States –some temporarily, others for good. 279 290 1 10.4171/089-1/10 http://www.ems-ph.org/doi/10.4171/089-1/10

André Haefliger Université de Genève, GENÈVE 4, SWITZERLAND History and biography General 291 302 1 10.4171/089-1/11 http://www.ems-ph.org/doi/10.4171/089-1/11

Friedrich Hirzebruch , BONN, GERMANY History and biography General I was released as a prisoner of war on July 1, 1945, still 17 years old, and was lucky enough to be able to begin my study of mathematics, mathematical logic, and physics at the University of Münster already in the winter semester 1945/46. The University and the town were terribly destroyed, but the University began to operate. My teachers were Heinrich Behnke and Heinrich Scholz, later Karl Stein and Friedrich Karl Schmidt joined the faculty. In 1948 I participated in a Swiss program for German students (three weeks of hard work on a farm, followed by one week to be spent freely in Switzerland). Upon recommendation of Behnke I was invited by Heinz and Anja Hopf to stay with them in their house in Zollikon near Zurich for this fourth week. I learnt a lot from Heinz Hopf about algebraic topology and complex manifolds and returned to Münster full of new ideas which I told Heinrich Behnke and Karl Stein. I spoke in their seminars. This was already basic for my dissertation under Behnke, but much inspired by Hopf. I received a stipend for the ETH Zurich for three semesters (summer 1949 to summer 1950) and became a student of Heinz Hopf, Beno Eckmann and Paul Bernays at the ETH. For a German student who had lived through the war, Switzerland was a paradise academically and otherwise. I enjoyed the international atmosphere with visitors from many countries and the excellent courses and seminars offered. But most important were my frequent discussions with Heinz Hopf at the ETH and at his home. My dissertation developed, won a prize of the ETH and was accepted in Münster by Behnke. In July 1950 I was promoted to Dr. rer. nat. by the University of Münster. In my report I try to explain how important my study in Zurich was for my academic career and how fortunate I was that such a chance was given to me shortly after the war. My dissertation was published in two parts in Mathematische Annalen 1951 and 1953. These are the first two papers in my Collected Papers, Springer-Verlag 1987. In my report I indicate the contents of Part II of the dissertation. 303 315 1 10.4171/089-1/12 http://www.ems-ph.org/doi/10.4171/089-1/12

Norbert Hungerbühler ETH Zentrum, HG E63.1, ZÜRICH, SWITZERLAND Martine Schmutz Université de Fribourg, FRIBOURG, SWITZERLAND General Michel Plancherel was one of the leading figures of Swiss mathematics of the 20th century. He is best known for his fundamental results in harmonic analysis and applications in PDE theory and the calculus of variations. A milestone in mathematical physics was his proof that mechanical systems cannot be ergodic. He also left his traces in algebra, in particular in the context of quadratic forms and the theory of commutative Hilbert algebras. Not less remarkable is Plancherel’s unfatiguing dedication to the community: He served as a president of the Swiss Mathematical Society, as vice-president of the International Congress of Mathematicians 1932 in Zurich, he was rector of ETH, president and co-founder of the foundation for the advancement of the mathematical sciences in Switzerland, and he served in many other institutions. He raised funds to help the 550 Hungarian students who escaped in 1956 to Switzerland, he was president of the Swiss Winterhilfe, an organization to help people during hard winters, and he presided the Mission Catholique Française in Zurich. With his wife Cécile, née Tercier, he had 9 children, five boys and four girls. 317 342 1 10.4171/089-1/13 http://www.ems-ph.org/doi/10.4171/089-1/13

General The story of the development of the Institute of Mathematics of the University of Fribourg begins 1896 when Mathias Lerch and Matthieu Franz Daniëls were nominated professors at the newly founded Faculty of Science. There is a long list of mathematicians who spent their career, or at least part of it, in Fribourg. Many began their professional activity in Fribourg and changed later to greener meadows at another university. Some spent only their last few active years at the Fribourg Mathematics Department. However they all left their traces and helped to put Fribourg on the mathematical map. Among the former was Michel Plancherel who was one of the great Swiss mathematician of the 20th century. He held a position in Fribourg from 1911 to 1921. Among the latter I should mention Peter Thullen who in his youth was already a distinguished mathematician in the field of complex analysis. He emigrated 1933 from Germany to Italy and then to South America where he changed his research to statistics and became an expert in Social Security. He held the position of professor of mathematical statistics in Fribourg from 1972 until his retirement in 1976. 343 350 1 10.4171/089-1/14 http://www.ems-ph.org/doi/10.4171/089-1/14

Jürg Kramer Humboldt-Universität zu Berlin, BERLIN, GERMANY General In this paper, the life and work of Martin Eichler, who taught at the University of Basel from 1958 until his retirement in 1980, is presented. It gives an overview of his fundamental contributions to the arithmetic of algebras and quadratic forms, the theory of modular forms and Jacobi forms, as well as the Riemann–Roch Theorem. In particular, one of his most original contributions, namely the confirmation the Taniyama–Shimura conjecture by first non-trivial examples, is discussed, which experienced a renaissance in connection with A. Wiles’ proof of the Fermat Conjecture. The article contains a complete list of Martin Eichler’s publications. 351 371 1 10.4171/089-1/15 http://www.ems-ph.org/doi/10.4171/089-1/15

Peter Mani , Hünibach/BE, SWITZERLAND History and biography General 373 388 1 10.4171/089-1/16 http://www.ems-ph.org/doi/10.4171/089-1/16

Martin Raussen Aalborg Universitet, AALBORG 0ST, DENMARK Alain Valette Université de Neuchâtel, NEUCHÂTEL, SWITZERLAND History and biography General 389 401 1 10.4171/089-1/17 http://www.ems-ph.org/doi/10.4171/089-1/17

Christine Riedtmann Universität Bern, BERN, SWITZERLAND History and biography General 403 421 1 10.4171/089-1/18 http://www.ems-ph.org/doi/10.4171/089-1/18

Alain Robert Université de Neuchâtel, NEUCHÂTEL, SWITZERLAND History and biography General In the second half of the 20th century, mathematics at the University of Neuchâtel occupied four different sites. Its teaching staff increased from two to five. The author of this article relates a few personal recollections concerning the actors and the atmosphere during this exceptional period. 423 439 1 10.4171/089-1/19 http://www.ems-ph.org/doi/10.4171/089-1/19

Urs Stammbach ETH Zürich, ZÜRICH, SWITZERLAND History and biography General The year 1930 was a year of great significance for the ETH in Zurich, for the development of mathematics in Switzerland, and also for mathematics in general. Firstly, in this year the ETH celebrated its 75th anniversary; secondly, Hermann Weyl, left Zurich after 17 fruitful years at ETH and accepted the call to succeed David Hilbert in Göttingen; thirdly, to replace Weyl the ETH appointed Heinz Hopf, who subsequently built up an important school in algebraic topology; and finally, just before accepting the call to ETH in 1930, Heinz Hopf completed his influential paper, in which he described and classified the essential maps from the three dimensional sphere onto the two dimensionals sphere. The present text is concerned with these four significant developments of the year 1930; central to the presentation are the eminent personalities of Hermann Weyl and Heinz Hopf. 441 469 1 10.4171/089-1/20 http://www.ems-ph.org/doi/10.4171/089-1/20

Kurt Strebel Universität Zürich, ZÜRICH, SWITZERLAND History and biography General The article starts with a proof of Denjoy’s conjecture by Lars Ahlfors who was then a twenty years old student of Rolf Nevanlinna. This is followed by an account of the first year of Nevanlinna in Zürich. He stimulated the mathematical and even the musical life of Zurich. In 1963 he retired and went back to Finland, where he became chancellor of the University of Turcu. In order not to lose him permanently Künzi and Strebel founded the Nevanlinna Colloqium, which took place about every other year and to which Nevanlinna was always invited. 471 485 1 10.4171/089-1/21 http://www.ems-ph.org/doi/10.4171/089-1/21

Claude Weber Université de Genève, GENÈVE 4, SWITZERLAND History and biography General In this paper the author recalls, in a very subjective way, some events which were at the origin of the “Troisième Cycle Romand de Mathématiques” and of the “Séminaire des Plans-sur-Bex”. It deals with the main actors who worked for their success. The paper ends with the list of the subjects and of the participants of Les Plans-sur-Bex. 487 503 1 10.4171/089-1/22 http://www.ems-ph.org/doi/10.4171/089-1/22

Eduard Zehnder ETH Zürich, LEO D 3, ZÜRICH, SWITZERLAND General This is a brief account on Moser’s important contributions to mathematics and physics, ranging from dynamical systems, celestial mechanics, spectral theory, calculus of variations, partial differential equations to differential geometry and complex analysis. 505 517 1 10.4171/089-1/23 http://www.ems-ph.org/doi/10.4171/089-1/23

Lectures on Logic Jean-Yves Girard Institut de Mathématiques de Luminy, Marseille, France Mathematical logic and foundations Category theory; homological algebra Computer science 03-01, 03A05, 03B40, 03F(03-05-07-10-20-40-52-99), 18A15, 68Q(05-10-12-15-55) Logic, proof-theory, incompleteness, sequent calculus, natural deduction, lambda-calculus, Curry–Howard isomorphism, system F, coherent spaces, linear logic, proof-nets, ludics, implicit complexity, geometry of interaction These lectures on logic, more specifically proof theory, are basically intended for postgraduate students and researchers in logic. The question at stake is the nature of mathematical knowledge and the difference between a question and an answer, i.e., the implicit and the explicit. The problem is delicate mathematically and philosophically as well: the relation between a question and its answer is a sort of equality where one side is “more equal than the other”: one thus discovers essentialist blind spots. Starting with Gödel’s paradox (1931) – so to speak, the incompleteness of answers with respect to questions – the book proceeds with paradigms inherited from Gentzen’s cut-elimination (1935). Various settings are studied: sequent calculus, natural deduction, lambda calculi, category-theoretic composition, up to geometry of interaction (GoI), all devoted to explicitation, which eventually amounts to inverting an operator in a von Neumann algebra. Mathematical language is usually described as referring to a preexisting reality. Logical operations can be given an alternative procedural meaning: typically, the operators involved in GoI are invertible, not because they are constructed according to the book, but because logical rules are those ensuring invertibility. Similarly, the durability of truth should not be taken for granted: one should distinguish between imperfect (perennial) and perfect modes. The procedural explanation of the infinite thus identifies it with the unfinished, i.e., the perennial. But is perenniality perennial? This questioning yields a possible logical explanation for algorithmic complexity. This highly original course on logic by one of the world’s leading proof theorists challenges mathematicians, computer scientists, physicists and philosophers to rethink their views and concepts on the nature of mathematical knowledge in an exceptionally profound way. 9 25 2011 978-3-03719-088-3 978-3-03719-588-8 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/088 http://www.ems-ph.org/doi/10.4171/088 Athanase Papadopoulos IRMA, Strasbourg, France History and biography Geometry Differential geometry 01-02, 51-02; 01A55, 51-01, 51-03, 53-02, 53-01, 53-03, 53A35 Lobachevsky geometry, hyperbolic geometry, non-Euclidean geometry, pangeometry Lobachevsky wrote his Pangeometry in 1855, the year before his death. This memoir is a résumé of his work on non-Euclidean geometry and its applications, and it can be considered as his clearest account on the subject. It is also the conclusion of his lifework, and the last attempt he made to acquire recognition. The treatise contains basic ideas of hyperbolic geometry, including the trigonometric formulae, the techniques of computation of arc length, of area and of volume, with concrete examples. It also deals with the applications of hyperbolic geometry to the computation of new definite integrals. The techniques are different from those found in most modern books on hyperbolic geometry since they do not use models. Besides its historical importance, Lobachevsky’s Pangeometry is a beautiful work, written in a simple and condensed style. The material that it contains is still very alive, and reading this book will be most useful for researchers and for students in geometry and in the history of science. It can be used as a textbook, as a source book and as a repository of inspiration. The present edition provides the first complete English translation of the Pangeometry that appears in print. It contains facsimiles of both the Russian and the French original versions. The translation is accompanied by notes, followed by a biography of Lobachevky and an extensive commentary. 12 1 2010 978-3-03719-087-6 978-3-03719-587-1 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/087 http://www.ems-ph.org/doi/10.4171/087 With Appendices by Michael Müger and Alexis Virelizier Vladimir Turaev Indiana University, Bloomington, USA Manifolds and cell complexes Associative rings and algebras Category theory; homological algebra Quantum theory 57-02, 81-02; 16-02, 18-02, 57M27 Topological Quantum Field Theory, surfaces, 3-manifolds, Frobenius algebras, monoidal categories, sections of fiber bundles Homotopy Quantum Field Theory (HQFT) is a branch of Topological Quantum Field Theory founded by E. Witten and M. Atiyah. It applies ideas from theoretical physics to study principal bundles over manifolds and, more generally, homotopy classes of maps from manifolds to a fixed target space. This book is the first systematic exposition of Homotopy Quantum Field Theory. It starts with a formal definition of an HQFT and provides examples of HQFTs in all dimensions. The main body of the text is focused on 2-dimensional and 3-dimensional HQFTs. A study of these HQFTs leads to new algebraic objects: crossed Frobenius group-algebras, crossed ribbon group-categories, and Hopf group-coalgebras. These notions and their connections with HQFTs are discussed in detail. The text ends with several appendices including an outline of recent developments and a list of open problems. Three appendices by M. Müger and A. Virelizier summarize their work in this area. The book is addressed to mathematicians, theoretical physicists, and graduate students interested in topological aspects of quantum field theory. The exposition is self-contained and well suited for a one-semester graduate course. Prerequisites include only basics of algebra and topology. 6 1 2010 978-3-03719-086-9 978-3-03719-586-4 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/086 http://www.ems-ph.org/doi/10.4171/086 10 Hans Triebel University of Jena, Germany Functional analysis Approximations and expansions Fourier analysis Computer science 46-02, 46E35, 42C40, 42B35, 68Q17, 41A55 Function spaces, Haar bases, Faber bases, sampling, numerical integration, discrepancy The first chapters of this book deal with Haar bases, Faber bases and some spline bases for function spaces in Euclidean n-space and n-cubes. This is used in the subsequent chapters to study sampling and numerical integration preferably in spaces with dominating mixed smoothness. The subject of the last chapter is the symbiotic relationship between numerical integration and discrepancy, measuring the deviation of sets of points from uniformity. This book is addressed to graduate students and mathematicians having a working knowledge of basic elements of function spaces and approximation theory, and who are interested in the subtle interplay between function spaces, complexity theory and number theory (discrepancy). 6 1 2010 978-3-03719-085-2 978-3-03719-585-7 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/085 http://www.ems-ph.org/doi/10.4171/085 11 Volume II: Standard Information for Functionals Erich Novak University of Jena, Germany Henryk Woźniakowski Columbia University, New York, USA, and University of Warsaw, Poland Numerical analysis Computer science 65Y20, 68Q17, 68Q25, 41A63, 65-02, 46E22, 28C20, 46E30, 11K38, 65D32 Discrepancy, lattice rules, Smolyak/sparse grid algorithms, decomposable reproducing kernels, curse of dimensionality, tractability, high-dimensional numerical problems, worst case, average case, probabilistic and randomized settings, weighted spaces This three-volume set is a comprehensive study of the tractability of multivariate problems. The present second volume deals with algorithms using standard information consisting of function values for the approximation of linear and selected nonlinear functionals. An important example is numerical multivariate integration. The proof techniques used in volumes I and II are quite different. It is especially hard to establish meaningful lower error bounds for the approximation of functionals by using finitely many function values. Here, the concept of decomposable reproducing kernels is helpful, allowing it to find matching lower and upper error bounds for some linear functionals. It is then possible to conclude tractability results from such error bounds. Tractability results even for linear functionals are very rich in variety. There are infinite-dimensional Hilbert spaces for which the approximation with an arbitrarily small error of all linear functionals requires only one function value. There are Hilbert spaces for which all nontrivial linear functionals suffer from the curse of dimensionality. This holds for unweighted spaces, where the role of all variables and groups of variables is the same. For weighted spaces one can monitor the role of all variables and groups of variables. Necessary and sufficient conditions on the decay of the weights are given to obtain various notions of tractability. The text contains extensive chapters on discrepancy and integration, decomposable kernels and lower bounds, the Smolyak/sparse grid algorithms, lattice rules and the CBC (component-by-component) algorithms. This is done in various settings. Path integration and quantum computation are also discussed. The book is of interest for researchers working in computational mathematics, especially in approximation of high-dimensional problems. It is also well suited for graduate courses and seminars. 61 open problems are listed to stimulate future research in tractability. 7 3 2010 978-3-03719-084-5 978-3-03719-584-0 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/084 http://www.ems-ph.org/doi/10.4171/084 12 Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids Ronald Brown Bangor University, UK Philip Higgins Durham University, UK Rafael Sivera Universitat de València, Spain Algebraic topology Category theory; homological algebra 55-02, 18B40, 18G50, 55P15, 55U50; 18A30, 18A40, 18D15, 18D30, 18D30, 18G55, 19D23, 20F65, 20L05,55Q45, 55S37, 55S45, 55U10, 55U15, 55U35, 57M05, 57M10, 57M15, 57M20 Algebraic topology, homotopy theory, nonabelian methods, van Kampen theorem, groupoids, cubical homotopy groupoids, crossed complexes, filtered spaces, crossed modules, double groupoids, cubical sets with connections, monoidal closed categories, higher category theory, homotopy classification of maps, classifying spaces The main theme of this book is that the use of filtered spaces rather than just topological spaces allows the development of basic algebraic topology in terms of higher homotopy groupoids; these algebraic structures better reflect the geometry of subdivision and composition than those commonly in use. Exploration of these uses of higher dimensional versions of groupoids has been largely the work of the first two authors since the mid 1960s. The structure of the book is intended to make it useful to a wide class of students and researchers for learning and evaluating these methods, primarily in algebraic topology but also in higher category theory and its applications in analogous areas of mathematics, physics and computer science. Part I explains the intuitions and theory in dimensions 1 and 2, with many figures and diagrams, and a detailed account of the theory of crossed modules. Part II develops the applications of crossed complexes. The engine driving these applications is the work of Part III on cubical ω-groupoids, their relations to crossed complexes, and their homotopically defined examples for filtered spaces. Part III also includes a chapter suggesting further directions and problems, and three appendices give accounts of some relevant aspects of category theory. Endnotes for each chapter give further history and references. 8 11 2011 978-3-03719-083-8 978-3-03719-583-3 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/083 http://www.ems-ph.org/doi/10.4171/083 15 Laurent Bessières Université Joseph Fourier, Grenoble, France Gérard Besson Université Joseph Fourier, Grenoble, France Michel Boileau Université Paul Sabatier, Toulouse, France Sylvain Maillot Université Montpellier II, France Joan Porti Universitat Autònoma de Barcelona, Spain Manifolds and cell complexes Differential geometry 57-02; 57M50, 53C44 Ricci flow, geometrisation, hyperbolisation, three-dimensional topology, geometric structure, Haken manifolds, JSJ decomposition The Geometrisation Conjecture was proposed by William Thurston in the mid 1970s in order to classify compact 3-manifolds by means of a canonical decomposition along essential, embedded surfaces into pieces that possess geometric structures. It contains the famous Poincaré Conjecture as a special case. In 2002, Grigory Perelman announced a proof of the Geometrisation Conjecture based on Richard Hamilton’s Ricci flow approach, and presented it in a series of three celebrated arXiv preprints. Since then there has been an ongoing effort to understand Perelman’s work by giving more detailed and accessible presentations of his ideas or alternative arguments for various parts of the proof. This book is a contribution to this endeavour. Its two main innovations are first a simplified version of Perelman’s Ricci flow with surgery, which is called Ricci flow with bubbling-off, and secondly a completely different and original approach to the last step of the proof. In addition, special effort has been made to simplify and streamline the overall structure of the argument, and make the various parts independent of one another. A complete proof of the Geometrisation Conjecture is given, modulo pre-Perelman results on Ricci flow, Perelman’s results on the ℒ-functional and κ-solutions, as well as the Colding–Minicozzi extinction paper. The book can be read by anyone already familiar with these results, or willing to accept them as black boxes. The structure of the proof is presented in a lengthy introduction, which does not require knowledge of geometric analysis. The bulk of the proof is the existence theorem for Ricci flow with bubbling-off, which is treated in parts I and II. Part III deals with the long time behaviour of Ricci flow with bubbling-off. Part IV finishes the proof of the Geometrisation Conjecture. 9 29 2010 978-3-03719-082-1 978-3-03719-582-6 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/082 http://www.ems-ph.org/doi/10.4171/082 13 Hamiltonian Vector Fields and Symplectic Capacities Eduard Zehnder ETH Zurich, Switzerland Dynamical systems and ergodic theory Ordinary differential equations Differential geometry Mechanics of particles and systems 37-01; 37A25, 37B25, 37C10, 37C20, 37C27, 37C29,37C50, 37C57, 37D05, 37J05, 37J35, 37J45, 37N05, 34C25, 34C28, 34C37, 34C45, 34D20, 70H15,70H05, 70H25, 53D35, 53C24, 53D05, 58E05, 58E30 Dynamical systems, ergodicity, transitivity, structural stability, stable and unstable manifolds, hyperbolic sets, shadowing lemma, homoclinic and heteroclinic orbits, Bernoulli systems, chaos, limit sets, Lyapunov function, gradient systems, Morse inequalities, symplectic maps, Hamiltonian vector fields, Hamiltonian formalism, integrable systems, symplectic capacities, Hofer–Zehnder capacity, action functional, mini-max principles, periodic motions on prescribed energy surfaces This book originated from an introductory lecture course on dynamical systems given by the author for advanced students in mathematics and physics at the ETH Zurich. The first part centres around unstable and chaotic phenomena caused by the occurrence of homoclinic points. The existence of homoclinic points complicates the orbit structure considerably and gives rise to invariant hyperbolic sets nearby. The orbit structure in such sets is analyzed by means of the shadowing lemma, whose proof is based on the contraction principle. This lemma is also used to prove S. Smale’s theorem about the embedding of Bernoulli systems near homoclinic orbits. The chaotic behavior is illustrated in the simple mechanical model of a periodically perturbed mathematical pendulum. The second part of the book is devoted to Hamiltonian systems. The Hamiltonian formalism is developed in the elegant language of the exterior calculus. The theorem of V. Arnold and R. Jost shows that the solutions of Hamiltonian systems which possess sufficiently many integrals of motion can be written down explicitly and for all times. The existence proofs of global periodic orbits of Hamiltonian systems on symplectic manifolds are based on a variational principle for the old action functional of classical mechanics. The necessary tools from variational calculus are developed. There is an intimate relation between the periodic orbits of Hamiltonian systems and a class of symplectic invariants called symplectic capacities. From these symplectic invariants one derives surprising symplectic rigidity phenomena. This allows a first glimpse of the fast developing new field of symplectic topology. 5 19 2010 978-3-03719-081-4 978-3-03719-581-9 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/081 http://www.ems-ph.org/doi/10.4171/081 Yurii Neretin University of Vienna, Austria, and Moscow State University, Russia Topological groups, Lie groups Several complex variables and analytic spaces Operator theory Differential geometry 22-02; 22E46, 53C35, 22E50, 47G10, 32M15, 51N30, 14K25, 47B50 Semisimple Lie groups and their representations, classical groups, symmetric spaces, Fourier transform, p-adic groups, integral operators, Hilbert spaces of holomorphic functions, holomorphic discrete series, reproducing kernels, theta-functions, Cartier This book is an elementary self-contained introduction to some constructions of representation theory and related topics of differential geometry and analysis. Topics covered include the theory of various Fourier-like integral operators as Segal–Bargmann transforms, Gaussian integral operators in $L^2$ and in the Fock space, integral operators with theta-kernels, the geometry of real and $p$-adic classical groups and symmetric spaces. The heart of the book is the Weil representation of the symplectic group (real and complex realizations, relations with theta-functions and modular forms, $p$-adic and adelic constructions) and representations in Hilbert spaces of holomorphic functions of several complex variables. The book is addressed to graduate students and researchers in representation theory, differential geometry, and operator theory. The reader is supposed to be familiar with standard university courses in linear algebra, functional analysis, and complex analysis. 2 25 2011 978-3-03719-080-7 978-3-03719-580-2 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/080 http://www.ems-ph.org/doi/10.4171/080 Vicente Cortés University of Hamburg, Germany Differential geometry Quantum theory Relativity and gravitational theory 53-00; 53C26, 53C50, 81T60, 83E30, 83E50, 53C25, 53C29, 53C35, 53C10 Pseudo-Riemannian manifold, supersymmetry, special geometry, quaternionic Kähler manifold, c-map, generalized geometry, sigma-model, skew-symmetric torsion, nearly Kähler manifold, Euclidian supersymmetry, black hole, para-geometries The purpose of this handbook is to give an overview of some recent developments in differential geometry related to supersymmetric field theories. The main themes covered are: special geometry and supersymmetry generalized geometry geometries with torsion para-geometries holonomy theory symmetric spaces and spaces of constant curvature conformal geometry wave equations on Lorentzian manifolds D-branes and K-theory The intended audience consists of advanced students and researchers working in differential geometry, string theory and related areas. The emphasis is on geometrical structures occurring on target spaces of supersymmetric field theories. Some of these structures can be fully described in the classical framework of pseudo-Riemannian geometry. Others lead to new concepts relating various fields of research, such as special Kähler geometry or generalized geometry. 6 10 2010 978-3-03719-079-1 978-3-03719-579-6 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/079 http://www.ems-ph.org/doi/10.4171/079 16

Martin Roček Stony Brook University, STONY BROOK, UNITED STATES Cumrun Vafa Harvard University, Cambridge, UNITED STATES Stefan Vandoren Universiteit Utrecht, UTRECHT, NETHERLANDS Differential geometry Quantum theory General Under the action of the c-map, special Kähler manifolds are mapped into a class of quaternion-Kähler spaces. We explicitly construct the corresponding Swann bundle or hyper-Kähler cone, and determine the hyper-Kähler potential in terms of the prepotencies of the special Kähler geometry. 3 13 1 10.4171/079-1/1 http://www.ems-ph.org/doi/10.4171/079-1/1

Gregor Weingart Universidad Nacional Autónoma de México, CUERNAVACA, MORELOS, MEXICO Differential geometry General The bundle of differential forms on a hyper-Kähler manifold is a representation of a natural Lie algebra so4,1ℝ of parallel operators. After constructing this algebra from the point of view of representation theory we calculate the minimal parallel decomposition of the bundle of differential forms on hyper-Kähler and quaternionic Kähler manifolds. 15 37 1 10.4171/079-1/2 http://www.ems-ph.org/doi/10.4171/079-1/2

Charles Boyer University of New Mexico, ALBUQUERQUE, UNITED STATES Krzysztof Galicki† University of New Mexico, ALBUQUERQUE, UNITED STATES Differential geometry Quantum theory General The main purpose of this review is to describe geometric properties of Sasaki–Einstein manifolds which make them interesting in modern theoretical physics. In spite of the fact that it is supersymmetry that connects Sasaki–Einstein spaces to physics, it is not the purpose of this article to describe what this concept really means to either physicists or mathematicians. There have been many recent attempts to frame these important notions of theoretical physics in precise mathematical terms. This enormous task is far beyond the scope of this article. Here we content ourselves with providing the main theorems and results concerning Killing spinors. 39 83 1 10.4171/079-1/3 http://www.ems-ph.org/doi/10.4171/079-1/3

María Lledó Universidad de Valencia, BURJASSOT (VALENCIA), SPAIN Óscar Maciá Universidad de Valencia, BURJASSOT (VALENCIA), SPAIN Antoine Van Proeyen Katholieke Universiteit Leuven, LEUVEN, BELGIUM Veeravalli Varadarajan UCLA, LOS ANGELES, UNITED STATES Differential geometry General In this paper we generalize special geometry to arbitrary signatures in target space. We formulate the definitions in a precise mathematical setting and give a translation to the coordinate formalism used in physics. For the projective case, we first discuss in detail projective Kähler manifolds, appearing in N = 1 supergravity. We develop a new point of view based on the intrinsic construction of the line bundle. The topological properties are then derived and the Levi-Civita connection in the projective manifold is obtained as a particular projection of a Levi-Civita connection in a ‘mother’ manifold with one extra complex dimension. The origin of this approach is in the superconformal formalism of physics, which is also explained in detail. Finally, we specialize these results to projective special Kähler manifolds and provide explicit examples with different choices of signature. 85 147 1 10.4171/079-1/4 http://www.ems-ph.org/doi/10.4171/079-1/4

Thomas Mohaupt University of Liverpool, Liverpool, UNITED KINGDOM Differential geometry Relativity and gravitational theory Quantum theory General We review recent developments in special geometry and explain its role in the theory of supersymmetric black holes. To make this article self-contained, a short introduction to black holes is given, with emphasis on the laws of black hole mechanics and black hole entropy. We also summarize the existing results on the para-complex version of special geometry, which occurs in Euclidean supersymmetry. The role of real coordinates in special geometry is illustrated, and we briefly indicate how Euclidean supersymmetry can be used to study stationary black hole solutions via dimensional reduction over time. 149 181 1 10.4171/079-1/5 http://www.ems-ph.org/doi/10.4171/079-1/5

Nigel Hitchin University of Oxford, OXFORD, UNITED KINGDOM Several complex variables and analytic spaces Algebraic geometry Differential geometry General “Generalized geometry” is an approach to differential geometric structures which seems remarkably well-adapted to some of the concepts in String Theory and Supergravity, for example: 3-form flux, gauged sigma-models, D-branes, connections with skew torsion. It also incorporates in a natural way the role of the B-field as a symmetry. Here we shall offer an introduction to this geometry and a few of its applications, showing how in particular some relatively old results in the physics literature acquire a natural meaning within this new setting. 185 208 1 10.4171/079-1/6 http://www.ems-ph.org/doi/10.4171/079-1/6

Alexei Kotov Université du Luxembourg, LUXEMBOURG, LUXEMBOURG Thomas Strobl Université Claude Bernard Lyon 1, VILLEURBANNE CEDEX, FRANCE General In this contribution we review some of the interplay between sigma models in theoretical physics and novel geometrical structures such as Lie (n-)algebroids. The first part of the article contains the mathematical background, the definition of various algebroids as well as of Dirac structures, a joint generalization of Poisson, presymplectic, but also complex structures. Proofs are given in detail. The second part deals with sigma models. Topological ones, in particular the AKSZ and the Dirac sigma models, as generalizations of the Poisson sigma models to higher dimensions and to Dirac structures, respectively, but also physical ones, that reduce to standard Yang Mills theories for the “flat” choice of a Lie algebra: Lie algebroid Yang–Mills theories and possible action functionals for nonabelian gerbes and general higher gauge theories. Characteristic classes associated to Dirac structures and to higher principal bundels are also mentioned. 209 262 1 10.4171/079-1/7 http://www.ems-ph.org/doi/10.4171/079-1/7

Ulf Lindström Uppsala Universitet, UPPSALA, SWEDEN Martin Roček Stony Brook University, STONY BROOK, UNITED STATES Rikard von Unge Masaryk University, BRNO, CZECH REPUBLIC Maxim Zabzine Uppsala Universitet, UPPSALA, SWEDEN Differential geometry General We show that, locally, in the neighbourhood of a regular point, all geometric objects of Generalized Kähler Geometry can be derived from a function K, the “generalized Kähler potential”. The metric g and two-form B are determined as nonlinear functions of second derivatives of K. These nonlinearities are shown to arise via a quotient construction from an auxiliary local product (ALP) space. 263 273 1 10.4171/079-1/8 http://www.ems-ph.org/doi/10.4171/079-1/8

Ilka Agricola Philipps-Universität, MARBURG, GERMANY Differential geometry Quantum theory General This review article intends to introduce the reader to non-integrable geometric structures on Riemannian manifolds and invariant metric connections with torsion, and to discuss recent aspects of mathematical physics – in particular superstring theory – where these naturally appear. 277 346 1 10.4171/079-1/9 http://www.ems-ph.org/doi/10.4171/079-1/9

Paul-Andi Nagy University of Auckland, AUCKLAND, NEW ZEALAND Differential geometry General We study almost Hermitian structures admitting a Hermitian connection with totally skew-symmetric torsion or equivalently, those almost Hermitian structures with totally skew-symmetric Nijenhuis tensor. We investigate up to what extent the Nijenhuis tensor fails to be parallel with respect to the characteristic connection. This is naturally described by means of an extension of the notion of Killing form to almost Hermitian geometry. In this context, we also make an essentially self-contained survey of nearly-Kähler geometry, from the perspective of non-integrable holonomy systems. 347 398 1 10.4171/079-1/10 http://www.ems-ph.org/doi/10.4171/079-1/10

Jean-Baptiste Butruille Université de Cergy-Pontoise, CERGY-PONTOISE CEDEX, FRANCE Differential geometry General We classify six-dimensional homogeneous nearly Kähler manifolds and give a positive answer to Gray and Wolf’s conjecture: every homogeneous nearly Kähler manifold is a Riemannian 3-symmetric space equipped with its canonical almost Hermitian structure. The only four examples in dimension 6 are S3 × S3, the complex projective space ℂP3, the flag manifold 399 423 1 10.4171/079-1/11 http://www.ems-ph.org/doi/10.4171/079-1/11

Lars Schäfer Leibniz Universität Hannover, HANNOVER, GERMANY Fabian Schulte-Hengesbach Universität Hamburg, HAMBURG, GERMANY Differential geometry General The subject of this chapter is six-dimensional nearly (para-)Kähler geometry with pseudo-Riemannian metrics. Firstly, we derive the analogue of the well-known exterior differential system characterising a nearly Kähler manifold and prove applications to the automorphism group of a nearly (para-)Kähler structure. Secondly, we prove existence and uniqueness results for left-invariant nearly (para-)Kähler structures on Lie groups G × G where G is three-dimensional and simple. 425 453 1 10.4171/079-1/12 http://www.ems-ph.org/doi/10.4171/079-1/12

Andrew Swann University of Southern Denmark, ODENSE M, DENMARK Differential geometry Quantum theory General The geometry of hyper-Kähler manifolds with torsion is surveyed, particularly in the cases arising from Michelson and Strominger’s description of superconformal symmetry of one-dimensional quantum mechanics. 455 474 1 10.4171/079-1/13 http://www.ems-ph.org/doi/10.4171/079-1/13

Stefan Ivanov University of Sofia, SOFIA, BULGARIA Ivan Minchev University of Sofia, SOFIA, BULGARIA Simeon Zamkovoy University of Sofia, SOFIA, BULGARIA Differential geometry General We investigate the integrability of almost complex structures on the twistor space of an almost para-quaternionic manifold as well as the integrability of natural almost paracomplex structures on the reflector space of an almost para-quaternionic manifold constructed with the help of an arbitrary para-quaternionic connection. We show that if there exists an integrable structure then it is independent on the para-quaternionic connection. In dimension four, we express the ant-self-duality condition in terms of the Riemannian Ricci forms with respect to the associated para-quaternionic structure. 477 496 1 10.4171/079-1/14 http://www.ems-ph.org/doi/10.4171/079-1/14

Matthias Krahe , Frankfurt, GERMANY Differential geometry General Twistor theory goes back to the 1960s, where twistor methods were introduced by R. Penrose to provide an approach to quantum gravity. The general idea is to translate a problem on a differentiable manifold M endowed with a certain geometric structure to a problem on a complex manifold Z, the twistor space. The purpose of this work is to transfer some of these methods to paracomplex geometry. 497 557 1 10.4171/079-1/15 http://www.ems-ph.org/doi/10.4171/079-1/15

Dmitri Alekseevsky University of Edinburgh, EDINBURGH, UNITED KINGDOM Costantino Medori Universit, PARMA, ITALY Adriano Tomassini Università degli Studi di Parma, PARMA, ITALY Differential geometry Global analysis, analysis on manifolds General An almost para-CR structure on a manifold M is given by a distribution HM ⊂ TM together with a field K ∈ (End(HM)) of involutive endomorphisms of HM. If K satisfies an integrability condition, then (HM,K) is called a para-CR structure. The notion of maximally homogeneous para-CR structure of a semisimple type is given. A classification of such maximally homogeneous para-CR structures is given in terms of appropriate gradations of real semisimple Lie algebras. 559 577 1 10.4171/079-1/16 http://www.ems-ph.org/doi/10.4171/079-1/16

Anton Galaev Masaryk University, BRNO, CZECH REPUBLIC Thomas Leistner The University of Adelaide, , AUSTRALIA Differential geometry General We review recent results in the theory of holonomy groups of pseudo-Riemannian manifolds, i.e., manifolds with indefinite metrics. First we present the classification of Lorentzian holonomy groups, that is a list of possible groups and metrics which realise these as holonomy groups. This is followed by applications and some remarks about holonomy related structures on Lorentzian manifolds. Then we review partial results in signature (2, n + 2), in particular the classification of unitary holonomy groups, again presenting the groups and the realising metrics. Then we turn to results in neutral signature (n, n), focussing on the situation of a para-Kähler structure. Finally, the classification in signature (2, 2) obtained by Bérard-Bergery and Ikemakhen is presented. As a new result we prove the existence of metrics in cases for which the realisation as holonomy group was left open in their article. 581 627 1 10.4171/079-1/17 http://www.ems-ph.org/doi/10.4171/079-1/17

Antonio Di Scala Politecnico di Torino, TORINO, ITALY Thomas Leistner The University of Adelaide, , AUSTRALIA Thomas Neukirchner Humboldt-Universität zu Berlin, BERLIN, GERMANY Differential geometry Group theory and generalizations Topological groups, Lie groups General In this note we give proofs of the following three algebraic facts which have applications in the theory of holonomy groups and homogeneous spaces: Any irreducibly acting connected subgroup G ⊂ GL(n,ℝ) is closed. Moreover, if G admits an invariant bilinear form of Lorentzian signature, G is maximal, i.e. it is conjugated to SO(1,n − 1)0. We calculate the vector space of G-invariant symmetric bilinear forms, show that it is at most 3-dimensional, and determine the maximal stabilizers for each dimension. Finally, we give some applications and present some open problems. 629 651 1 10.4171/079-1/18 http://www.ems-ph.org/doi/10.4171/079-1/18

Konrad Waldorf University of California, BERKELEY, UNITED STATES Differential geometry Quantum theory General The notion of holonomy of a line bundle (endowed with a connection) around a loop is extended to the holonomy of a gerbe along a closed oriented surface, which corresponds to the interaction of a string with a three-form gauge field. 653 682 1 10.4171/079-1/19 http://www.ems-ph.org/doi/10.4171/079-1/19

Ines Kath Ernst-Moritz-Arndt-Universität Greifswald, GREIFSWALD, GERMANY Differential geometry General We give an introduction to the long-standing classification problem for pseudo- Riemannian symmetric spaces and summarise some old and new approaches and results. 685 702 1 10.4171/079-1/20 http://www.ems-ph.org/doi/10.4171/079-1/20

Dmitri Alekseevsky University of Edinburgh, EDINBURGH, UNITED KINGDOM Differential geometry General We study simply connected pseudo-Kähler and para-Kähler symmetric spaces that is simply connected pseudo-Riemannian symmetric spaces (M,g) which admit a parallel skew-symmetric complex or para-complex structure J ( such that J2 = −Id or J2 = Id respectively.) The classification of para-Kähler symmetric spaces reduces to description of 3-graded Lie algebras g = g−1 + g0 + g1 such that g0-modules g−1 and g1 are dual. The classification of pseudo-Kähler symmetric spaces reduces to description of some anti-involutive automorphisms of complex 3-graded Lie algebras. We classify para-Kähler and pseudo-Kähler Ricci-flat symmetric spaces of dimension ≤ 6. A class of Ricci-flat para-Kähler and pseudo-Kähler symmetric spaces with commutative holonomy group is described in terms of rank four tensors. 703 729 1 10.4171/079-1/21 http://www.ems-ph.org/doi/10.4171/079-1/21

Oliver Baues Georg-August Universität Göttingen, GÖTTINGEN, GERMANY Differential geometry Number theory Group theory and generalizations Topological groups, Lie groups The theory of flat pseudo-Riemannian manifolds and flat affine manifolds is closely connected to the topic of prehomogeneous affine representations of Lie groups. In this article, we exhibit several aspects of this correspondence. At the heart of our presentation is a development of the theory of characteristic classes and characters of prehomogeneous affine representations. We give applications concerning flat affine, as well as pseudo-Riemannian and symplectic affine flat manifolds. 731 817 1 10.4171/079-1/22 http://www.ems-ph.org/doi/10.4171/079-1/22

Helga Baum Humboldt-Universität zu Berlin, BERLIN, GERMANY Differential geometry General This survey intends to introduce the reader to holonomy theory of Cartan connections. Special attention is given to the normal conformal Cartan connection, uniquely defined for a class of conformally equivalent metrics, and to its holonomy group – the ‘conformal holonomy group’. We explain the relation between conformal holonomy group and existence of Einstein metrics in the conformal class as well as the relation between conformal holonomy group and existence of conformal Killing spinors. In particular, we describe Lorentzian manifolds with conformal holonomy group in SU(1, m), which can be viewed as conformal analog of Calabi–Yau manifolds. Such Lorentzian metrics, known as Fefferman metrics, appear on S1-bundles over strictly pseudoconvex CR spin manifolds. 821 861 1 10.4171/079-1/23 http://www.ems-ph.org/doi/10.4171/079-1/23

Yoshinobu Kamishima Tokyo Metropolitan University, TOKYO, JAPAN Differential geometry General In this expository article, we study integrable, nondegenerate geometric structures on odd dimensional smooth manifolds simultaneously. We are mainly interested in conformal structure on n-manifolds, CR structure on (2n + 1)-manifolds and a new structure, pseudo-conformal quaternionic CR structure on (4n + 3)-manifolds. 863 893 1 10.4171/079-1/24 http://www.ems-ph.org/doi/10.4171/079-1/24

Christian Bär Universität Potsdam, POTSDAM, GERMANY Global analysis, analysis on manifolds Quantum theory General We summarize the analytic theory of linear wave equations on globally hyperbolic Lorentzian manifolds. 897 914 1 10.4171/079-1/25 http://www.ems-ph.org/doi/10.4171/079-1/25

Daniel Freed University of Texas at Austin, AUSTIN, UNITED STATES Differential geometry Quantum theory General In this chapter we explain the relation between D-branes in string theory and K-theory. We will not review the purely mathematical developments, but will instead recount the justification for positing K-theory as the home for D-brane charges. 915 925 1 10.4171/079-1/26 http://www.ems-ph.org/doi/10.4171/079-1/26

Analysis and MATLAB programs Helge Holden Norwegian University of Science and Technology, Trondheim, Norway Kenneth Karlsen University of Oslo, Norway Knut-Andreas Lie University of Oslo, Norway Nils Henrik Risebro University of Oslo, Norway Numerical analysis Partial differential equations Operator theory 35L65, 35K65, 65M99, 65M15, 65M12; 35L67, 35D30, 47N40, 35A35 Operator splitting, nonlinear partial differential equations, evolution equations, hyperbolic conservation laws, degenerate convection-diffusion equations, finite difference methods, finite volume methods, front tracking, Matlab codes Operator splitting (or the fractional steps method) is a very common tool to analyze nonlinear partial differential equations both numerically and analytically. By applying operator splitting to a complicated model one can often split it into simpler problems that can be analyzed separately. In this book one studies operator splitting for a family of nonlinear evolution equations, including hyperbolic conservation laws and degenerate convection-diffusion equations. Common for these equations is the prevalence of rough, or non-smooth, solutions, e.g., shocks. Rigorous analysis is presented, showing that both semi-discrete and fully discrete splitting methods converge. For conservation laws, sharp error estimates are provided and for convection-diffusion equations one discusses a priori and a posteriori correction of entropy errors introduced by the splitting. Numerical methods include finite difference and finite volume methods as well as front tracking. The theory is illustrated by numerous examples. There is a dedicated web page that provides MATLAB codes for many of the examples. The book is suitable for graduate students and researchers in pure and applied mathematics, physics, and engineering. 4 30 2010 978-3-03719-078-4 978-3-03719-578-9 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/078 http://www.ems-ph.org/doi/10.4171/078 A.C.M. Ran Vrije Universiteit Amsterdam, The Netherlands Herman te Riele Centrum Wiskunde en Informatica, Amsterdam, The Netherlands Jan Wiegerinck University of Amsterdam, The Netherlands 00Bxx The European Congress of Mathematics, held every four years, has established itself as a major international mathematical event. Following those in Paris (1992), Budapest (1996), Barcelona (2000) and Stockholm (2004), the Fifth European Congress of Mathematics (5ECM) took place in Amsterdam, The Netherlands, July 14–18, 2008, with about 1000 participants from 68 different countries. Ten plenary and thirty-three invited lectures were delivered. Three science lectures outlined applications of mathematics in other sciences: climate change, quantum information theory and population dynamics. As in the four preceding EMS congresses, ten EMS prizes were granted to very promising young mathematicians. In addition, the Felix Klein Prize was awarded, for the second time, for an application of mathematics to a concrete and difficult industrial problem. There were twenty-two minisymposia, spread over the whole mathematical area. Two round table meetings were organized: one on industrial mathematics and one on mathematics and developing countries. As part of the 44th Nederlands Mathematisch Congres, which was embedded in 5ECM, the so-called Brouwer lecture was presented. It is the Netherland's most prestigious award in mathematics, organized every three years by the Royal Dutch Mathematical Society. Information about Brouwer was given in an invited historical lecture during the congress. These proceedings contain a selection of the contributions to the congress, providing a permanent record of the best what mathematics offers today. 4 24 2010 978-3-03719-077-7 978-3-03719-577-2 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/077 http://www.ems-ph.org/doi/10.4171/077

José Carrillo Imperial College London, LONDON, UNITED KINGDOM Jesús Rosado Universidad Autonoma de Barcelona, BELLATERRA, SPAIN Partial differential equations General We show how to extend the method used in [22] to prove uniqueness of solutions to a family of several nonlocal equations containing aggregation terms and aggregation/diffusion competition. They contain several mathematical biology models proposed in macroscopic descriptions of swarming and chemotaxis for the evolution of mass densities of individuals or cells. Uniqueness is shown for bounded nonnegative mass-preserving weak solutions without diffusion. In diffusive cases, we use a coupling method [16], [33], and thus we need a stochastic representation of the solution to hold. In summary, our results show, modulo certain technical hypotheses, that nonnegative mass-preserving solutions remain unique as long as their L∞-norm is controlled in time. 3 16 1 10.4171/077-1/1 http://www.ems-ph.org/doi/10.4171/077-1/1

Bas Edixhoven Universiteit Leiden, LEIDEN, NETHERLANDS Number theory Algebraic geometry General An overview for a non-specialised audience is given of joint work with Jean-Marc Couveignes, Robin de Jong, Franz Merkl, and Johan Bosman. This joint work concerns fast computation of coefficients of modular forms, via the computation of associated Galois representations. For example, for p prime, Ramanujan’s τ(p) can be computed in time polynomial in log p. The overview focuses on the main results and ideas. Developments since 2006 are included: more examples by Johan Bosman, generalisation to forms of level one and of arbitrary weight, and an application to theta functions of lattices. Some future developments are mentioned. 17 29 1 10.4171/077-1/2 http://www.ems-ph.org/doi/10.4171/077-1/2

Manfred Einsiedler ETH Zürich, HG G 64.2, ZÜRICH, SWITZERLAND Dynamical systems and ergodic theory Number theory General In these notes we discuss some equidistribution problems with the aim to give reasonable error rates, i.e., we are interested in effective statements. We motivate some arguments by studying a concrete problem on a two-torus, and then describe recent results on the equidistribution of semisimple orbits obtained in joint work with G. Margulis and A. Venkatesh. We end by studying the relationship between equidistribution of closed orbits and mixing properties. This leads to a way of transporting spectral gap from one group – via an effective equidistribution result on a quotient by an irreducible lattice – to another group. The latter topic is ongoing joint work with G. Margulis and A. Venkatesh. 31 51 1 10.4171/077-1/3 http://www.ems-ph.org/doi/10.4171/077-1/3

Wolfgang Lück Universität Bonn, BONN, GERMANY Manifolds and cell complexes $K$-theory Group theory and generalizations General This is a survey on known results and open problems about closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. Many examples come from certain kinds of non-positive curvature conditions. The property aspherical, which is a purely homotopy theoretical condition, implies many striking results about the geometry and analysis of the manifold or its universal covering, and the ring theoretic properties and the K- and L-theory of the group ring associated to its fundamental group. The Borel Conjecture predicts that closed aspherical manifolds are topologically rigid. The article contains new results about product decompositions of closed aspherical manifolds and an announcement of a result joint with Arthur Bartels and Shmuel Weinberger about hyperbolic groups with spheres of dimension ≥ 6 as boundary. At the end we describe (winking) our universe of closed manifolds. 53 82 1 10.4171/077-1/4 http://www.ems-ph.org/doi/10.4171/077-1/4

Sergei Merkulov Stockholm University, STOCKHOLM, SWEDEN Category theory; homological algebra Functional analysis General Wheeled props is one the latest species found in the world of operads and props. We attempt to give an elementary introduction to the main ideas of the theory of wheeled props for beginners, and also a survey of its most recent major applications (ranging from algebra and geometry to deformation theory and Batalin–Vilkovisky quantization) which might be of interest to experts. 83 114 1 10.4171/077-1/5 http://www.ems-ph.org/doi/10.4171/077-1/5

Oleg Musin University of Texas at Brownsville, BROWNSVILLE, UNITED STATES Fourier analysis Convex and discrete geometry General I. J. Schoenberg proved that a function is positive definite in the unit sphere if and only if this function is a nonnegative linear combination of Gegenbauer polynomials. This fact plays a crucial role in Delsarte’s method for finding bounds for the density of sphere packings on spheres and Euclidean spaces. One of the most exciting applications of Delsarte’s method is a solution of the kissing number problem in dimensions 8 and 24. However, 8 and 24 are the only dimensions in which this method gives a precise result. For other dimensions (for instance, three and four) the upper bounds exceed the lower. We have found an extension of Delsarte’s method that allows to solve the kissing number problem (as well as the one-sided kissing number problem) in dimensions three and four. In this paper we also will discuss the maximal cardinalities of spherical two-distance sets. With help of the so-called polynomial method and Delsarte’s method these cardinalities can be determined for all dimensions n < 40. Recently, extensions were found of Schoenberg’s theorem for multivariate positive-definite functions. With help of these extensions and semidefinite programming some upper bounds for spherical codes can be improved. 115 134 1 10.4171/077-1/6 http://www.ems-ph.org/doi/10.4171/077-1/6

Jaroslav Nešetřil Charles University, PRAHA 1, CZECH REPUBLIC Patrice Ossona de Mendez Ecole des hautes études en sciences sociales, PARIS, FRANCE Combinatorics General This paper surveys the recent research related to the structural properties of sparse graphs and general finite relational structures. We provide a classification of classes of finite relational structures which is defined by means of a sequence of derived classes (defined by local changes). The resulting classification is surprisingly robust and it leads to many equivalent formulations which play an important role in applications in model theory, algorithmic graph theory and structural theory. One of these equivalent formulations guarantees a quick test for a finite type decomposition (called low tree depth decomposition). We give numerous examples from geometry and extremal graph theory and apply the result to dualities for special classes of graphs. Finally we characterize the existence of all restricted dualities in terms of limit objects induced by the convergence defined on the homomorphism order of graphs. 135 165 1 10.4171/077-1/7 http://www.ems-ph.org/doi/10.4171/077-1/7

Jürgen Fuchs Karlstads Universitet, KARLSTAD, SWEDEN Thomas Nikolaus Universität Regensburg, REGENSBURG, GERMANY Christoph Schweigert Universität Hamburg, HAMBURG, GERMANY Konrad Waldorf University of California, BERKELEY, UNITED STATES Differential geometry General Hermitian bundle gerbes with connection are geometric objects for which a notion of surface holonomy can be defined for closed oriented surfaces. We systematically introduce bundle gerbes by closing the pre-stack of trivial bundle gerbes under descent. Inspired by structures arising in a representation theoretic approach to rational conformal field theories, we introduce geometric structure that is appropriate to define surface holonomy in more general situations: Jandl gerbes for unoriented surfaces, D-branes for surfaces with boundaries, and bi-branes for surfaces with defect lines. 167 195 1 10.4171/077-1/8 http://www.ems-ph.org/doi/10.4171/077-1/8

Constantin Teleman University of California, BERKELEY, UNITED STATES Manifolds and cell complexes Algebraic geometry Quantum theory General We discuss a formality result for 2-dimensional topological field theories which are based on a semi-simple Frobenius algebra: namely, when sufficiently constrained by structural axioms, the complete theory is determined by the Frobenius algebra and the grading information. The structural constraints apply to Gromov–Witten theory of a variety whose quantum cohomology is semi-simple. Some open questions about semi-simple field theories are mentioned in the final section. 197 210 1 10.4171/077-1/9 http://www.ems-ph.org/doi/10.4171/077-1/9

Dirk van Dalen Utrecht University, UTRECHT, NETHERLANDS History and biography Mathematical logic and foundations General Brouwer’s dissertation contains both the germs of his topological and his foundational work. We concentrate here on the latter. The extraordinary rich thesis contains comments and critique on his contemporaries, and a novel approach to many foundational issues. Between the lines one finds the genesis of the continuum and the natural numbers via the ur-intuition, the constructive interpretation of logic, choice sequences, and a precise discussion of the language, logic, and mathematics levels. The present paper provides a survey of the material and comments on Brouwer’s innovations. 213 230 1 10.4171/077-1/10 http://www.ems-ph.org/doi/10.4171/077-1/10

Jean Bourgain Institute for Advanced Study, PRINCETON, UNITED STATES Number theory Group theory and generalizations General This is a survey of a line of research in arithmetic combinatorics. It is centered around so-called sum-product phenomena in various settings and its applications to problems in number theory, computer science, spectral and ergodic theory. More specifically, the sum-product results in finite fields and residue rings lead to new bounds on exponential sums of various types and in fact provide the first non-trivial estimates. A typical result in this spirit are bounds on Gauss sums for small multiplicative subgroups. Product theorems in matrix spaces derived from the scalar theory enable one to prove various conjectures on the expansion of Cayley graphs and the existence of spectral gaps for Hecke operators, most notably in SL2(q) and SU(2). Those in turn lead to an extension of Selberg’s theorem for congruence subgroups and new results on prime sieving in non-elementary subgroups of SL2(ℤ). Finally, Furstenberg’s “stiffness problem” for toral actions of subgroups of SL2(ℤ) as well as a quantitative equidistibution property of the orbits are described. 233 251 1 10.4171/077-1/11 http://www.ems-ph.org/doi/10.4171/077-1/11

Jean-François Le Gall Université Paris-Sud, ORSAY CEDEX, FRANCE Probability theory and stochastic processes Combinatorics General We discuss scaling limits of random planar maps chosen uniformly at random in a certain class. This leads to a universal limiting space called the Brownian map, which is viewed as a random compact metric space. The Brownian map can be obtained as a quotient of the continuous random tree called the CRT, for an equivalence relation which is defined in terms of Brownian labels assigned to the vertices of the CRT. We discuss the known properties of the Brownian map. In particular, we give a complete description of the geodesics starting from the distinguished point called the root. We also discuss applications to various properties of large random planar maps. 253 276 1 10.4171/077-1/12 http://www.ems-ph.org/doi/10.4171/077-1/12

François Loeser Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE Number theory Algebraic geometry Several complex variables and analytic spaces General Non-archimedean integrals are ubiquitous in various parts of mathematics. Motivic integration allows to understand them geometrically and to get strong uniformity statements. In these notes, intended for a general audience, we start by giving various examples of situations where one can get new geometric results by using p-adic or motivic integrals. We then present some more recent results in this area, in particular a Transfer Principle allowing to transfer identities involving functions defined by integrals from one class of local fields to another. Orbital integrals occurring in the Fundamental Lemma of Langlands Theory form a natural family of functions falling within the range of application of this Transfer Principle. 277 292 1 10.4171/077-1/13 http://www.ems-ph.org/doi/10.4171/077-1/13

Matilde Marcolli California Institute of Technology, PASADENA, UNITED STATES Quantum theory $K$-theory General This article gives an overview of recent results on the relation between quantum field theory and motives, with an emphasis on two different approaches: a “bottom-up” approach based on the algebraic geometry of varieties associated to Feynman graphs, and a “top-down” approach based on the comparison of the properties of associated categorical structures. This survey is mostly based on joint work of the author with Paolo Aluffi, along the lines of the first approach, and on previous work of the author with Alain Connes on the second approach. 293 332 1 10.4171/077-1/14 http://www.ems-ph.org/doi/10.4171/077-1/14

Nicolai Reshetikhin University of California, BERKELEY, UNITED STATES Manifolds and cell complexes Quantum theory General This article is an overview of the developments in topological quantum field theory, and, in particular on the progress in the Chern–Simons theory. 333 377 1 10.4171/077-1/15 http://www.ems-ph.org/doi/10.4171/077-1/15

Olga Holtz University of California, BERKELEY, UNITED STATES Noam Shomron Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES Numerical analysis Algebraic geometry Group theory and generalizations Computer science We survey classical and recent developments in numerical linear algebra, focusing on two issues: computational complexity, or arithmetic costs, and numerical stability, or performance under roundoff error. We present a brief account of the algebraic complexity theory as well as the general error analysis for matrix multiplication and related problems. We emphasize the central role played by the matrix multiplication problem and discuss historical and modern approaches to its solution. 381 400 1 10.4171/077-1/16 http://www.ems-ph.org/doi/10.4171/077-1/16

Bo'az Klartag Tel-Aviv University, TEL-AVIV, ISRAEL Functional analysis Convex and discrete geometry Probability theory and stochastic processes General We review recent advances in the understanding of probability measures with geometric characteristics on Rn , for large n. These advances include the central limit theorem for convex sets, according to which the uniform measure on a high-dimensional convex body has marginals that are approximately gaussian. 401 417 1 10.4171/077-1/17 http://www.ems-ph.org/doi/10.4171/077-1/17

Laure Saint-Raymond Ecole Normale Superieure, PARIS CEDEX 05, FRANCE Partial differential equations Fluid mechanics Statistical mechanics, structure of matter General The behaviour of a gas can be described by different models depending on the time and space scales to be considered. A natural question is therefore to know if there exist continuous transitions between these models: this is precisely the matter of the sixth problem asked by Hilbert on the occasion of the International Congress of Mathematicians held in Paris in 1900. Recent works allow to understand rigorously the passage from Boltzmann’s equation (which provides a statistical description of the microscopic state of the gas) to continuous models of fluid mechanics. The goal of the present paper is to present briefly the mathematical framework of the study, and to exhibit some difficulties of the asymptotic analysis related to the physics of the problem. 419 439 1 10.4171/077-1/18 http://www.ems-ph.org/doi/10.4171/077-1/18

Agata Smoktunowicz University of Edinburgh, EDINBURGH, UNITED KINGDOM Associative rings and algebras General It is shown that the associated graded algebras of affine nil algebras need not be nil. A similar argument shows that filtered algebras generated in degree one of nil algebras need not be nil. This answers a question of Riley [5]. 441 449 1 10.4171/077-1/19 http://www.ems-ph.org/doi/10.4171/077-1/19

Emil Straube Texas A&M University, College Station, USA Several complex variables and analytic spaces Partial differential equations 32-02; 32W05, 35N15 ∂-Neumann – problem, ℒ2-theory, pseudoconvex domains, compactness, global regularity This book provides a thorough and self-contained introduction to the ∂-Neumann problem, leading up to current research, in the context of the ℒ2-Sobolev theory on bounded pseudoconvex domains in ℂn. It grew out of courses for advanced graduate students and young researchers given by the author at the Erwin Schrödinger International Institute for Mathematical Physics and at Texas A&M University. The introductory chapter provides an overview of the contents and puts them in historical perspective. The second chapter presents the basic ℒ2-theory. Following is a chapter on the subelliptic estimates on strictly pseudoconvex domains. The two final chapters on compactness and on regularity in Sobolev spaces bring the reader to the frontiers of research. Prerequisites are a solid background in basic complex and functional analysis, including the elementary ℒ2-Sobolev theory and distributions. Some knowledge in several complex variables is helpful. Concerning partial differential equations, not much is assumed. The elliptic regularity of the Dirichlet problem for the Laplacian is quoted a few times, but the ellipticity results needed for elliptic regularization in the third chapter are proved from scratch. 2 26 2010 978-3-03719-076-0 978-3-03719-576-5 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/076 http://www.ems-ph.org/doi/10.4171/076 Robert Penner Centre for Quantum Geometry of Moduli Spaces, Aarhus, Denmark, and Caltech, Pasadena, USA Functions of a complex variable Several complex variables and analytic spaces 30-02, 30F60, 32G15, 30F10, 30Fxx Teichmüller space, mapping class group, Riemann’s moduli space, quantum Teichmüller theory There is an essentially “tinker-toy” model of a trivial bundle over the classical Teichmüller space of a punctured surface, called the decorated Teichmüller space, where the fiber over a point is the space of all tuples of horocycles, one about each puncture. This model leads to an extension of the classical mapping class groups called the Ptolemy groupoids and to certain matrix models solving related enumerative problems, each of which has proved useful both in mathematics and in theoretical physics. These spaces enjoy several related parametrizations leading to a rich and intricate algebro-geometric structure tied to the already elaborate combinatorial structure of the tinker-toy model. Indeed, the natural coordinates give the prototypical examples not only of cluster algebras but also of tropicalization. This interplay of combinatorics and coordinates admits further manifestations, for example, in a Lie theory for homeomorphisms of the circle, in the geometry underlying the Gauss product, in profinite and pronilpotent geometry, in the combinatorics underlying conformal and topological quantum field theories, and in the geometry and combinatorics of macromolecules. This volume gives the story and wider context of these decorated Teichmüller spaces as developed by the author over the last two decades in a series of papers, some of them in collaboration. Sometimes correcting errors or typos, sometimes simplifying proofs and sometimes articulating more general formulations than the original research papers, this volume is self-contained and requires little formal background. Based on a master’s course at Aarhus University, it gives the first treatment of these works in monographic form. 1 2 2012 978-3-03719-075-3 978-3-03719-575-8 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/075 http://www.ems-ph.org/doi/10.4171/075 Gebhard Böckle Interdisciplinary Center for Scientific Computing, Heidelberg, Germany Richard Pink ETH Zürich, Switzerland Number theory Algebraic geometry 11-02, 14-02 Characteristic $p$ cohomologies, Drinfeld modules, $A$-motives, $L$-functions, Weil conjecture This book develops a new cohomological theory for schemes in positive characteristic p and it applies this theory to give a purely algebraic proof of a conjecture of Goss on the rationality of certain L-functions arising in the arithmetic of function fields. These L-functions are power series over a certain ring A, associated to any family of Drinfeld A-modules or, more generally, of A-motives on a variety of finite type over the finite field Fp. By analogy to the Weil conjecture, Goss conjectured that these L-functions are in fact rational functions. In 1996 Taguchi and Wan gave a first proof of Goss’s conjecture by analytic methods à la Dwork. The present text introduces A-crystals, which can be viewed as generalizations of families of A-motives, and studies their cohomology. While A-crystals are defined in terms of coherent sheaves together with a Frobenius map, in many ways they actually behave like constructible étale sheaves. A central result is a Lefschetz trace formula for L-functions of A-crystals, from which the rationality of these L-functions is immediate. Beyond its application to Goss’s L-functions, the theory of A-crystals is closely related to the work of Emerton and Kisin on unit root F-crystals, and it is essential in an Eichler–Shimura type isomorphism for Drinfeld modular forms as constructed by the first author. The book is intended for researchers and advanced graduate students interested in the arithmetic of function fields and/or cohomology theories for varieties in positive characteristic. It assumes a good working knowledge in algebraic geometry as well as familiarity with homological algebra and derived categories, as provided by standard textbooks. Beyond that the presentation is largely self-contained. 10 1 2009 978-3-03719-074-6 978-3-03719-574-1 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/074 http://www.ems-ph.org/doi/10.4171/074 9 Alain Connes IHÉS, Bures-sur-Yvette, France Frédéric Fauvet IRMA, Strasbourg, France Jean-Pierre Ramis Institut de Mathématiques de Toulouse, France Algebraic geometry Quantum theory 14F42, 34M30, 34M35, 34M37, 34M40, 81T15, 81T18, 81T75; 11S40, 33B30, 34M50 Renormalization, motives, noncommutative geometry, Hopf algebras, differential Galois groups, Borel summation, Gevrey asymptotics, Stokes phenomenon, resurgent functions This volume is the outcome of a CIRM Workshop on Renormalization and Galois Theories held in Luminy, France, in March 2006. The subject of this workshop was the interaction and relationship between four currently very active areas: renormalization in quantum field theory (QFT), differential Galois theory, noncommutative geometry, motives and Galois theory. The last decade has seen a burst of new techniques to cope with the various mathematical questions involved in QFT, with notably the development of a Hopf-algebraic approach and insights into the classes of numbers and special functions that systematically appear in the calculations of perturbative QFT (pQFT). The analysis of the ambiguities of resummation of the divergent series of pQFT, an old problem, has been renewed, using recent results on Gevrey asymptotics, generalized Borel summation, Stokes phenomenon and resurgent functions. The purpose of the present book is to highlight, in the context of renormalization, the convergence of these various themes, orchestrated by diverse Galois theories. It contains three lecture courses together with five research articles and will be useful to both reseachers and graduate students in mathematics and physics. 9 24 2009 978-3-03719-073-9 978-3-03719-573-4 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/073 http://www.ems-ph.org/doi/10.4171/073 15

Caterina Consani The Johns Hopkins University, BALTIMORE, UNITED STATES Global analysis, analysis on manifolds Number theory Algebraic geometry General This paper gives a short and historical survey on the theory of pure motives in algebraic geometry and reviews some of the recent developments of this theory in noncommutative geometry. The second part of the paper outlines the new theory of endomotives and some of its relevant applications in number theory. 1 37 1 10.4171/073-1/1 http://www.ems-ph.org/doi/10.4171/073-1/1

Vincent Rivasseau Université Paris Sud-XI, ORSAY CEDEX, FRANCE Fabien Vignes-Tourneret Université Claude Bernard Lyon 1, VILLEURBANNE CEDEX, FRANCE General The first renormalisable quantum field theories on non-commutative space have been found recently. We review this rapidly growing subject. 39 82 1 10.4171/073-1/2 http://www.ems-ph.org/doi/10.4171/073-1/2

David Sauzin , PARIS, FRANCE General This article is an introduction to some aspects of Écalle’s mould calculus, a powerful combinatorial tool which yields surprisingly explicit formulas for the normalising series attached to an analytic germ of singular vector field or of map. This is illustrated on the case of the saddle-node, a two-dimensional vector field which is formally conjugate to Euler’s vector field x2 ∂/∂x + (x + y) ∂/∂y, and for which the formal normalisation is shown to be resurgent in 1/x. Resurgence monomials adapted to alien calculus are also described as another application of mould calculus. 83 163 1 10.4171/073-1/3 http://www.ems-ph.org/doi/10.4171/073-1/3

Yves André Projet de théorie des nombres, tour 25 4ème étage, PARIS, FRANCE General From its early beginnings up to nowadays, algebraic number theory has evolved in symbiosis with Galois theory: indeed, one could hold that it consists in the very study of the absolute Galois group of the field of rational numbers. Nothing like that can be said of transcendental number theory. Nevertheless, could one not associate conjugates and a Galois group to transcendental numbers such as π? Beyond, can one not envision an appropriate Galois theory in the field of transcendental number theory? In which role? The aim of this text is to indicate what Grothendieck’s theory of motives has to say, at least conjecturally, on these questions. 165 177 1 10.4171/073-1/4 http://www.ems-ph.org/doi/10.4171/073-1/4

Kurusch Ebrahimi-Fard Université de Haute Alsace, MULHOUSE CEDEX 07, FRANCE Dominique Manchon Université Blaise Pascal, AUBIÈRE CEDEX, FRANCE General We describe various combinatorial aspects of the Birkhoff–Connes–Kreimer factorization in perturbative renormalization. The analog of Bogoliubov’s preparation map on the Lie algebra of Feynman graphs is identified with the pre-Lie Magnus expansion. Our results apply to any connected filtered Hopf algebra, based on the pro-nilpotency of the Lie algebra of infinitesimal characters. 179 207 1 10.4171/073-1/5 http://www.ems-ph.org/doi/10.4171/073-1/5

Michael Hoffman United States Naval Academy, ANNAPOLIS, UNITED STATES Combinatorics Associative rings and algebras Quantum theory General The Connes–Kreimer Hopf algebra of rooted trees, its dual, and the Foissy Hopf algebra of planar rooted trees are related to each other and to the well-known Hopf algebras of symmetric and quasi-symmetric functions via a pair of commutative diagrams. We show how this point of view can simplify computations in the Connes–Kreimer Hopf algebra and its dual, particularly for combinatorial Dyson–Schwinger equations. 209 227 1 10.4171/073-1/6 http://www.ems-ph.org/doi/10.4171/073-1/6

Frédéric Menous Université Paris-Sud, ORSAY CEDEX, FRANCE General The study of solutions of differential equations (analytic or formal) can often be reduced to a conjugacy problem, namely the conjugation of a given equation to a much simpler one, using identity-tangent diffeomorphisms. On the one hand, following Écalle’s work (with a different terminology), such diffeomorphisms are given by characters on a given Hopf algebra (here a shuffle Hopf algebra). On the other hand, for some equations, the obstacles in the formal conjugacy are reflected in the fact that the associated characters appear to be ill-defined. The analogy with the need for a renormalization scheme (dimensional regularization, Birkhoff decomposition) in quantum field theory becomes obvious for such equations and deliver a wide range of toy models. We discuss here the case of a simple class of differential equations where a renormalization scheme yields meaningful results. 229 246 1 10.4171/073-1/7 http://www.ems-ph.org/doi/10.4171/073-1/7

Stefan Weinzierl Johannes Gutenberg-Universität, MAINZ, GERMANY General In this article I review the connections between Feynman integrals and multiple polylogarithms. After an introductory section on loop integrals I discuss the Mellin–Barnes transformation and shuffle algebras. In a subsequent section multiple polylogarithms are introduced. Finally, I discuss how certain Feynman integrals evaluate to multiple polylogarithms. 247 270 1 10.4171/073-1/8 http://www.ems-ph.org/doi/10.4171/073-1/8

Jochen Blath Technical University of Berlin, Germany Peter Imkeller Humboldt University of Berlin, Germany Sylvie Rœlly University of Potsdam, Germany Probability theory and stochastic processes Primary: 60-06, Secondary 60Gxx, 60Jxx Stochastic processes, stochastic finance, stochastic analysis,statistical physics, stochastic differential equations The 33rd Bernoulli Society Conference on “Stochastic Processes and Their Applications” was held in Berlin from July 27 to July 31, 2009. It brought together more than 600 researchers from 49 countries, who communicated recent progress in the mathematical research related to stochastic processes, with applications ranging from biology to statistical mechanics, finance and climatology. The present book collects survey articles highlighting new trends and focal points in the area written by plenary speakers of the conference, all of them outstanding international experts. A particular aim of this collection is to inspire young scientists in setting up research goals within the wide scope of fields represented in this volume. 8 10 2011 978-3-03719-072-2 978-3-03719-572-7 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/072 http://www.ems-ph.org/doi/10.4171/072

Saïd Hamadène Université du Maine, LE MANS CEDEX 9, FRANCE Probability theory and stochastic processes Statistics Game theory, economics, social and behavioral sciences Systems theory; control This paper deals with the optimal switching problem with multiple regimes. We first provide a Verification Theorem, expressed by Snell envelope of processes, and later we show that it has a solution. We then study the connection with systems of reflected BSDEs with oblique reflection. In the case when randomness stems from a diffusion process, we study the relation of the problem with systems of variational inequalities with inter-connected obstacles. We prove that this latter has a unique continuous solution in viscosity sense. 1 27 1 10.4171/072-1/1 http://www.ems-ph.org/doi/10.4171/072-1/1

Claudia Klüppelberg TU München, GARCHING, GERMANY Ross Maller Australian National University, CANBERRA, AUSTRALIA Alexander Szimayer Universität Hamburg, HAMBURG, GERMANY Probability theory and stochastic processes Statistics Game theory, economics, social and behavioral sciences General Continuous time models have been elevated to great importance in the modelling of time series data, in response to the successful options pricing model of Black and Scholes (1973), among other things. In 2004, Klüppelberg, Lindner, and Maller introduced the “COGARCH” model as a continuous-time analogue to the enormously influential and successful discrete time GARCH stochastic volatility model of Engle and Bollerslev. Like the GARCH model, the COGARCH is based on a single source of random variability, in this case, on a single background driving Lévy process. Since its inception, the original COGARCH model has been studied intensively and generalised and extended in various ways. In the present paper we formulate the model using stochastic differential equations and review some of its important properties as well as some recent developments, including some statistical issues. As a new contribution we present a COGARCH option pricing model including the possibility of default, in which the underlying stock price process is taken as a stochastic exponential of a COGARCH model with drift. We give a preliminary analysis of this model in its risk-neutral dynamics, and as a prominent example, compute European option prices in the Variance-Gamma COGARCH model. For practical implementation, we must discretise the continuous-time COGARCH onto a discrete grid over a finite time interval. We go on to review ways of doing this by means of various approximation schemes, in particular, via a “first jump” approximation to the underlying Lévy process, which preserves features of the process important for optimal stopping problems. Some other applications of the technology, especially to the modelling of irregularly spaced time series data, are discussed too. 29 58 1 10.4171/072-1/2 http://www.ems-ph.org/doi/10.4171/072-1/2

Servet Martínez Universidad de Chile, SANTIAGO, CHILE Jaime San Martín Universidad de Chile, SANTIAGO, CHILE Probability theory and stochastic processes General We analyze the quasi-stationary distributions for discrete time finite state Markov chains, we give some known properties and supply new elementary results, including a lumping property. 59 66 1 10.4171/072-1/3 http://www.ems-ph.org/doi/10.4171/072-1/3

Peter Mörters University of Bath, BATH, UNITED KINGDOM Probability theory and stochastic processes Statistical mechanics, structure of matter General The parabolic Anderson model is the Cauchy problem for the heat equation with random potential. It offers a case study for the effects that a random, or irregular, environment can have on a diffusion process. The main focus in the present survey is on phenomena that are due to a highly irregular potential, which we model by a spatially independent, identically distributed random field with heavy tails. Among the effects we discuss are random fluctuations in the growth of the total mass, localisation in the weak and almost sure sense, and ageing. 67 85 1 10.4171/072-1/4 http://www.ems-ph.org/doi/10.4171/072-1/4

Etienne Pardoux Université de Provence, MARSEILLE CEDEX 13, FRANCE Anton Wakolbinger J. W. Goethe-Universität, FRANKFURT, GERMANY Probability theory and stochastic processes General Based on an intuitive approach to the Ray–Knight representation of Feller’s branching diffusion in terms of Brownian excursions we survey a few recent developments around exploration and mass excursions. One of these is Bertoin’s “tree of alleles with rare mutations” [6], seen as a tree of excursions of Feller’s branching diffusion. Another one is a model of a population with individual reproduction, pairwise fights and emigration to ever new colonies, conceived as a tree of excursions of Feller’s branching diffusion with logistic growth [14]. Finally, we report on a Ray–Knight representation of Feller’s branching diffusion with logistic growth in terms of a reflected Brownian motion whose drift depends on the local time accumulated at its current level [19]. 87 106 1 10.4171/072-1/5 http://www.ems-ph.org/doi/10.4171/072-1/5

Gesine Reinert University of Oxford, OXFORD, UNITED KINGDOM Probability theory and stochastic processes General Combining Malliavin calculus and Stein’s method has recently lead to a new framework for normal and for chi-square approximation, and for second-order Poincaré inequalities. Applications include functionals of Gaussian random fields as well as functionals of infinite Poisson and Rademacher sequences. Here we present the framework and an extension which leads to invariance principles for multilinear homogeneous sums of general centered independent random variables. In the spirit of Stein’s method, the key is weak dependence, which can be quantified by the influence function of a variable. In particular we see that Wiener chaos is universal, in the sense that the normal approximations of any homogeneous sum can be completely characterised by first switching to its Wiener chaos counterpart. Then the Malliavin–Stein framework yields simple upper bounds and convergence criteria to the normal (and to the chi-square) distribution. These results are joint work with Ivan Nourdin and Giovanni Peccati. 107 126 1 10.4171/072-1/6 http://www.ems-ph.org/doi/10.4171/072-1/6

Laurent Saloff-Coste Cornell University, ITHACA, UNITED STATES Jessica Zúñiga Stanford University, STANFORD, UNITED STATES Probability theory and stochastic processes General We discuss problems posed by the quantitative study of time inhomogeneous Markov chains. The two main notions for our purpose are merging and stability. Merging (also called weak ergodicity) occurs when the chain asymptotically forgets where it started. It is a loss of memory property. Stability relates to the question of whether or not, despite temporary variations, there is a rough shape describing the long time behavior of the chain. For instance, we will discuss an example where the long time behavior is roughly described by a binomial, with temporal variations. 127 151 1 10.4171/072-1/7 http://www.ems-ph.org/doi/10.4171/072-1/7

Alexander Schied Universität Mannheim, MANNHEIM, GERMANY Alla Slynko , GARCHING-HOCHBRÜCK, GERMANY Partial differential equations Calculus of variations and optimal control; optimization Probability theory and stochastic processes General Market impact models describe the feedback of trading strategies on the underlying asset prices. The resulting feedback effects lead to interesting questions of stability and regularity, which are closely related to the existence and behavior of strategies for optimal order execution. In this paper, we give a survey on some recent results that were obtained in this context. In the first part, we explain in particular the stochastic control approach to the maximization of the expected utility of revenues in the Almgren–Chriss framework. In the second part, we describe stability issues that arise when market impact is allowed to be transient. 153 179 1 10.4171/072-1/8 http://www.ems-ph.org/doi/10.4171/072-1/8

Gordon Slade University of British Columbia, VANCOUVER, CANADA Probability theory and stochastic processes Statistical mechanics, structure of matter General Simple random walk is well understood. However, if we condition a random walk not to intersect itself, so that it is a self-avoiding walk, then it is much more difficult to analyse and many of the important mathematical problems remain unsolved. This paper provides an overview of some of what is known about the critical behaviour of the self-avoiding walk, including some old and some more recent results, using methods that touch on combinatorics, probability, and statistical mechanics. 181 199 1 10.4171/072-1/9 http://www.ems-ph.org/doi/10.4171/072-1/9

Masayoshi Takeda Tohoku University, SENDAI, JAPAN Probability theory and stochastic processes Partial differential equations General The theory of Dirichlet forms is an $L^2$-theory, while the theory of Markov processes is, in a sense, an $L^1$-theory. To bridge this gap, we study the $L^p$-independence of growth bounds of Markov semigroups, more generally, of generalized Feynman–Kac (Schrödinger) semigroups. A key idea for the proof of the $L^p$-independence is to employ arguments in the Donsker-Varadhan large deviation theory. 201 226 1 10.4171/072-1/10 http://www.ems-ph.org/doi/10.4171/072-1/10

Feng-Yu Wang Beijing Normal University, BEIJING, CHINA Probability theory and stochastic processes Operator theory General In this survey paper we introduce some recent progresses on functional inequalities and applications. We start from a Poincaré type inequality on Hilbert spaces which is corresponding to upper bound estimates of the essential spectrum for the infinitesimal generator, then clarify the inequality and further applications in the framework of Dirichlet form. Applications to the exponential decay of semigroups in the tail norm and transportation-cost inequalities with super cost functions will be presented. Finally, functional inequalities on Riemannian manifolds with unbounded below curvature or non-convex boundary will be investigated. 227 245 1 10.4171/072-1/11 http://www.ems-ph.org/doi/10.4171/072-1/11

Generating Functions, Boundary Theory, Random Walks on Trees Wolfgang Woess Graz University of Technology, Austria Probability theory and stochastic processes 60-01; 60J10, 60J50, 60J80, 60G50, 05C05, 94C05, 15A48 Markov chain, discrete time, denumerable state space, recurrence, transience, reversible Markov chain, electric network, birth-and-death chains, Galton–Watson process, branching Markov chain, harmonic functions, Martin compactification, Poisson boundary, random walks on trees Markov chains are the first and most important examples of random processes. This book is about time-homogeneous Markov chains that evolve with discrete time steps on a countable state space. Measure theory is not avoided, careful and complete proofs are provided. A specific feature is the systematic use, on a relatively elementary level, of generating functions associated with transition probabilities for analyzing Markov chains. Basic definitions and facts include the construction of the trajectory space and are followed by ample material concerning recurrence and transience, the convergence and ergodic theorems for positive recurrent chains. There is a side-trip to the Perron–Frobenius theorem. Special attention is given to reversible Markov chains and to basic mathematical models of “population evolution” such as birth-and-death chains, Galton–Watson process and branching Markov chains. A good part of the second half is devoted to the introduction of the basic language and elements of the potential theory of transient Markov chains. Here the construction and properties of the Martin boundary for describing positive harmonic functions are crucial. In the long final chapter on nearest neighbour random walks on (typically infinite) trees the reader can harvest from the seed of methods laid out so far, in order to obtain a rather detailed understanding of a specific, broad class of Markov chains. The level varies from basic to more advanced, addressing an audience from master’s degree students to researchers in mathematics, and persons who want to teach the subject on a medium or advanced level. A specific characteristic of the book is the rich source of classroom-tested exercises with solutions. 8 15 2009 978-3-03719-071-5 978-3-03719-571-0 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/071 http://www.ems-ph.org/doi/10.4171/071 A Path Integral Approach Sergio Albeverio University of Bonn, Germany Yuri Kondratiev University of Bielefeld, Germany Yuri Kozitsky Maria Curie-Skłodowska University, Lublin, Poland Michael Röckner University of Bielefeld, Germany Statistical mechanics, structure of matter Functional analysis 82-02, 82B10; 46G12, 46T12, 60G60, 60J25 Gibbs random field, Dobruchin–Lanford–Ruelle approach, quantum oscillator, Schrödinger operator, KMS state, Green function, algebra of observables, Euclidean Gibbs measure, phase transition, quantum effect, decay of correlations, critical point Quantum statistical mechanics plays a major role in many fields such as, for instance, thermodynamics, plasma physics, solid-state physics, and the study of stellar structure. While the theory of quantum harmonic oscillators is relatively simple, the case of anharmonic oscillators, a mathematical model of a localized quantum particle, is more complex and challenging. Moreover, infinite systems of interacting quantum anharmonic oscillators possess interesting ordering properties with respect to quantum stabilization. This book presents a rigorous approach to the statistical mechanics of such systems, in particular with respect to their actions on a crystal lattice. The text is addressed to both mathematicians and physicists, especially those who are concerned with the rigorous mathematical background of their results and the kind of problems that arise in quantum statistical mechanics. The reader will find here a concise collection of facts, concepts, and tools relevant for the application of path integrals and other methods based on measure and integration theory to problems of quantum physics, in particular the latest results in the mathematical theory of quantum anharmonic crystals. The methods developed in the book are also applicable to other problems involving infinitely many variables, for example, in biology and economics. 7 8 2009 978-3-03719-070-8 978-3-03719-570-3 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/070 http://www.ems-ph.org/doi/10.4171/070 8 Erich Novak University of Jena, Germany Ian Sloan University of New South Wales, Sydney, Australia Joseph Traub Columbia University, New York, USA Henryk Woźniakowski Columbia University, New York, USA Numerical analysis History and biography This book contains five essays on the complexity of continuous problems, written for a wider audience. Henryk Woźniakowski and the complexity of continuous problems Complexity as a new challenge for mathematicians A brief history of information-based complexity How high is high-dimensional? What is information-based complexity? The first four essays are based on talks presented in 2008 when Henryk Woźniakowski received an honorary doctoral degree of the Friedrich Schiller University of Jena. The focus is on introduction and history of the complexity of continuous problems, as well as on recent progress concerning the complexity of high-dimensional numerical problems. The last essay provides a brief and informal introduction to the basic notions and concepts of information-based complexity addressed to a general readership. 6 24 2009 978-3-03719-069-2 978-3-03719-569-7 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/069 http://www.ems-ph.org/doi/10.4171/069

Erich Novak Friedrich-Schiller-Universität Jena, JENA, GERMANY Numerical analysis History and biography General Henryk Woźniakowski was presented with an honorary doctoral degree by the University of Jena. The celebration took place on June 6, 2008, in the auditorium of the Friedrich Schiller University of Jena. This paper is an expanded and translated version of the laudatio. 1 37 1 10.4171/069-1/1 http://www.ems-ph.org/doi/10.4171/069-1/1

Henryk Woźniakowski Columbia University, NEW YORK, UNITED STATES Numerical analysis General We discuss the computational complexity of three problems: matrix multiplication, multivariate integration of smooth periodic function, and multivariate approximation of smooth functions. The first two problems are studied in the worst case setting whereas the third problem is studied in the average case setting with the folded Wiener sheet measure. These three problems serve as an illustration that computational complexity presents a new set of questions whose answers probably require new proof techniques. That is why even problems which were thoroughly studied in mathematics need to be revisited when we want to find sharp bounds on their complexity. 39 59 1 10.4171/069-1/2 http://www.ems-ph.org/doi/10.4171/069-1/2

Joseph Traub Columbia University, NEW YORK, UNITED STATES Numerical analysis History and biography General The note describes the history of information-based complexity (IBC), from the early beginnings (the earliest precursor mentioned is R. von Mises with a paper from 1933) till today. 61 71 1 10.4171/069-1/3 http://www.ems-ph.org/doi/10.4171/069-1/3

Ian Sloan University of New South Wales, SYDNEY, NSW, AUSTRALIA Numerical analysis General supp { vertical-align: 1.6ex; font-size:90%; } subb { vertical-align: -0.8ex; font-size:90%; } .ions { line-height: 2.8; } .ions supp {position: absolute; top: 1; left: 1; .ions subb {position: relative; top: 1; left: 1; } Consider the problem of evaluating the d-dimensional integral Id(f) := ∫10 · · · ∫10 f(x1,…, xd) dx1 … dxd = ∫[0,1]d f(x) dx, where f is a continuous function over the d-dimensional unit cube. To take up immediately the question in the title, let us ask: how large can we reasonably take d to be? 73 87 1 10.4171/069-1/4 http://www.ems-ph.org/doi/10.4171/069-1/4

Henryk Woźniakowski Columbia University, NEW YORK, UNITED STATES Numerical analysis General The purpose of this short note is to informally introduce information-based complexity (IBC). The basic notions of IBC for the approximate solution of continuous mathematically posed problems are described. 89 95 1 10.4171/069-1/5 http://www.ems-ph.org/doi/10.4171/069-1/5

Demetrios Christodoulou ETH Zürich, Switzerland Relativity and gravitational theory Partial differential equations Global analysis, analysis on manifolds In 1965 Penrose introduced the fundamental concept of a trapped surface, on the basis of which he proved a theorem which asserts that a spacetime containing such a surface must come to an end. The presence of a trapped surface implies, moreover, that there is a region of spacetime, the black hole, which is inaccessible to observation from infinity. A major challenge since that time has been to find out how trapped surfaces actually form, by analyzing the dynamics of gravitational collapse. The present monograph achieves this aim by establishing the formation of trapped surfaces in pure general relativity through the focusing of gravitational waves. The theorems proved in the present monograph constitute the first foray into the long-time dynamics of general relativity in the large, that is, when the initial data are no longer confined to a suitable neighborhood of trivial data. The main new method, the short pulse method, applies to general systems of Euler–Lagrange equations of hyperbolic type, and provides the means to tackle problems which have hitherto seemed unapproachable. This monograph will be of interest to people working in general relativity, geometric analysis, and partial differential equations. 1 10 2009 978-3-03719-068-5 978-3-03719-568-0 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/068 http://www.ems-ph.org/doi/10.4171/068 Christian Lubich University of Tübingen, Germany Numerical analysis Quantum theory 65M70, 65Z05, 81-08 Quantum dynamics, Schrödinger equation, model reduction, numerical methods Quantum dynamics of molecules poses a variety of computational challenges that are presently at the forefront of research efforts in numerical analysis in a number of application areas: high-dimensional partial differential equations, multiple scales, highly oscillatory solutions, and geometric structures such as symplecticity and reversibility that are favourably preserved in discretizations. This text addresses such problems in quantum mechanics from the viewpoint of numerical analysis, illustrating them to a large extent on intermediate models between the Schrödinger equation of full many-body quantum dynamics and the Newtonian equations of classical molecular dynamics. The fruitful interplay between quantum dynamics and numerical analysis is emphasized. 9 24 2008 978-3-03719-067-8 978-3-03719-567-3 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/067 http://www.ems-ph.org/doi/10.4171/067 Second Edition Stanley Payne University of Colorado Denver, Denver, USA Joseph Thas Ghent University, Ghent, Belgium Combinatorics Geometry 05-02, 51-02; 05B05, 05B25, 51B10, 51B15, 51B20, 51E05, 51E12, 51E14, 51E20, 51E21, 51E30 Finite generalized quadrangle, finite projective space, generalized polygon, polar space, incidence geometry, circle geometry Generalized quadrangles (GQ) were formally introduced by J. Tits in 1959 in order to describe geometric properties of simple groups of Lie type of rank 2. After its appearance in 1984, Finite Generalized Quadrangles (FGQ) quickly became the standard reference for finite GQ. It presents the whole story of the subject from the very beginning in a book of modest length. This second edition is essentially a reprint of the first edition. It is a careful rendering into LaTeX of the original, along with an appendix that introduces major new results pertaining to GQ, especially in those areas in which the authors of this work have made a contribution. The first edition has been out of print for many years, and the new edition makes again available this classical reference in the rapidly increasing field of finite geometries. 4 8 2009 978-3-03719-066-1 978-3-03719-566-6 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/066 http://www.ems-ph.org/doi/10.4171/066 Alexander Schmitt Freie Universität Berlin, Germany Algebraic geometry Commutative rings and algebras 14L24, 14H60; 13A50, 14D20, 14F05, 14L30 Geometric invariant theory, reductive group, representation, group action, categorical, good, geometric quotient, principal bundle, decorated bundle, moduli space The book starts with an introduction to Geometric Invariant Theory (GIT). The fundamental results of Hilbert and Mumford are exposed as well as more recent topics such as the instability flag, the finiteness of the number of quotients, and the variation of quotients. In the second part, GIT is applied to solve the classification problem of decorated principal bundles on a compact Riemann surface. The solution is a quasi-projective moduli scheme which parameterizes those objects that satisfy a semistability condition originating from gauge theory. The moduli space is equipped with a generalized Hitchin map. Via the universal Kobayashi–Hitchin correspondence, these moduli spaces are related to moduli spaces of solutions of certain vortex type equations. Potential applications include the study of representation spaces of the fundamental group of compact Riemann surfaces. The book concludes with a brief discussion of generalizations of these findings to higher dimensional base varieties, positive characteristic, and parabolic bundles. The text is fairly self-contained (e.g., the necessary background from the theory of principal bundles is included) and features numerous examples and exercises. It addresses students and researchers with a working knowledge of elementary algebraic geometry. 7 4 2008 978-3-03719-065-4 978-3-03719-565-9 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/065 http://www.ems-ph.org/doi/10.4171/065 A Classical View of Algebraic Geometry First corrected reprint, June 2012 Mauro Beltrametti University of Genova, Italy Ettore Carletti University of Genova, Italy Dionisio Gallarati University of Genova, Italy Giacomo Monti Bragadin University of Genova, Italy Algebraic geometry 14-01; 14E05, 14E07, 14H50, 14J26, 14J70, 14M99, 14N05 Algebraic geometry, projective varieties, curves, surfaces, special varieties This book offers a wide-ranging introduction to algebraic geometry along classical lines. It consists of lectures on topics in classical algebraic geometry, including the basic properties of projective algebraic varieties, linear systems of hypersurfaces, algebraic curves (with special emphasis on rational curves), linear series on algebraic curves, Cremona transformations, rational surfaces, and notable examples of special varieties like the Segre, Grassmann, and Veronese varieties. An integral part and special feature of the presentation is the inclusion of many exercises, not easy to find in the literature and almost all with complete solutions. The text is aimed at students of the last two years of an undergraduate program in mathematics. It contains some rather advanced topics suitable for specialized courses on the advanced undergraduate or beginning graduate level, as well as interesting topics for a senior thesis. The prerequisites have been deliberately limited to basic elements of projective geometry and abstract algebra. Thus, for example, some knowledge of the geometry of subspaces and properties of fields is assumed. The book will be welcomed by teachers and students of algebraic geometry who are seeking a clear and panoramic path leading from the basic facts about linear subspaces, conics and quadrics to a systematic discussion of classical algebraic varieties and the tools needed to study them. The text provides a solid foundation for approaching more advanced and abstract literature. 8 8 2009 978-3-03719-064-7 978-3-03719-564-2 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/064 http://www.ems-ph.org/doi/10.4171/064 Paul Seidel Massachusetts Institute of Technology, Cambridge, USA Differential geometry Associative rings and algebras Several complex variables and analytic spaces 53D40; 32Q65, 53D12, 16E45 Floer homology, Fukaya category, homological algebra, Lagrangian submanifolds, Lefschetz fibrations, mirror symmetry, pseudo-holomorphic curve, symplectic geometry The central objects in the book are Lagrangian submanifolds and their invariants, such as Floer homology and its multiplicative structures, which together constitute the Fukaya category. The relevant aspects of pseudo-holomorphic curve theory are covered in some detail, and there is also a self-contained account of the necessary homological algebra. Generally, the emphasis is on simplicity rather than generality. The last part discusses applications to Lefschetz fibrations, and contains many previously unpublished results. The book will be of interest to graduate students and researchers in symplectic geometry and mirror symmetry. Winner 2010 AMS Veblen Prize in Geometry 6 26 2008 978-3-03719-063-0 978-3-03719-563-5 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/063 http://www.ems-ph.org/doi/10.4171/063 Andrzej Skowroński Nicolaus Copernicus University, Toru&#324;, Poland Associative rings and algebras Primary 16Gxx; Secondary 03Cxx, 05Exx, 13Axx, 13Cxx, 13Hxx, 14Dxx ,14Gxx, 14Lxx, 14Mxx, 16Dxx, 16Exx, 16Sxx, 17Bxx, 18Exx, 18Gxx, 20Cxx, 20Gxx, 20Jxx, 32Sxx, 55Pxx, 57Mxx, 81Rxx This book is concerned with recent trends in the representation theory of algebras and its exciting interaction with geometry, topology, commutative algebra, Lie algebras, quantum groups, homological algebra, invariant theory, combinatorics, model theory and theoretical physics. The collection of articles, written by leading researchers in the field, is conceived as a sort of handbook providing easy access to the present state of knowledge and stimulating further development. The topics under discussion include diagram algebras, Brauer algebras, cellular algebras, quasi-hereditary algebras, Hall algebras, Hecke algebras, symplectic reflection algebras, Cherednik algebras, Kashiwara crystals, Fock spaces, preprojective algebras, cluster algebras, rank varieties, varieties of algebras and modules, moduli of representations of quivers, semi-invariants of quivers, Cohen–Macaulay modules, singularities, coherent sheaves, derived categories, spectral representation theory, Coxeter polynomials, Auslander–Reiten theory, Calabi–Yau triangulated categories, Poincaré duality spaces, selfinjective algebras, periodic algebras, stable module categories, Hochschild cohomologies, deformations of algebras, Galois coverings of algebras, tilting theory, algebras of small homological dimensions, representation types of algebras, model theory. The book consists of fifteen self-contained expository survey articles and is addressed to researchers and graduate students in algebra as well as a broader mathematical community. They contain a large number of open problems and give new perspectives for research in the field. 9 24 2008 978-3-03719-062-3 978-3-03719-562-8 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/062 http://www.ems-ph.org/doi/10.4171/062

Susumu Ariki Kyoto University, KYOTO, JAPAN Nonassociative rings and algebras Combinatorics Group theory and generalizations General This article surveys developments on finite dimensional Hecke algebras in the last decade. In the first part, we explain results on canonical basic sets by Geck and Jacon and propose a categorification framework which is suitable for our example of Hecke algebras. In the second part, we review basics of Kashiwara’s crystal theory and explain the Fock space theory of cyclotomic Hecke algebras and its applications. In the third part, we explain Rouquier’s theory of quasihereditary covers of cyclotomic Hecke algebras. We add detailed explanation of the proofs here. The third part is based on my intensive course given at Nagoya University in January 2007. 1 48 1 10.4171/062-1/1 http://www.ems-ph.org/doi/10.4171/062-1/1

Grzegorz Bobiński Nicolaus Copernicus University, TORUN, POLAND Christine Riedtmann Universität Bern, BERN, SWITZERLAND Andrzej Skowroński Nicolaus Copernicus University, TORU&#323;, POLAND General We survey old and new results on semi-invariant polynomial functions on affine varieties of quivers with respect to conjugate actions of products of general linear groups over an algebraically closed field and the geometry of the sets of common zeros of the nonconstant semi-invariants. 49 99 1 10.4171/062-1/2 http://www.ems-ph.org/doi/10.4171/062-1/2

Igor Burban Universität zu Köln, KÖLN, GERMANY Yuriy Drozd National Academy of Science of Ukraine, KYIV, UKRAINE Associative rings and algebras Commutative rings and algebras Several complex variables and analytic spaces General This is a survey article about properties of Cohen–Macaulay modules over surface singularities. We discuss results on the Macaulayfication functor, reflexive modules over simple, quotient and minimally elliptic singularities, geometric and algebraic McKay correspondence. Finally, we describe matrix factorizations corresponding to indecomposable Cohen–Macaulay modules over the non-isolated singularities A∞ and D∞. 101 166 1 10.4171/062-1/3 http://www.ems-ph.org/doi/10.4171/062-1/3

Jon Carlson University of Georgia, ATHENS, UNITED STATES Group theory and generalizations General This article is a survey of the theory of rank varieties for modules over finite groups and finite group schemes. We follow the development of the ideas over the nearly thirty years of the subject. The discussion also touches on modules of constant Jordan type and recent results on rank varieties for truncated polynomial rings. 167 200 1 10.4171/062-1/4 http://www.ems-ph.org/doi/10.4171/062-1/4

Karin Erdmann Oxford University, OXFORD, UNITED KINGDOM Andrzej Skowroński Nicolaus Copernicus University, TORU&#323;, POLAND Associative rings and algebras Category theory; homological algebra General We discuss periodicity of modules and algebras with respect to the action of syzygy operators. In particular, we exhibit the known classes of periodic algebras and their roots in the theory of finite groups, algebraic topology and commutative algebra. 201 251 1 10.4171/062-1/5 http://www.ems-ph.org/doi/10.4171/062-1/5

Christof Geiss Universidad Nacional Autonoma de México, MÉXICO, D.F., MEXICO Bernard Leclerc Université de Caen, CAEN CEDEX, FRANCE Jan Schröer Rheinische Friedrich-Wilhelms-Universität Bonn, BONN, GERMANY Algebraic geometry Associative rings and algebras Nonassociative rings and algebras Group theory and generalizations We use the representation theory of preprojective algebras to construct and study certain cluster algebras related to semisimple algebraic groups. 253 283 1 10.4171/062-1/6 http://www.ems-ph.org/doi/10.4171/062-1/6

Iain Gordon University of Edinburgh, EDINBURGH, UNITED KINGDOM Algebraic geometry Associative rings and algebras Several complex variables and analytic spaces General We survey recent results on the representation theory of symplectic reflection algebras, focusing particularly on connections with symplectic quotient singularities and their resolutions, with category O, and with spaces of representations of quivers. 285 347 1 10.4171/062-1/7 http://www.ems-ph.org/doi/10.4171/062-1/7

Osamu Iyama Nagoya University, NAGOYA, JAPAN Associative rings and algebras General We recall several results in Auslander–Reiten theory for finite-dimensional algebras over fields and orders over complete local rings. Then we introduce n-cluster tilting subcategories and higher theory of almost split sequences and Auslander algebras. Several examples are explained. 349 397 1 10.4171/062-1/8 http://www.ems-ph.org/doi/10.4171/062-1/8

Peter Jørgensen University of Newcastle, NEWCASTLE UPON TYNE, UNITED KINGDOM Associative rings and algebras Algebraic topology General The singular cochain complex of a topological space is a classical object. It is a differential graded algebra which has been studied intensively with a range of methods, not least within rational homotopy theory. More recently, the tools of Auslander–Reiten theory have also been applied to the singular cochain complex. One of the highlights is that by these methods, each Poincaré duality space gives rise to a Calabi–Yau category. This paper is a review of the theory. 399 431 1 10.4171/062-1/9 http://www.ems-ph.org/doi/10.4171/062-1/9

Stanisław Kasjan Nicolaus Copernicus University, TORUN, POLAND Associative rings and algebras Mathematical logic and foundations General We present results on representation types of algebras obtained by applying classical model theoretical tools, like ultraproducts and quantifier elimination, to classes of algebras defined in representation theoretical terms, in particular, by the representation types. 433 465 1 10.4171/062-1/10 http://www.ems-ph.org/doi/10.4171/062-1/10

Bernhard Keller Université Paris Diderot - Paris 7, PARIS CEDEX 13, FRANCE Category theory; homological algebra Associative rings and algebras General We review the definition of a Calabi–Yau triangulated category and survey examples coming from the representation theory of quivers and finite-dimensional algebras. Our main motivation comes from the links between quiver representations and Fomin–Zelevinsky’s cluster algebras. 467 489 1 10.4171/062-1/11 http://www.ems-ph.org/doi/10.4171/062-1/11

Steffen Koenig Universität Stuttgart, STUTTGART, GERMANY Associative rings and algebras Nonassociative rings and algebras Category theory; homological algebra Group theory and generalizations This survey aims to provide an elementary introduction to the structure and representation theory of diagram algebras, by a slightly expanded version of the author’s lectures given at the workshop preceding ICRA XII at Toruń . The goal is to inform experts in representation theory of finite dimensional algebras about current results and problems in the emerging area of diagram algebras. 491 540 1 10.4171/062-1/12 http://www.ems-ph.org/doi/10.4171/062-1/12

Helmut Lenzing Universität Paderborn, PADERBORN, GERMANY Luz de Teresa Universidad Nacional Autonoma de México, MÉXICO, D.F., MEXICO Associative rings and algebras Algebraic geometry Category theory; homological algebra General For a finite dimensional algebra A of finite global dimension the bounded derived category of finite dimensional A-modules admits Auslander–Reiten triangles such that the Auslander–Reiten translation τ is an equivalence. On the level of the Grothendieck group τ induces the Coxeter transformation ΦA. More generally this extends to a homologically finite triangulated category T admitting Serre duality. In both cases the Coxeter polynomial, that is, the characteristic polynomial of the Coxeter transformation, yields an important homological invariant of A or T. Spectral analysis is the study of this interplay, it often reveals unexpected links between apparently different subjects. This paper gives a summary on spectral techniques and studies the links to singularity theory. In particular, it offers a contribution to the categorification of the Milnor lattice through triangulated categories which are naturally attached to a weighted projective line. 541 588 1 10.4171/062-1/13 http://www.ems-ph.org/doi/10.4171/062-1/13

Markus Reineke Ruhr-Universität Bochum, BOCHUM, GERMANY Associative rings and algebras Algebraic geometry General An introduction to moduli spaces of representations of quivers is given, and results on their global geometric properties are surveyed. In particular, the geometric approach to the problem of classification of quiver representations is motivated, and the construction of moduli spaces is reviewed. Topological, arithmetic and algebraic methods for the study of moduli spaces are discussed. 589 637 1 10.4171/062-1/14 http://www.ems-ph.org/doi/10.4171/062-1/14

Andrzej Skowroński Nicolaus Copernicus University, TORU&#323;, POLAND Kunio Yamagata Tokyo University of Agriculture and Technology, TOKYO, JAPAN Associative rings and algebras Category theory; homological algebra General We survey old and new results on the representation theory of selfinjective algebras of quasitilted type, that is, the finite dimensional selfinjective algebras over a field with Galois coverings by the repetitive categories of quasitilted algebras (endomorphism algebras of tilting objects in hereditary abelian categories). 639 708 1 10.4171/062-1/15 http://www.ems-ph.org/doi/10.4171/062-1/15

Guillermo Cortiñas Universidad de Buenos Aires, Argentina Joachim Cuntz University of Münster, Germany Max Karoubi Université Paris 7, France Ryszard Nest University of Copenhagen, Denmark Charles Weibel Rutgers University, USA $K$-theory Global analysis, analysis on manifolds 19-06, 58-06; 14A22, 14Fxx, 46Lxx, 53D55, 58Bxx Isomorphism conjectures,Waldhausen K-theory, twisted K-theory, equivariant cyclic homology, index theory, Bost–Connes algebra, C*-manifolds, T-duality, deformation of gerbes, torsion theories, Parshin conjecture, Bloch–Kato conjecture Since its inception 50 years ago, K-theory has been a tool for understanding a wide-ranging family of mathematical structures and their invariants: topological spaces, rings, algebraic varieties and operator algebras are the dominant examples. The invariants range from characteristic classes in cohomology, determinants of matrices, Chow groups of varieties, as well as traces and indices of elliptic operators. Thus K-theory is notable for its connections with other branches of mathematics. Noncommutative geometry develops tools which allow one to think of noncommutative algebras in the same footing as commutative ones: as algebras of functions on (noncommutative) spaces. The algebras in question come from problems in various areas of mathematics and mathematical physics; typical examples include algebras of pseudodifferential operators, group algebras, and other algebras arising from quantum field theory. To study noncommutative geometric problems one considers invariants of the relevant noncommutative algebras. These invariants include algebraic and topological K-theory, and also cyclic homology, discovered independently by Alain Connes and Boris Tsygan, which can be regarded both as a noncommutative version of de Rham cohomology and as an additive version of K-theory. There are primary and secondary Chern characters which pass from K-theory to cyclic homology. These characters are relevant both to noncommutative and commutative problems, and have applications ranging from index theorems to the detection of singularities of commutative algebraic varieties. The contributions to this volume represent this range of connections between K-theory, noncommmutative geometry, and other branches of mathematics. 10 6 2008 978-3-03719-060-9 978-3-03719-560-4 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/060 http://www.ems-ph.org/doi/10.4171/060

Ralf Meyer Georg-August-Universität Göttingen, GÖTTINGEN, GERMANY General This survey article on bivariant Kasparov theory and E-theory is mainly intended for readers with a background in homotopical algebra and category theory. We approach both bivariant K-theories via their universal properties and equip them with extra structure such as a tensor product and a triangulated category structure. We discuss the construction of the Baum–Connes assembly map via localisation of categories and explain how this is related to the purely topological construction by Davis and Lück. 1 39 1 10.4171/060-1/1 http://www.ems-ph.org/doi/10.4171/060-1/1

Arthur Bartels Universität Münster, MÜNSTER, GERMANY Siegfried Echterhoff Universität Münster, MÜNSTER, GERMANY Wolfgang Lück Universität Bonn, BONN, GERMANY $K$-theory Category theory; homological algebra Algebraic topology General We investigate when isomorphism conjectures, such as the ones due to Baum–Connes, Bost and Farrell–Jones, are stable under colimits of groups over directed sets (with not necessarily injective structure maps). We show in particular that both the K-theoretic Farrell–Jones Conjecture and the Bost Conjecture with coefficients hold for those groups for which Higson, Lafforgue and Skandalis have disproved the Baum–Connes Conjecture with coefficients. 41 70 1 10.4171/060-1/2 http://www.ems-ph.org/doi/10.4171/060-1/2

Heath Emerson University of Victoria, VICTORIA, B.C., CANADA Ralf Meyer Georg-August-Universität Göttingen, GÖTTINGEN, GERMANY $K$-theory Functional analysis General We study an equivariant co-assembly map that is dual to the usual Baum–Connes assembly map and closely related to coarse geometry, equivariant Kasparov theory, and the existence of dual Dirac morphisms. As applications, we prove the existence of dual Dirac morphisms for groups with suitable compactifications, that is, satisfying the Carlsson–Pedersen condition, and we study a K-theoretic counterpart to the proper Lipschitz cohomology of Connes, Gromov and Moscovici. 71 89 1 10.4171/060-1/3 http://www.ems-ph.org/doi/10.4171/060-1/3

Fernando Muro Universidad de Sevilla, SEVILLA, SPAIN Andrew Tonks London Metropolitan University, LONDON, UNITED KINGDOM $K$-theory Associative rings and algebras Category theory; homological algebra General We give a simple representation of all elements in K1 of a Waldhausen category and prove relations between these representatives which hold in K1. 91 115 1 10.4171/060-1/4 http://www.ems-ph.org/doi/10.4171/060-1/4

Max Karoubi Université Paris 7, Denis Diderot, PARIS CEDEX 05, FRANCE Algebraic topology $K$-theory Functional analysis General Twisted K-theory has its origins in the author’s PhD thesis [28] and in a paper with P. Donovan [19]. The objective of this paper is to revisit the subject in the light of new developments inspired by mathematical physics. See for instance E. Witten [43], J. Rosenberg [38], C. Laurent-Gentoux, J.-L. Tu, P. Xu [35] and M. F. Atiyah, G. Segal [8], among many authors. We also prove some new results in the subject: a Thom isomorphism, explicit computations in the equivariant case and new cohomology operations. 117 149 1 10.4171/060-1/5 http://www.ems-ph.org/doi/10.4171/060-1/5

Christian Voigt University of Glasgow, GLASGOW, UNITED KINGDOM $K$-theory Associative rings and algebras Quantum theory General We define equivariant periodic cyclic homology for bornological quantum groups. Generalizing corresponding results from the group case, we show that the theory is homotopy invariant, stable and satisfies excision in both variables. Along the way we prove Radford’s formula for the antipode of a bornological quantum group. Moreover we discuss anti-Yetter–Drinfeld modules and establish an analogue of the Takesaki–Takai duality theorem in the setting of bornological quantum groups. 151 179 1 10.4171/060-1/6 http://www.ems-ph.org/doi/10.4171/060-1/6

Paulo Carrillo Rouse Université Paris 7 Denis Diderot, PARIS, FRANCE Global analysis, analysis on manifolds $K$-theory General We construct an algebra of smooth functions over the tangent groupoid associated to any Lie groupoid. This algebra is a field of algebras over the closed interval [0,1], where the fiber at zero is the algebra of Schwartz functions over the Lie algebroid, and any fiber out of zero is the convolution algebra of the initial groupoid. Our motivation comes from index theory for Lie groupoids. In fact, our construction gives an intermediate algebra between the enveloping C*-algebra and the convolution algebra of compactly supported functions of the tangent groupoid; and it will allows us, in a further work, to define other analytic index morphisms as a sort of “deformations”. 181 199 1 10.4171/060-1/7 http://www.ems-ph.org/doi/10.4171/060-1/7

Joachim Cuntz Universität Münster, MÜNSTER, GERMANY Global analysis, analysis on manifolds Functional analysis General We present a C*-algebra which is naturally associated to the ax + b-semigroup over ℕ. It is simple and purely infinite and can be obtained from the algebra considered by Bost and Connes by adding one unitary generator which corresponds to addition. Its stabilization can be described as a crossed product of the algebra of continuous functions, vanishing at infinity, on the space of finite adeles for ℚ by the natural action of the ax + b-group over ℚ. 201 215 1 10.4171/060-1/8 http://www.ems-ph.org/doi/10.4171/060-1/8

Wend Werner Universität Münster, MÜNSTER, GERMANY Nonassociative rings and algebras Topological groups, Lie groups Functional analysis Differential geometry This paper deals with a special class of ‘non-compact’ hermitian infinite dimensional symmetric spaces, generically denoted by U. We calculate their invariant connection very explicitly and use the concept of a Hilbert C*-manifold so that the Banach manifold in question is of the form Aut U/H, where Aut U is the automorphism group of the Hilbert C*-manifold. We finally use results previously obtained with D. Blecher to characterize causal structure on U that comes from interpreting the elements of U as bounded Hilbert space operators. 218 225 1 10.4171/060-1/9 http://www.ems-ph.org/doi/10.4171/060-1/9

Ulrich Bunke Universität Regensburg, REGENSBURG, GERMANY Thomas Schick Georg-August-Universität Göttingen, GÖTTINGEN, GERMANY Markus Spitzweck Georg-August-Universität Göttingen, GÖTTINGEN, GERMANY Andreas Thom Technische Universität Dresden, DRESDEN, GERMANY General We extend Pontrjagin duality from topological abelian groups to certain locally compact group stacks. To this end we develop a sheaf theory on the big site of topological spaces S in order to prove that the sheaves ExtiShAbS(G,T), i = 1, 2, vanish, where G is the sheaf represented by a locally compact abelian group and T is the circle. As an application of the theory we interpret topological T-duality of principal Tn-bundles in terms of Pontrjagin duality of abelian group stacks. 227 347 1 10.4171/060-1/10 http://www.ems-ph.org/doi/10.4171/060-1/10

Paul Bressler Institute for Advanced Study, PRINCETON, UNITED STATES Alexander Gorokhovsky University of Colorado at Boulder, BOULDER, UNITED STATES Ryszard Nest University of Copenhagen, COPENHAGEN, DENMARK Boris Tsygan Northwestern University, EVANSTON, UNITED STATES Differential geometry Category theory; homological algebra Global analysis, analysis on manifolds General We identify the 2-groupoid of deformations of a gerbe on a C∞ manifold with the Deligne 2-groupoid of a corresponding twist of the DGLA of local Hochschild cochains on C∞ functions. 349 392 1 10.4171/060-1/11 http://www.ems-ph.org/doi/10.4171/060-1/11

Grigory Garkusha University of Wales Swansea, SWANSEA, UNITED KINGDOM Mike Prest University of Manchester, MANCHESTER, UNITED KINGDOM Category theory; homological algebra General Given a commutative ring R (respectively a positively graded commutative ring A = ⊕j ≥ 0 Aj which is finitely generated as an A0-algebra), a bijection between the torsion classes of finite type in Mod R (respectively tensor torsion classes of finite type in QGr A) and the set of all subsets Y ⊆ spec R (respectively Y ⊆ Proj A) of the form Y = ∪i ∈ ΩYi, with Spec R \Yi (respectively Proj A \Yi) quasi-compact and open for all i ∈ Ω, is established. Using these bijections, there are constructed isomorphisms of ringed spaces (Spec R,OR) → (Spec(Mod R),OModR) and (Proj A,OProj A) → (Spec(QGr A),OQGr A), where (Spec(Mod R),OModR) and (Spec(QGr A),OQGr A) are ringed spaces associated to the lattices Ltor(Mod R) and Ltor(QGr A) of torsion classes of finite type. Also, a bijective correspondence between the thick subcategories of perfect complexes Dper(R) and the torsion classes of finite type in Mod R is established. 393 412 1 10.4171/060-1/12 http://www.ems-ph.org/doi/10.4171/060-1/12

Thomas Geisser Nagoya University, NAGOYA, JAPAN $K$-theory General Parshin’s conjecture states that Ki(X)ℚ = 0 for i > 0 and X smooth and projective over a finite field Fq. The purpose of this paper is to break up Parshin’s conjecture into several independent statements, in the hope that each of them is easier to attack individually. If CHn(X,i) is Bloch’s higher Chow group of cycles of relative dimension n, then in view of Ki(X)ℚ ≅ ⊕n CHn(X,i)ℚ, Parshin’s conjecture is equivalent to Conjecture P(n) for all n, stating that CHn(X,i)ℚ = 0 for i > 0, and all smooth and projective X. We show, assuming resolution of singularities, that Conjecture P(n) is equivalent to the conjunction of three conjectures A(n), B(n) and C(n), and give several equivalent versions of these conjectures. 413 425 1 10.4171/060-1/13 http://www.ems-ph.org/doi/10.4171/060-1/13

Charles Weibel Rutgers University, PISCATAWAY, UNITED STATES Algebraic geometry $K$-theory Algebraic topology General We give an axiomatic framework for proving that the norm residue map is an isomorphism (i.e., for settling the motivic Bloch–Kato conjecture). This framework is a part of the Voevodsky–Rost program. 427 435 1 10.4171/060-1/14 http://www.ems-ph.org/doi/10.4171/060-1/14

Janet Beery University of Redlands, USA Jacqueline Stedall University of Oxford, UK History and biography 01-02 Harriot, seventeenth century, figurate numbers, interpolation Thomas Harriot (c. 1560–1621) was a mathematician and astronomer, known not only for his work in algebra and geometry, but also for his wide-ranging interests in ballistics, navigation, and optics (he discovered the sine law of refraction now known as Snell’s law). By about 1614, Harriot had developed finite difference interpolation methods for navigational tables. In 1618 (or slightly later) he composed a treatise entitled ‘De numeris triangularibus et inde de progressionibus arithmeticis, Magisteria magna’, in which he derived symbolic interpolation formulae and showed how to use them. This treatise was never published and is here reproduced for the first time. Commentary has been added to help the reader to follow Harriot’s beautiful but almost completely nonverbal presentation. The introductory essay preceding the treatise gives an overview of the contents of the ‘Magisteria’ and describes its influence on Harriot’s contemporaries and successors over the next sixty years. Harriot’s method was not superseded until Newton, apparently independently, made a similar discovery in the 1660s. The ideas in the ‘Magisteria’ were spread primarily through personal communication and unpublished manuscripts, and so, quite apart from their intrinsic mathematical interest, their survival in England during the seventeenth century provides an important case study in the dissemination of mathematics through informal networks of friends and acquaintances. 12 1 2008 978-3-03719-059-3 978-3-03719-559-8 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/059 http://www.ems-ph.org/doi/10.4171/059 Tonny Springer Utrecht University, The Netherlands Dirk van Dalen Utrecht University, The Netherlands History and biography 00B60, 01A75 Hans Freudenthal (1905–1990) was a Dutch mathematician, born in Luckenwalde, Germany. His scientific activities were of a rich variety. Enrolling at the University of Berlin as a student in the 1920s, he followed in the footsteps of his teachers, and became a topologist, but with a lively interest in group theory. Later in life, after a long journey through the realm of mathematics, working on almost all subjects that drew his interest, he turned towards the practical and methodological issues of the didactics of mathematics. The present Selecta are devoted to Freudenthal’s mathematical oeuvre, they contain a selection of his major contributions. Included are fundamental contributions to topology such as the foundation of the theory of ends (in the thesis of 1931), the introduction (in 1937) of the suspension and its use in stability results for homotopy groups of spheres. In group theory there is work on topological groups (of the 1930s) and on various aspects of the theory of Lie groups, such as a paper on automorphisms of 1941. From the later work of the 1950s and 1960s, papers on geometric aspects of Lie theory (geometries associated to exceptional groups, space problems) have been included. Freudenthal’s versatility is further demonstrated by a choice from his foundational and historical work: papers on intuitionistic logic and topology, a paper on axiomatic geometry reappraising Hilbert’s Grundlagen, and a paper summarizing his development of Lincos, a universal (“cosmic”) language. The book also contains a sketch of Freudenthal’s life. Most of the selected papers are accompanied by brief comments. 10 30 2009 978-3-03719-058-6 978-3-03719-558-1 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/058 http://www.ems-ph.org/doi/10.4171/058 Giovanni Gallavotti Università di Roma “La Sapienza”, Italy Wolfgang Reiter University of Vienna, Austria Jakob Yngvason University of Vienna, Austria History and biography Statistical mechanics, structure of matter 01-02, 82-02; 82C03, 82B40 Foundations of statistical physics; Ludwig Boltzmann Ludwig Eduard Boltzmann (1844–1906) was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics. He was one of the most important advocates for atomic theory when that scientific model was still highly controversial. To commemorate the 100th anniversary of his death in Duino, the International Symposium “Boltzmann's Legacy” was held at the Erwin Schrödinger International Institute for Mathematical Physics in June 2006. This text covers a wide spectrum of topics ranging from equilibrium statistical and nonequilibrium statistical physics, ergodic theory and chaos to basic questions of biology and historical accounts of Boltzmann's work. Besides the lectures presented at the symposium the volume also contains contributions specially written for this occasion. The articles give a broad overview of Boltzmann's legacy to the sciences from the standpoint of some of present day's leading scholars in the field. The book addresses students and researchers in mathematics, physics and the history of science. 10 22 2008 978-3-03719-057-9 978-3-03719-557-4 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/057 http://www.ems-ph.org/doi/10.4171/057

Giovanni Gallavotti Università di Roma La Sapienza, ROMA, ITALY History and biography Statistical mechanics, structure of matter General 1 6 1 10.4171/057-1/1 http://www.ems-ph.org/doi/10.4171/057-1/1

Jürgen Renn , BERLIN, GERMANY History and biography Statistical mechanics, structure of matter General 7 25 1 10.4171/057-1/2 http://www.ems-ph.org/doi/10.4171/057-1/2

Elliott Lieb Princeton University, PRINCETON, UNITED STATES Statistical mechanics, structure of matter History and biography General 27 36 1 10.4171/057-1/3 http://www.ems-ph.org/doi/10.4171/057-1/3

Giovanni Gallavotti Università di Roma La Sapienza, ROMA, ITALY Statistical mechanics, structure of matter History and biography General 39 60 1 10.4171/057-1/4 http://www.ems-ph.org/doi/10.4171/057-1/4

Joel Lebowitz Rutgers University, PISCATAWAY, UNITED STATES Statistical mechanics, structure of matter History and biography General Time-asymmetric behavior as embodied in the second law of thermodynamics is observed in individual macroscopic systems. It can be understood as arising naturally from time-symmetric microscopic laws when account is taken of a) the great disparity between microscopic and macroscopic scales, b) a low entropy state of the early universe, and c) the fact that what we observe is the behavior of systems coming from such an initial state – not all possible systems. The explanation of the origin of the second law based on these ingredients goes back to Maxwell, Thomson and particularly Boltzmann. Common alternate explanations, such as those based on the ergodic or mixing properties of probability distributions (ensembles) already present for chaotic dynamical systems having only a few degrees of freedom or on the impossibility of having a truly isolated system, are either unnecessary, misguided or misleading. Specific features of macroscopic evolution, such as the diffusion equation, do however depend on the dynamical instability (deterministic chaos) of trajectories of isolated macroscopic systems. The extensions of these classical notions to the quantum world is in many ways fairly direct. It does however also bring in some new problems. These are discussed but not resolved. 63 87 1 10.4171/057-1/5 http://www.ems-ph.org/doi/10.4171/057-1/5

David Ruelle Le Bois-Marie, BURES-SUR-YVETTE, FRANCE Statistical mechanics, structure of matter History and biography General Statistical mechanics away from equilibrium is in a formative stage, where general concepts slowly emerge. We restrict ourselves here to the study of transport phenomena, especially heat transport, and consider several well-studied classes of systems: finite systems with isokinetic thermostats, infinite classical Hamiltonian systems, infinite quantum spin systems. For those classes we discuss how various physical quantities can be defined. Our review however leaves several basic questions quite open. 89 97 1 10.4171/057-1/6 http://www.ems-ph.org/doi/10.4171/057-1/6

Donald Ornstein Stanford University, STANFORD, UNITED STATES Dynamical systems and ergodic theory Statistical mechanics, structure of matter General 99 104 1 10.4171/057-1/7 http://www.ems-ph.org/doi/10.4171/057-1/7

Carlo Cercignani Politecnico di Milano, MILANO, ITALY Statistical mechanics, structure of matter History and biography General We recall the early contributions by Boltzmann with a particular attention to his basic paper appeared in 1872. Then we survey the developments in the mathematical theory of the Boltzmann equation and its applications. 107 127 1 10.4171/057-1/8 http://www.ems-ph.org/doi/10.4171/057-1/8

Cédric Villani École Normale Supérieure de Lyon, LYON CEDEX 07, FRANCE Statistical mechanics, structure of matter History and biography General Some of the objects introduced by Boltzmann, entropy in particular, have proven to be of great inspiration in mathematics, and not only in problems related to physics. Here I will describe, in a slightly informal style, a few striking examples of the beauty and power of the notions cooked up by Boltzmann. 129 143 1 10.4171/057-1/9 http://www.ems-ph.org/doi/10.4171/057-1/9

Herbert Spohn TU München, GARCHING BEI MÜNCHEN, GERMANY General We explain how the kinetic theory of Ludwig Boltzmann is applied to weakly nonlinear wave equations. 145 159 1 10.4171/057-1/10 http://www.ems-ph.org/doi/10.4171/057-1/10

E.G.D. Cohen The Rockefeller University, NEW YORK, UNITED STATES Statistical mechanics, structure of matter General Boltzmann’s struggle with a derivation of the Second Law of Thermodynamics is sketched. So is his first derivation of the connection between entropy and probability in 1877. Planck’s derivation and quantum mechanical modifications of Boltzmann’s connection between entropy probability are given next. Then Einstein’s objections to a purely probabilistic rather than a dynamical interpretation of entropy are discussed. Finally, the dynamical basis of the Sinai–Ruelle–Bowen distribution for very chaotic systems is sketched and appears to be an example of Einstein’s dynamical interpretation of entropy. 161 170 1 10.4171/057-1/11 http://www.ems-ph.org/doi/10.4171/057-1/11

Christoph Dellago Universität Wien, WIEN, AUSTRIA Harald Posch Universität Wien, WIEN, AUSTRIA Statistical mechanics, structure of matter General Today computer simulations belong to the most important and powerful theoretical tools for the investigation of statistical mechanical systems particularly in the field of condensed matter. Many simulation methods may directly, or indirectly, be traced back to ideas of Ludwig Boltzmann or are used to carry on the program he had envisioned. Using several illustrative examples, we discuss the role of computer simulation in modern statistical mechanics and, in particular, its relation to Boltzmann's legacy. 171 201 1 10.4171/057-1/12 http://www.ems-ph.org/doi/10.4171/057-1/12

Andrei Gabrielov Purdue University, WEST LAFAYETTE, UNITED STATES Vladimir Keilis-Borok University of California Los Angeles, LOS ANGELES, UNITED STATES Yakov Sinai Princeton University, PRINCETON, UNITED STATES Ilya Zaliapin University of Nevada, RENO, UNITED STATES Statistical mechanics, structure of matter General Cluster dynamics is the property of dynamics of systems of statistical mechanics when for given time the whole array of particles is decomposed onto finite clusters which move independently during a random interval of time. In this paper we study statistical properties and critical indices related to cluster dynamics. 203 215 1 10.4171/057-1/13 http://www.ems-ph.org/doi/10.4171/057-1/13

Peter Schuster Universität Wien, WIEN, AUSTRIA Statistical mechanics, structure of matter Biology and other natural sciences General Ludwig Boltzmann’s visionary view of life being understandable on the molecular level as the interplay of complex molecules performing highly sophisticated chemical syntheses is discussed in the light of current molecular biology. Boltzmann’s high esteem for Charles Darwin’s theory of evolution is complemented by an account of the current view of selection as a general phenomenon not restricted to cellular life. In two more technical sections we present a brief overview of a molecular theory of evolution that provides a conceptual frame for understanding evolution and a tool for quantitative description of evolutionary phenomena. 217 241 1 10.4171/057-1/14 http://www.ems-ph.org/doi/10.4171/057-1/14

Wolfgang Reiter Universität Wien, WIEN, AUSTRIA History and biography General 243 260 1 10.4171/057-1/15 http://www.ems-ph.org/doi/10.4171/057-1/15

Invited Lectures Rolf Jeltsch ETH Zürich, Switzerland Gerhard Wanner University of Geneva, Switzerland Numerical analysis Partial differential equations Computer science Biology and other natural sciences 00Bxx The International Council for Industrial and Applied Mathematics (ICIAM) is the worldwide organisation of societies which are dedicated primarily or significantly to applied and/or industrial mathematics. The ICIAM Congresses, held every 4 years, are run under the auspices of the Council with the aim to advance the applications of mathematics in all parts of the world. The 6th ICIAM Congress was held in Zürich, Switzerland, 16–20 July 2007, and was attended by more than 3000 scientists from 47 countries. This volume collects the invited lectures of this Congres, the appreciations of the ICIAM Prize winners’ achievements and the Euler Lecture celebrating the 300th anniversary of Euler. The authors of these papers are leading researchers of their fields, rigorously selected by a distinguished international program committee. The book presents an overview of contemporary applications of mathematics, new perspectives and open problems. Topics embrace analysis of and numerical methods for: linear and nonlinear partial differential equations multiscale modeling nonlinear problems involving integral operators controllability and observability asymptotic solutions of Hamilton–Jacobi equations contact problems in solid mechanics topology optimization of structures dissipation inequalities in systems theory greedy algorithms sampling in function space order-value optimization parabolic partial differential equations and deterministic games Moreover, particular applications involve risk in financial markets, radar imaging, brain dynamics, complex geometric optics applied to acoustics and electromagnetics. 6 15 2009 978-3-03719-056-2 978-3-03719-556-7 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/056 http://www.ems-ph.org/doi/10.4171/056

Grégoire Allaire Ecole Polytechnique, PALAISEAU, FRANCE François Jouve Université Denis Diderot - Paris 7, PARIS CEDEX 05, FRANCE Nicolas Van Goethem Universidade de Lisboa, LISBOA, PORTUGAL Mechanics of deformable solids Calculus of variations and optimal control; optimization General This paper is devoted to the numerical simulation of the evolution of damage in brittle materials following the Francfort–Marigo model. This model is based on a Griffith energy criterion for the competition between the two phases, healthy and damaged, separated by a sharp interface. In a quasi-static and irreversible framework, the damage configuration is obtained by minimizing a total energy using a gradient descent method. The interface is modeled by a level set function which is advected by the energy gradient issued from a shape derivation. The nucleation of the damaged zone is obtained by using the so-called topological derivative. Several numerical examples in 2-d and 3-d are discussed. 3 22 1 10.4171/056-1/1 http://www.ems-ph.org/doi/10.4171/056-1/1

Christian Ebenbauer Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES Tobias Raff Universität Stuttgart, STUTTGART, GERMANY Frank Allgöwer Universität Stuttgart, STUTTGART, GERMANY Systems theory; control General The aim of this article is to present an introduction to dissipation inequalities and to present some well known and some recent results in this area. 23 42 1 10.4171/056-1/2 http://www.ems-ph.org/doi/10.4171/056-1/2

Luis Caffarelli University of Texas at Austin, AUSTIN, UNITED STATES Partial differential equations General Recently, a lot of attention has focused on phenomena that involve integral diffusion processes: Phase transition, turbulent flows, constrained processes involving discontinuous(Levy) walks, etc. In this survey we discuss recentproblems anddevelopments in the field. 43 56 1 10.4171/056-1/3 http://www.ems-ph.org/doi/10.4171/056-1/3

Claudio Canuto Politecnico di Torino, TORINO, ITALY Numerical analysis Probability theory and stochastic processes General We provide a condensed presentation of the main features of high-order discretization methods for PDEs, based on variational formulations (h-N spectral element methods, h-p finite element methods). We also illustrate an application of high-order methods to the numerical discretization of stochastic PDEs by fully deterministic techniques. 57 87 1 10.4171/056-1/4 http://www.ems-ph.org/doi/10.4171/056-1/4

Margaret Cheney Rensselaer Polytechnic Institute, TROY, UNITED STATES Partial differential equations General Radar imaging is a technology that has been developed, very successfully, within the engineering community during the last 50 years. Radar systems on satellites now make beautiful images of regions of our earth and of other planets such as Venus. One of the key components of this impressive technology is mathematics, and many of the open problems are mathematical ones. This paper explains, from first principles, some of the basics of radar and the mathematics involved in producing high-resolution radar images. 89 98 1 10.4171/056-1/5 http://www.ems-ph.org/doi/10.4171/056-1/5

Albert Cohen Université Pierre et Marie Curie, PARIS, FRANCE Numerical analysis General We consider two problems of nonlinear approximation: adaptive triangulations and best N-term approximations. For both problems we present simple greedy algorithms which aim to provide near-optimal solutions at a reasonable computational cost. We also present several results on the convergence rate of these algorithms. 99 111 1 10.4171/056-1/6 http://www.ems-ph.org/doi/10.4171/056-1/6

Weinan E Princeton University, PRINCETON, UNITED STATES Partial differential equations Mechanics of deformable solids General We discuss multiscale problems that arise in the analysis of the electronic structure of materials using density functional theory. In contrast to the traditional approach which views Kohn–Sham density functional theory as a nonlinear system of eigenvalue problems, we take the viewpoint that what we are really interested in is the eigen-subspace which represents the subspace occupied by the electrons, and we discuss localized bases for such eigen-subspaces. We then discuss the continuum limit of density functional theory, sub-linear scaling algorithms based on multiscale analysis and coupled formulation between density functional theory and continuum models. 113 130 1 10.4171/056-1/7 http://www.ems-ph.org/doi/10.4171/056-1/7

Michel Fortin Université Laval, QUEBEC (QUEBEC), CANADA Carl Robitaille Université Laval, QUEBEC (QUEBEC), CANADA André Fortin Université Laval, QUEBEC (QUEBEC), CANADA Ali Rezgui Michelin, CLERMONT-FERRAND Cedex 09, FRANCE Numerical analysis Mechanics of deformable solids General We present a formulation of frictional contact problems. We discuss the choice of finite element approximations and describe some algorithms for the computation of contact pressure and tractions. Numerical results illustrate the presentation. 131 154 1 10.4171/056-1/8 http://www.ems-ph.org/doi/10.4171/056-1/8

Roland Glowinski University of Houston, HOUSTON, UNITED STATES Numerical analysis Partial differential equations General In this article we address the numerical solution of fully nonlinear elliptic equations, a typical one being the canonical Monge–Ampère equation det D2ψ = f, where D2ψ is the Hessian of the unknown function D2ψ, and where f is a given positive function. We will start ourdiscussionby a short review of methods which have been used by various authors for the solution of fully nonlinear elliptic equations. Then we will give a relatively detailed description of one of these approaches when applied to the solution of the above Monge–Ampère equation and of the following two-dimensional Pucci’s equation α λ+ + λ- = 0, where λ+ and λ- are, respectively, the largest and smallest eigenvalues of D2ψ, and α > 1. The results of numerical experiments will be presented. 155 192 1 10.4171/056-1/9 http://www.ems-ph.org/doi/10.4171/056-1/9

Kenji Nishihara Waseda University, TOKYO, JAPAN Partial differential equations General We discuss the recent developments related to the large-time asymptotic behavior of solutions of Hamilton–Jacobi equations. 193 217 1 10.4171/056-1/10 http://www.ems-ph.org/doi/10.4171/056-1/10

Barbara Lee Keyfitz Ohio State University, COLUMBUS, UNITED STATES Partial differential equations General The field of conservation laws (quasilinear hyperbolic partial differential equations) has captured the attention of mathematics researchers, computational fluid dynamicists, and modelers of physical and engineering phenomena for over 70 years. This talk will survey some of the power, and some of the limitations, of the conservation law approach to modeling. It will also expound some of the major achievements in establishing a mathematical theory, and discuss why theoretical advances have been so slow, and why so much still seems to remain out of reach. Finally,I will describe recent work of a number of people on establishing mathematical properties of shock reflection phenomena. 219 238 1 10.4171/056-1/11 http://www.ems-ph.org/doi/10.4171/056-1/11

Robert Kohn New York University, NEW YORK, UNITED STATES Sylvia Serfaty Université Pierre et Marie Curie (Paris VI), PARIS CEDEX 05, FRANCE Partial differential equations Calculus of variations and optimal control; optimization Game theory, economics, social and behavioral sciences General We provide a brief, expository introduction to our recent work on deterministic game interpretations of second-order parabolic and elliptic PDE. 239 249 1 10.4171/056-1/12 http://www.ems-ph.org/doi/10.4171/056-1/12

Tatsien Li Fudan University, SHANGHAI, CHINA Systems theory; control Partial differential equations General By means of the semi-global classical solution, a simple and direct constructive method is presented to give a complete result with applications on the exact boundary controllability and the exact boundary observability for 1-D quasilinear hyperbolic systems with general nonlinear boundary conditions. 251 278 1 10.4171/056-1/13 http://www.ems-ph.org/doi/10.4171/056-1/13

José Mario Martínez UNICAMP, CAMPINAS, BRAZIL Operations research, mathematical programming General Order-Value Optimization (OVO) is a family of nonsmooth optimization problems for which many practical applications exist. In this paper the main algorithms are reviewed and several applications are described. 279 296 1 10.4171/056-1/14 http://www.ems-ph.org/doi/10.4171/056-1/14

Christof Schütte Freie Universität Berlin, BERLIN, GERMANY Frank Noe Freie Universität Berlin, BERLIN, GERMANY Eike Meerbach Freie Universität Berlin, BERLIN, GERMANY Philipp Metzner New York University, NEW YORK, UNITED STATES Carsten Hartmann Freie Universität Berlin, BERLIN, GERMANY Biology and other natural sciences Probability theory and stochastic processes General This article surveys the present state of the transfer operator approach to the effective dynamics of metastable dynamical systems and the variety of algorithms associated with it. 297 335 1 10.4171/056-1/15 http://www.ems-ph.org/doi/10.4171/056-1/15

Alexandros Beskos University of Warwick, COVENTRY, UNITED KINGDOM Andrew Stuart University of Warwick, COVENTRY, UNITED KINGDOM Partial differential equations Numerical analysis General Applied mathematics is concerned with developing models with predictive capability, and with probing those models to obtain qualitative and quantitative insight into the phenomena being modelled. Statistics is data-driven and is aimed at the devel opment of methodologies to optimize the information derived from data. The increasing complexity of phenomena that scientists and engineers wish to model, together with our increased ability to gather, store and interrogate data, mean that the subjects of applied mathematics and statistics are increasingly required to work in conjunction in order to significantly progress understanding. This article is concerned with a research program at the interface between these two disciplines, aimed at problems in differential equations where profusion of data and the sophisticated model combinetoproducethemathematicalproblem of obtaining informa tion from a probability measure on function space. In this context there is an array of problems with a common mathematical structure, namely that the probability measure in question is a change of measure from a Gaussian. We illustrate the wide-ranging ap plicability of this structure. For problems whose solution is determined by a probability measure on function space, information about the solution can be obtained by sampling from this probability measure. One way to do this is through the use of Markov chain Monte-Carlo(MCMC) methods. We showhowthe common mathematical structure of the aforementioned problems can be exploited in the design of effective MCMC methods. 337 364 1 10.4171/056-1/16 http://www.ems-ph.org/doi/10.4171/056-1/16

Ichiro Tsuda Hokkaido University, SAPPORO, JAPAN Ordinary differential equations Biology and other natural sciences General The hippocampus, which is a part of the old brain, has been considered responsible for the formation of episodic memory. It is also known that atrophy of the hippocampus can lead to Alzheimer’s disease. Motivated by the rehabilitation process of Alzheimer’s patients, we mathematically studied the coding scheme in the hippocampus. Predictions obtained from the model study have been partially verified by experiments using tissue slices from the rat’s hippocampus. 365 380 1 10.4171/056-1/17 http://www.ems-ph.org/doi/10.4171/056-1/17

Gunther Uhlmann University of Washington, SEATTLE, UNITED STATES Partial differential equations Optics, electromagnetic theory General The first part of this article concerns visibility, that is the question of determining the internal properties of a medium by making electromagnetic measurements at the boundary of the medium. We concentrate on the problem of Electrical Impedance Tomography(EIT) which consistsindetermining the electrical conductivity of a medium by making voltage and current measurements at the boundary. We describe the use of complex geometrical optics solutions in EIT. In the second part of this article we will review recent theoretical and experimental progress on making objects invisible to electromagnetic waves. This is joint work with A. Greenleaf, Y. Kurylev and M.Lassas. Maxwell’s equations have transformation laws that allow for design of electromagnetic parameters that would steer light around a hidden region, returning it to its original path on the far side. Not only would observers be unaware of the contents of the hidden region, they would not even be aware that something was hidden. The object would have no shadow. New advances in meta-materials have given some experimental evidence that this indeed can be made possible at certain frequencies. 381 408 1 10.4171/056-1/18 http://www.ems-ph.org/doi/10.4171/056-1/18

Jinchao Xu The Pennsylvania State University, UNIVERSITY PARK, UNITED STATES Numerical analysis General This paper gives an overview of some recent works by the author and collaborators on numerical methods for partial differential equations. One focus is on the development and analysis of algebraic solvers (especially multigrid methods) for discretized partial differential equations(PDEs) by closely taking into account qualitative properties of PDEs and their finite element discretizations together with grid adaptation. We will give several examples to demonstrate that optimal PDE solvers can be designed to be user-friendly and practical, robust with respect to various physical parameters and applicable to a variety of linear and nonlinear problems. More specifically, we will present a sharp convergence theory for the method of subspace corrections (such as multigrid and domaindecomposition) for SPD systems. In particular, we demonstrate that the method of subspace correction will be robust for nearly singular systems if the space decomposition satisfies the near-null space recovery condition (namely, the near-null space can be fully recovered by the parts of near-null space contained in all subspaces). We present a solution of an open problem on robust multigrid and domain decomposition methods for problems with strongly discontinuous jumps. We discuss a new class of optimal and practical (AMG) preconditioners for H(curl) and H(div) systems. We present a methodology that reduces the solution of an indefinite system to one or very few (nearly singular) SPD systems (which can be solved efficiently by the method of subspace correction). Finally, we use a class of non-Newtonian models to illustrate how to design special discretization schemes that are solver-friendly and also satisfy other desirable properties. 409 444 1 10.4171/056-1/19 http://www.ems-ph.org/doi/10.4171/056-1/19

Walter Gautschi Purdue University, WEST LAFAYETTE, UNITED STATES History and biography General On the occasion of the 300th anniversary (on April 15, 2007) of Euler’s birth, an attempt is made to bring Euler’s genius to the attention of a broad segment of the educated public. The three stations of his life—Basel, St. Petersburg, and Berlin— are sketched and the principal works identified in more or less chronological order. To convey a flavor of his work and its impact on modern science, a few of Euler’s memorable contributions are selected and discussed in more detail. Remarks on Euler’s personality, intellect, and craftsmanship round out the presentation. 447 483 1 10.4171/056-1/20 http://www.ems-ph.org/doi/10.4171/056-1/20

Athanase Papadopoulos IRMA, Strasbourg, France Functions of a complex variable Several complex variables and analytic spaces Primary 30-00, 32-00, 32G15, 30F60; secondary 14A20, 14C17, 14D20,14H15, 14H60, 14L24, 14L35, 16R30, 20F36, 20F38, 20F65, 22F10, 20H10, 30F10, 30F15, 30F20, 30F25, 30C62, 51N15, 53A20, 53A35, 53B35, 53C24, 53C25, 53C50, 57M07, 57M60, 57R20 This multi-volume set deals with Teichmüller theory in the broadest sense, namely, as the study of moduli space of geometric structures on surfaces, with methods inspired or adapted from those of classical Teichmüller theory. The aim is to give a complete panorama of this generalized Teichmüller theory and of its applications in various fields of mathematics. The volumes consist of chapters, each of which is dedicated to a specific topic. The present volume has 19 chapters and is divided into four parts: The metric and the analytic theory (uniformization, Weil–Petersson geometry, holomorphic families of Riemann surfaces, infinite-dimensional Teichmüller spaces, cohomology of moduli space, and the intersection theory of moduli space). The group theory (quasi-homomorphisms of mapping class groups, measurable rigidity of mapping class groups, applications to Lefschetz fibrations, affine groups of flat surfaces, braid groups, and Artin groups). Representation spaces and geometric structures (trace coordinates, invariant theory, complex projective structures, circle packings, and moduli spaces of Lorentz manifolds homeomorphic to the product of a surface with the real line). The Grothendieck–Teichmüller theory (dessins d'enfants, Grothendieck's reconstruction principle, and the Teichmüller theory of the soleniod). This handbook is an essential reference for graduate students and researchers interested in Teichmüller theory and its ramifications, in particular for mathematicians working in topology, geometry, algebraic geometry, dynamical systems and complex analysis. The authors are leading experts in the field. 3 25 2009 978-3-03719-055-5 978-3-03719-555-0 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/055 http://www.ems-ph.org/doi/10.4171/055 13

Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Functions of a complex variable General This introduction can be considered as a sequel to the introduction that I wrote for Volume I of the Handbook, and I shall limit myself here to a general presentation of the material covered in the present volume. The exposition will follow the four-parts division of the volume, and for each part, its division in chapters. Beyond the information given on the content of this volume, I hope that the reader of this introduction will get an idea of the richness of the subject of Teichmüller theory. I have tried to give some necessary definitions to make the introduction as much self-contained as possible. 1 44 1 10.4171/055-1/1 http://www.ems-ph.org/doi/10.4171/055-1/1

Scott Wolpert University of Maryland, COLLEGE PARK, UNITED STATES Functions of a complex variable General The aim is to present a summary introduction for the current understanding of a Weil–Petersson metric space. We present an exposition following the development of concepts rather than an exposition following the order of results discovered. Selected readings and general attributions are provided at the end of each section. Teichmüller space and its augmentation are described in terms of the Fenchel–Nielsen coordinates. The Weil–Petersson metric space basic properties are described. Geodesic-length functions are introduced and formulas for their gradients and Hessians are presented. Beginning applications are considered. A description of the Weil–Petersson metric in Fenchel–Nielsen coordinates is presented. A model metric is discussed. Properties of Weil–Petersson metric and geodesics, as well as properties of geodesic-length functions are combined to describe the Alexandrov tangent cone at points of the augmentation. A comparison dictionary is presented between the geometry of the space of flat tori and Teichmüller space with the Weil–Petersson metric. 47 64 1 10.4171/055-1/2 http://www.ems-ph.org/doi/10.4171/055-1/2

Alastair Fletcher University of Glasgow, GLASGOW, UNITED KINGDOM Vladimir Markovic University of Warwick, COVENTRY, UNITED KINGDOM Functions of a complex variable Several complex variables and analytic spaces General This is a survey on the analytic properties of the Teichmüller space T(M) of a Riemann surface M, with particular attention to the case where the dimension of T(M) is infinite. Some important results described are due to the authors of this chapter. The main results presented are the following. The cotangent space of T(M) is described via the Bers embedding, a biholomorphic embedding of T(M) onto a subset of the Banach space Q(M) of holomorphic quadratic differentials on M. The tangent space is then identified with the Bergman space A1(M), the pre-dual of Q(M). If the surface M is of infinite analytic type, then A1(M) ≠ Q(M), unlike the situation where M is of finite type. The classification of biholomorphic maps between Teichmüller spaces is studied via the classification of the isometries between the cotangent spaces of the corresponding Riemann surfaces, with respect to the Kobayashi and Teichmüller metrics. If M and N are two Riemann surfaces (which a priori need not be homeomorphic), the existence of a surjective ℂ-linear isometry between A1(M) and A1(N) implies that M and N are conformally equivalent. This implies that every biholomorphic map between Teichmüller spaces T(M) and T(N) is induced by a quasiconformal mapping between the surfaces M and N. In particular, the automorphism group of T(M) is equal to the mapping class group of M. This is a complete generalization of a famous theorem of Royden stating the same result for surfaces of finite type. The Teichmüller spaces of any two Riemann surfaces of infinite analytic type are locally bi-Lipschitz equivalent. The chapter ends with open problems on the subject discussed. 65 91 1 10.4171/055-1/3 http://www.ems-ph.org/doi/10.4171/055-1/3

Yoichi Imayoshi Osaka City University, OSAKA, JAPAN Functions of a complex variable General We characterize topological monodromies of holomorphic families of Riemann surfaces over the punctured disk. A holomorphic family (M, π, Δ*) of type (g, n) over Δ* is a triplet of a two-dimensional complex manifold M, the punctured disk Δ* and a holomorphic map π : M → Δ* such that for every t ∈ Δ* the fiber St = π−1(t) is a Riemann surface of fixed finite type (g, n) with 2g − 2 + n > 0 and the complex structure of St depends holomorphically on the parameter t. The topological monodromy of (M, π, Δ*) around the origin is given by an element [ω] of the mapping class group Mg, n(Σ) of an oriented topological surface Σ of type (g, n). Our main result sates that an element [ω] of Mg, n(Σ) is a topological monodromy of a holomorphic family (M, π, Δ*) if and only if ω : Σ → Σ is a pseudo-periodic map of negative type. 93 130 1 10.4171/055-1/4 http://www.ems-ph.org/doi/10.4171/055-1/4

Robert Silhol Université de Montpellier II, MONTPELLIER CEDEX 5, FRANCE Functions of a complex variable General This is a brief and elementary introduction to the problem of finding a Fuchsian group uniformizing an algebraic curve and the inverse problem of finding the algebraic curve uniformized by a given Fuchsian group. 131 149 1 10.4171/055-1/5 http://www.ems-ph.org/doi/10.4171/055-1/5

Gabriele Mondello Università di Roma &#8220;La Sapienza&#8221;, Roma, ITALY Several complex variables and analytic spaces Functions of a complex variable General We begin by describing the duality between arc systems and ribbon graphs embedded in a punctured surface and explaining how to cellularize the moduli space of curves in two different ways: using Jenkins–Strebel differentials and using hyperbolic geometry. We also briefly discuss how these two methods are related. Next, we recall the definition of Witten cycles and we illustrate their connection with tautological classes and Weil–Petersson geometry. Finally, we exhibit a simple argument to prove that Witten classes are stable. 151 215 1 10.4171/055-1/6 http://www.ems-ph.org/doi/10.4171/055-1/6

Nariya Kawazumi University of Tokyo, TOKYO, JAPAN Functions of a complex variable General As was shown by Harer the second homology of Mg, the moduli space of compact Riemann surfaces of genusg, is of rank 1, provided that g ≥ 3. This means that a nontrivial second de Rham cohomology class on Mg is unique up to a constant factor. But several canonical 2-forms on the moduli space have been constructed in various geometric contexts and differ from each other. In this chapter we review some of constructions in order to provide material for future research on “secondary geometry” of the moduli space Mg. 217 237 1 10.4171/055-1/7 http://www.ems-ph.org/doi/10.4171/055-1/7

Koji Fujiwara Kyoto University, KYOTO, JAPAN Differential geometry General This is a survey on quasi-homomorphisms on mapping class groups based on the hyperbolic geometry in the sense of Gromov. We stress the analogy between word-hyperbolic groups and mapping class groups in many aspects. 242 269 1 10.4171/055-1/8 http://www.ems-ph.org/doi/10.4171/055-1/8

Mustafa Korkmaz Middle East Technical University, ANKARA, TURKEY András Stipsicz Hungarian Academy of Sciences, BUDAPEST, HUNGARY Differential geometry General Lefschetz pencils and fibrations were introduced for studying topological properties of smooth complex projective varieties. More recently, as an application of Donaldson’s asymptotically holomorphic methods [13], Lefschetz pencils have been found on all symplectic manifolds [14], [15]. Conversely, Gompf showed that a 4-manifold admitting a Lefschetz pencil/fibration carries a symplectic structure [23], [24]. Since symplectic 4-manifolds play a prominent role in modern low-dimensional topology, the Donaldson Gompf correspondence piqued interest in the study of Lefschetz fibrations. In addition, these structures provide a connection between symplectic topology and geometric group theory. In this chapter we collect the most basic definitions, describe the fundamental results and explain some of the consequences of the correspondence mentioned above. In some cases we provide a full or partial proof of the result. Most of the time, however, we restrict ourselves to quoting the theorems, and try to put them in perspective. The only new result is stated in Corollary 5.16. We would like to point out that although most of the fundamental results hold in any (even) dimension, we restrict our attention to the case of symplectic 4-manifolds, in which case the Lefschetz fibrations have surface fibers, and therefore the theory of mapping class groups enters in an essential way. 271 296 1 10.4171/055-1/9 http://www.ems-ph.org/doi/10.4171/055-1/9

Yoshikata Kida The University of Tokyo, TOKYO, JAPAN Differential geometry General In this survey, we present an outline of the proof of measure equivalence and orbit equivalence rigidity results for mapping class groups. We also describe all locally compact second countable groups containing a lattice isomorphic to a mapping class group. 297 367 1 10.4171/055-1/10 http://www.ems-ph.org/doi/10.4171/055-1/10

Martin Möller J.W. Goethe-Universität, FRANKFURT A.M., GERMANY Functions of a complex variable General According to Thurston’s classification, a diffeomorphism of a closed oriented surface is either elliptic, reducible or a pseudo-Anosov diffeomorphism. The structure of pseudo-Anosov diffeomorphisms, e.g., their dilatation coefficients and the corresponding measured foliations have been intensely investigated since. Rather than focusing on properties of a single diffeomorphism, the purpose of this chapter is to study the flat surfaces the pseudo-Anosov diffeomorphisms live on together with their whole group of affine diffeomorphisms. 369 387 1 10.4171/055-1/11 http://www.ems-ph.org/doi/10.4171/055-1/11

Luis Paris Université de Bourgogne, DIJON CEDEX, FRANCE Differential geometry General This chapter is a survey on the braid groups, the Artin groups, and the Garside groups. It is a presentation, accessible to non-experts, of various topological and algebraic aspects of these groups. It is also a report on three points of the theory: the faithful linear representations, the cohomology, and the geometrical representations. 389 451 1 10.4171/055-1/12 http://www.ems-ph.org/doi/10.4171/055-1/12

David Dumas University of Illinois at Chicago, CHICAGO, UNITED STATES Differential geometry General This is a survey of the theory of complex projective ℂℙ1-structures on compact surfaces. After some preliminary discussion and definitions, we concentrate on three main topics: Using the Schwarzian derivative to parameterize the moduli space Thurston’s parameterization of the moduli space using grafting Holonomy representations of ℂℙ1-structures We also discuss some results comparing the two parameterizations of the space of projective structures and relating these parameterizations to the holonomy map. 455 508 1 10.4171/055-1/13 http://www.ems-ph.org/doi/10.4171/055-1/13

Sadayoshi Kojima Tokyo Institute of Technology, TOKYO, JAPAN Convex and discrete geometry Functions of a complex variable Manifolds and cell complexes General We are interested in a circle packing on a projective Riemann surface, which will be a particular configuration of circles such that all complementary regions are curvilinear triangles. It enjoys both rigid and flexible properties in connection with Teichmüller spaces. The main purpose of this chapter is to discuss such interesting properties from geometric viewpoints. The organization is as follows. After reviewing the basics we discuss rigidity results together with density on constant curvature surfaces due to Koebe, Andreev, Thurston and Brooks. We then discuss flexibility by constructing moduli spaces and formulate a conjecture along with some supporting evidence. 509 531 1 10.4171/055-1/14 http://www.ems-ph.org/doi/10.4171/055-1/14

Riccardo Benedetti Università di Pisa, PISA, ITALY Francesco Bonsante Università di Pavia, PAVIA, ITALY Differential geometry General The aim of this chapter is to give an overview on the geometry of Einstein maximal globally hyperbolic 2+1 spacetimes of arbitrary curvature containing a complete Cauchy surface of finite type. In particular a specialization to the finite type case of the canonical Wick rotation-rescaling theory, previously developed by the authors, is provided. This includes, for arbitrary curvatures, parameterizations in terms of suitable measured geodesic laminations on open hyperbolic surfaces of finite type. The same geometric objects also parameterize complex projective structures on the surfaces. The coincidence of such parameter space is explained by means of geometric correlations between spacetimes of different curvatures and projective surfaces realized via canonical WR-rescaling along the cosmological times. We also specialize on AdS case mostly referring to recent results achieved by other authors. In particular we describe maximal causal extensions of AdS globally hyperbolic spacetimes and an AdS approach to the theory of earthquakes for hyperbolic surfaces of finite type. A general earthquake theorem is proved for the so called enhanced Teichmüller space. The case of spacetimes with conical timelike singularities is also treated. 533 609 1 10.4171/055-1/15 http://www.ems-ph.org/doi/10.4171/055-1/15

William Goldman University of Maryland, COLLEGE PARK, UNITED STATES Functions of a complex variable General The conjugacy class of a generic unimodular 2 × 2 complex matrix is determined by its trace, which may be an arbitrary complex number. In the 19th century, it was known that a generic pair (X,Y) of such pairs is determined up to conjugacy by the triple of traces (tr(X),tr(Y),tr(XY)), which may be an arbitary element of ℂ3. This chapter gives an elementary and detailed proof of this fact, which was published by Vogt in 1889. The folk theorem describing the extension of a representation to a representation of the index-two supergroup which is a free product of three groups of order two, is described in detail, and related to hyperbolic geometry. When n > 2, the classification of conjugacy classes of n-tuples in SL(2,ℂ) is more complicated. We describe it in detail when n = 3. The deformation spaces of hyperbolic structures on some simple surfaces S whose fundamental group is free of rank two or three are computed in trace coordinates. (We only consider the two orientable surfaces whose fundamental group has rank 3.) 611 684 1 10.4171/055-1/16 http://www.ems-ph.org/doi/10.4171/055-1/16

Sean Lawton The University of Texas-Pan American, Edinburg, UNITED STATES Elisha Peterson United States Military Academy, West Point, UNITED STATES Differential geometry General Denote the free group on 2 letters by F2 and the SL(2,ℂ)-representation variety of F2 by R = Hom(F2,SL(2,ℂ)). The group SL(2,ℂ) acts on R by conjugation. We construct an isomorphism between the coordinate ring ℂ[SL(2,ℂ)] and the ring of matrix coefficients, providing an additive basis of ℂ[R]SL(2,ℂ) in terms of spin networks. Using a graphical calculus, we determine the symmetries and multiplicative structure of this basis. This gives a canonical description of the regular functions on the SL(2,ℂ)-character variety of F2 and a new proof of a classical result of Fricke, Klein, and Vogt. 685 730 1 10.4171/055-1/17 http://www.ems-ph.org/doi/10.4171/055-1/17

Feng Luo Rutgers University, PISCATAWAY, UNITED STATES General We relate some ideas of Grothendieck in his work Esquisse d'un programme with some of the recent results on surface geometry and topology. 733 765 1 10.4171/055-1/18 http://www.ems-ph.org/doi/10.4171/055-1/18

Frank Herrlich Karlsruhe Institute of Technology, KARLSRUHE, GERMANY Gabriela Schmithüsen Universität Karlsruhe, KARLSRUHE, GERMANY Functions of a complex variable General This chapter is a detailed exposition of Grothendieck’s theory of dessins d’enfants and its relation with Teichmüller theory. In the first part, the authors consider several aspects of dessins d’enfants, including their appearance through Belyi pairs and their Galois actions, ribbon graphs and their monodromy maps, and the relation to the theory of subgroups of the free group in two generators. In th second part, the authors give a detailed account of origami curves, also commonly known as translation surfaces. These are special Teichmüller curves. The authors study Galois actions on origamis, and they show in particular how such objects are related to boundary points of origami curves. 767 809 1 10.4171/055-1/19 http://www.ems-ph.org/doi/10.4171/055-1/19

Dragomir Šarić City University of New York, FLUSHING, UNITED STATES General The compact solenoid was introduced by Sullivan in the 1990s as a universal object in the category of pointed unbranched coverings of a closed surface of genus at least two. It can be seen as a “universal closed surface”. Sullivan studied the compact solenoid in relation with his approach to the Ehrenpreis Conjecture, and this object turned out to be equipped with a variety of interseting structures. This chapter is a survey on the theory of the solenoid that includes its Teichmüller space equipped with its Teichmüller metric, its Teichmüller extremal maps, its modular group action, its decorated theory, with generalizations to the case of the non-compact solenoid. 811 858 1 10.4171/055-1/20 http://www.ems-ph.org/doi/10.4171/055-1/20

Michael Farber Durham University, United Kingdom Global analysis, analysis on manifolds Manifolds and cell complexes 58E05, 57R70; 57R30 Betti numbers, configuration space, mechanical linkage, navigational complexity, polyhedra, robotics The book discusses several selected topics of a new emerging area of research lying on the interface between topology and engineering. The first main topic of the book is topology of configuration spaces of mechanical linkages. These manifolds arise in various fields of mathematics and in other sciences, e.g. engineering, statistics, molecular biology. To compute Betti numbers of these configuration spaces we apply a new technique of Morse theory in the presence of an involution. A significant result of topology of linkages presented in the book is a solution of a conjecture of Kevin Walker which states that the relative sizes of bars of a linkage are determined, up to certain equivalence, by the cohomology algebra of the linkage configuration space. The book also describes a new probabilistic approach to topology of linkages which treats the bar lengths as random variables and studies mathematical expectations of Betti numbers. The second main topic of the book is topology of configuration spaces associated to polyhedra. The book gives an account of a beautiful work of S.R. Gal suggesting an explicit formula for the generating function encoding Euler characteristics of these spaces. Next we study the knot theory of a robot arm focusing on a recent important result of R. Connelly, E. Demain and G. Rote. Finally, the book investigates topological problems arising in the theory of robot motion planning algorithms and studies the homotopy invariant TC(X) measuring navigational complexity of configuration spaces. The book is intended as an appetizer and will introduce the reader to many fascinating topological problems motivated by engineering. 8 19 2008 978-3-03719-054-8 978-3-03719-554-3 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/054 http://www.ems-ph.org/doi/10.4171/054 Hans Ringström KTH Mathematics, Stockholm, Sweden Relativity and gravitational theory 83-02; 83C75 General relativity, Cauchy problem, strong cosmic censorship, non-linear wave equations The general theory of relativity is a theory of manifolds equipped with Lorentz metrics and fields which describe the matter content. Einstein’s equations equate the Einstein tensor (a curvature quantity associated with the Lorentz metric) with the stress energy tensor (an object constructed using the matter fields). In addition, there are equations describing the evolution of the matter. Using symmetry as a guiding principle, one is naturally led to the Schwarzschild and Friedmann–Lemaître–Robertson–Walker solutions, modelling an isolated system and the entire universe respectively. In a different approach, formulating Einstein’s equations as an initial value problem allows a closer study of their solutions. This book first provides a definition of the concept of initial data and a proof of the correspondence between initial data and development. It turns out that some initial data allow non-isometric maximal developments, complicating the uniqueness issue. The second half of the book is concerned with this and related problems, such as strong cosmic censorship. The book presents complete proofs of several classical results that play a central role in mathematical relativity but are not easily accessible to those wishing to enter the subject. Prerequisites are a good knowledge of basic measure and integration theory as well as the fundamentals of Lorentz geometry. The necessary background from the theory of partial differential equations and Lorentz geometry is included. 6 8 2009 978-3-03719-053-1 978-3-03719-553-6 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/053 http://www.ems-ph.org/doi/10.4171/053 Alexander Barvinok University of Michigan, Ann Arbor, USA Convex and discrete geometry Combinatorics Number theory 52C07; 52B20, 05A15, 52B45, 52B55, 52C45, 11H06 Algebra of polyhedra, Berline–Vergne local formula, continued fractions, Ehrhart polynomial, Euler characteristic, generating functions, integer points, Lenstra–Lenstra–Lovász lattice reduction algorithm, Minkowski Convex Body Theorem This is a self-contained exposition of several core aspects of the theory of rational polyhedra with a view towards algorithmic applications to efficient counting of integer points, a problem arising in many areas of pure and applied mathematics. The approach is based on the consistent development and application of the apparatus of generating functions and the algebra of polyhedra. Topics range from classical, such as the Euler characteristic, continued fractions, Ehrhart polynomial, Minkowski Convex Body Theorem, and the Lenstra– Lenstra–Lovász lattice reduction algorithm, to recent advances such as the Berline–Vergne local formula. The text is intended for graduate students and researchers. Prerequisites are a modest background in linear algebra and analysis as well as some general mathematical maturity. Numerous figures, exercises of varying degree of difficulty as well as references to the literature and publicly available software make the text suitable for a graduate course. 9 1 2008 978-3-03719-052-4 978-3-03719-552-9 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/052 http://www.ems-ph.org/doi/10.4171/052 Dmitri Alekseevsky School of Mathematics, The University of Edinburgh, UK Helga Baum Humboldt-Universität, Berlin, Germany Differential geometry 53-00; 53C50 Pseudo-Riemannian manifold, Lorentzian manifold, spacetime, neutral signature, holonomy group, pseudo-Riemannian symmetric space, Cahen–Wallach space, hypersymplectic manifold, anti-self-dual conformal structure, integrable system, neutral Kähler surfaces, Einstein metric, Einstein universe, essential conformal structure, conformal transformation, causal hierarchy, geodesic, supergravity, generalized geometry, generalized G-structure, Killing spinor This book provides an introduction to and survey of recent developments in pseudo-Riemannian geometry, including applications in mathematical physics, by leading experts in the field. Topics covered are: Classification of pseudo-Riemannian symmetric spaces Holonomy groups of Lorentzian and pseudo-Riemannian manifolds Hypersymplectic manifolds Anti-self-dual conformal structures in neutral signature and integrable systems Neutral Kähler surfaces and geometric optics Geometry and dynamics of the Einstein universe Essential conformal structures and conformal transformations in pseudo-Riemannian geometry The causal hierarchy of spacetimes Geodesics in pseudo-Riemannian manifolds Lorentzian symmetric spaces in supergravity Generalized geometries in supergravity Einstein metrics with Killing leaves The book is addressed to advanced students as well as to researchers in differential geometry, global analysis, general relativity and string theory. It shows essential differences between the geometry on manifolds with positive definite metrics and on those with indefinite metrics, and highlights the interesting new geometric phenomena, which naturally arise in the indefinite metric case. The reader finds a description of the present state of art in the field as well as open problems, which can stimulate further research. 6 11 2008 978-3-03719-051-7 978-3-03719-551-2 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/051 http://www.ems-ph.org/doi/10.4171/051

Ines Kath Ernst-Moritz-Arndt-Universität Greifswald, GREIFSWALD, GERMANY Martin Olbrich Université du Luxembourg, LUXEMBOURG, LUXEMBOURG Differential geometry General Riemannian and pseudo-Riemannian symmetric spaces with semisimple transvection group are known and classified for a long time. Contrary to that the description of pseudo-Riemannian symmetric spaces with non-semisimple transvection group is an open problem. In the last years some progress on this problem was achieved. In this article we want to explain these results and some of their applications. 1 52 1 10.4171/051-1/1 http://www.ems-ph.org/doi/10.4171/051-1/1

Anton Galaev Masaryk University, BRNO, CZECH REPUBLIC Thomas Leistner The University of Adelaide, , AUSTRALIA Differential geometry General We review recent developments in the theory of Lorentzian holonomy groups focussing on the classification results. We present the list of indecomposable, non-irreducible Lorentzian holonomy groups, explain the idea of its proof, and describe a method of constructing metrics which realise all the possible groups. This method is then used to construct many examples of metrics. Finally, we give some applications for the existence of parallel spinors and a short outlook on other signatures. As a new result we obtain the holonomy classification for indecomposable, non-irreducible Lorentzian Einstein spaces. 53 96 1 10.4171/051-1/2 http://www.ems-ph.org/doi/10.4171/051-1/2

Andrew Dancer The University of Sydney, SYDNEY, NSW, AUSTRALIA Andrew Swann University of Southern Denmark, ODENSE M, DENMARK Differential geometry Manifolds and cell complexes General We survey aspects of hypersymplectic manifolds, especially comparing them with symplectic and hyperkähler manifolds. We consider hypersymplectic quotients of flat space by the action of abelian groups and describe a hypersymplectic cut construction. 97 111 1 10.4171/051-1/3 http://www.ems-ph.org/doi/10.4171/051-1/3

Maciej Dunajski University of Cambridge, CAMBRIDGE, UNITED KINGDOM Simon West University of Cambridge, CAMBRIDGE, UNITED KINGDOM Differential geometry General We review the subject of four dimensional anti-self-dual conformal structures with signature (++−−). Both local and global questions are discussed. Most of the material is well known in the literature and we present it in a way which underlines the connection with integrable systems. Some of the results – e.g. the Lax pair characterisation of the scalar-flat Kähler condition and a twistor construction of a conformal structure with twisting null Killing vector – are new. 113 148 1 10.4171/051-1/4 http://www.ems-ph.org/doi/10.4171/051-1/4

Brendan Guilfoyle Institute of Technology, TRALEE, Co. Kerry, IRELAND Wilhelm Klingenberg University of Durham, DURHAM, UNITED KINGDOM Differential geometry Optics, electromagnetic theory General The space L of oriented lines, or rays, in Euclidean 3-space E3 is a 4-dimensional space with an abundance of natural geometric structure. In particular, it boasts a neutral Kähler metric which is closely related to the Euclidean metric on E3. In this article we review recent work on this Kähler structure and consider its applications to geometric optics in a homogeneous isotropic medium. In particular, we discuss the complex geometry of reflection in a surface in E3 and the computation of focal sets of bundles of rays. To illustrate the method, we compute the focal set of the kth reflection of a point source off the inside of a cylinder. The focal sets, which we explicitly parameterize, exhibit unexpected symmetries, and are found to fit well with observable phenomena. We conclude with generalizations of the geometric construction. 149 178 1 10.4171/051-1/5 http://www.ems-ph.org/doi/10.4171/051-1/5

Thierry Barbot Université d'Avignon, AVIGNON, FRANCE Tomasz Kaszynski Université de Sherbrooke, SHERBROOKE, CANADA Todd Drumm Howard University, WASHINGTON, UNITED STATES William Goldman University of Maryland, COLLEGE PARK, UNITED STATES Karin Melnick University of Maryland, COLLEGE PARK, UNITED STATES Differential geometry Relativity and gravitational theory General The Einstein universe is the conformal compactification of Minkowski space. It also arises as the ideal boundary of anti-de Sitter space. The purpose of this article is to develop the synthetic geometry of the Einstein universe in terms of its homogeneous submanifolds and causal structure, with particular emphasis on dimension 2 + 1, in which there is a rich interplay with symplectic geometry. 179 229 1 10.4171/051-1/6 http://www.ems-ph.org/doi/10.4171/051-1/6

Charles Frances Université Paris Sud, ORSAY CEDEX, FRANCE Differential geometry General This article is devoted to pseudo-Riemannian essential structures, namely the pseudo-Riemannian manifolds, whose conformal group of transformations cannot be reduced to a group of isometries. A celebrated result of M. Obata and J. Ferrand states that for Riemannian manifolds, the only essential structures are the standard sphere, and the Euclidean space. We discuss what this result becomes in higher signature, focusing on the Lorentzian case. 231 260 1 10.4171/051-1/7 http://www.ems-ph.org/doi/10.4171/051-1/7

Wolfgang Kühnel Universität Stuttgart, STUTTGART, GERMANY Hans-Bert Rademacher Universität Leipzig, LEIPZIG, GERMANY Differential geometry Relativity and gravitational theory General This is a survey about conformal mappings between pseudo-Riemannian manifolds and, in particular, conformal vector fields defined on pseudo-Riemannian manifolds. 261 298 1 10.4171/051-1/8 http://www.ems-ph.org/doi/10.4171/051-1/8

Ettore Minguzzi Universita di Firenze, FIRENZE, ITALY Miguel Sánchez Universidad de Granada, GRANADA, SPAIN Differential geometry Relativity and gravitational theory General The full causal ladder of spacetimes is constructed, and their updated main properties are developed. Old concepts and alternative definitions of each level of the ladder are revisited, with emphasis in minimum hypotheses. The implications of the recently solved “folk questions on smoothability”, and alternative proposals (as recent isocausality), are also summarized. 299 358 1 10.4171/051-1/9 http://www.ems-ph.org/doi/10.4171/051-1/9

Anna Maria Candela Università degli Studi di Bari, BARI, ITALY Miguel Sánchez Universidad de Granada, GRANADA, SPAIN Differential geometry Global analysis, analysis on manifolds General Geodesics become an essential element of the geometry of a semi-Riemannian manifold. In fact, their differences and similarities with the (positive definite) Riemannian case, constitute the first step to understand semi-Riemannian Geometry. The progress in the last two decades has become impressive, being especially relevant the systematic introduction of (infinite-dimensional) variational methods. Our purpose is to give an overview, from refinements of classical results to updated variational settings. First, several properties (and especially completeness) of geodesics in some ambient spaces are studied. This includes heuristic constructions of compact incomplete examples, geodesics in warped, GRW or stationary spacetimes, properties in surfaces and spaceforms, or problems on stability of completeness. Then we study the variational framework, and focus on two fundamental problems of this approach, which regards geodesic connectedness. The first one deals with a variational principle for stationary manifolds, and its recent implementation inside causality theory. The second one concerns orthogonal splitting manifolds, and a reasonably self-contained development is provided, collecting some steps spread in the literature. 359 418 1 10.4171/051-1/10 http://www.ems-ph.org/doi/10.4171/051-1/10

José Miguel Figueroa-O’Farrill University of Edinburgh, EDINBURGH, UNITED KINGDOM Differential geometry Relativity and gravitational theory General I will discuss the emergence of Lorentzian symmetric spaces as supersymmetric supergravity backgrounds. I will focus on supergravity theories in dimension 11, 10, and 6, and will concentrate on the determination of the so-called maximally supersymmetric backgrounds, for which a classification exists up to local isometry. A special class of Lorentzian symmetric spaces also plays a rôle in the determination of parallelisable supergravity backgrounds in type II supergravity, which I will also summarise. 419 454 1 10.4171/051-1/11 http://www.ems-ph.org/doi/10.4171/051-1/11

Frederik Witt Universität Münster, MÜNSTER, GERMANY Differential geometry Relativity and gravitational theory General We study the geometry of type II supergravity compactifications in terms of an oriented vector bundle E, endowed with a bundle metric of split signature and further datum. The geometric structure is associated with a so-called generalised G-structure and characterised by an E-spinor ρ, which we can regard as a differential form of mixed degree. This enables us to reformulate the field equations of type II supergravity as an integrability condition of type dH ρ = 0, where dH = d + H∧ is the twisted differential on forms. Finally, we investigate some geometric properties of integrable structures and formulate various no-go theorems. 455 494 1 10.4171/051-1/12 http://www.ems-ph.org/doi/10.4171/051-1/12

Gaetano Vilasi Università degli Studi di Salerno, BARONISSI (SA), ITALY Differential geometry Relativity and gravitational theory General Solutions of vacuum Einstein’s field equations, for the class of pseudo-Riemannian 4-metrics admitting a non Abelian 2-dimensional Lie algebra of Killing fields, are explicitly described. When the distribution orthogonal to the orbits is completely integrable and the metric is not degenerate along the orbits, these solutions are parameterized either by solutions of a transcendental equation (the tortoise equation), or by solutions of a linear second order differential equation in two independent variables. Metrics, corresponding to solutions of the tortoise equation, are characterized as those that admit a 3-dimensional Lie algebra of Killing fields with 2-dimensional leaves. A formalism (ζ-complex analysis), allowing one to construct global Einstein metrics by matching together these solutions, is also described. It is shown that the remaining metrics, corresponding to the case in which the commutator of the two Killing fields is isotropic, represent nonlinear gravitational waves obeying to two nonlinear superposition laws. The energy and the polarization of this family of waves are explicitly evaluated; it is shown that they have spin-1 and their sources are also described. 495 526 1 10.4171/051-1/13 http://www.ems-ph.org/doi/10.4171/051-1/13

Iskander Taimanov Sobolev Institute of Mathematics, Novosibirsk, Russia Differential geometry 53-01; 20C05, 20G20, 37J05, 37J35, 53A10, 53D05, 53D17, 57S17, 58A05 Differential geometry, Riemannian geometry, smooth manifolds, minimal surfaces, Lie groups, representation theory, symplectic and Poisson geometry, finite-dimensional integrable systems Differential geometry studies geometrical objects using analytical methods. Like modern analysis itself, differential geometry originates in classical mechanics. For instance, geodesics and minimal surfaces are defined via variational principles and the curvature of a curve is easily interpreted as the acceleration with respect to the path length parameter. Modern differential geometry in its turn strongly contributed to modern physics. This book gives an introduction to the basics of differential geometry, keeping in mind the natural origin of many geometrical quantities, as well as the applications of differential geometry and its methods to other sciences. The text is divided into three parts. The first part covers the basics of curves and surfaces, while the second part is designed as an introduction to smooth manifolds and Riemannian geometry. In particular, Chapter 5 contains short introductions to hyperbolic geometry and geometrical principles of special relativity theory. Here, only a basic knowledge of algebra, calculus and ordinary differential equations is required. The third part is more advanced and introduces into matrix Lie groups and Lie algebras, representation theory of groups, symplectic and Poisson geometry, and applications of complex analysis in surface theory. The book is based on lectures the author held repeatedly at Novosibirsk State University. It is addressed to students as well as to anyone who wants to learn the basics of differential geometry. 4 8 2008 978-3-03719-050-0 978-3-03719-550-5 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/050 http://www.ems-ph.org/doi/10.4171/050 Marek Jarnicki Jagiellonian University, Kraków, Poland Peter Pflug University of Oldenburg, Germany Several complex variables and analytic spaces 32-01; 32A05, 32A07, 32A10, 32D05, 32E05, 32F12, 32F45 Reinhardt domains, domains of holomorphy, pseudoconvex domains, invariant metrics and distances This book provides a comprehensive introduction to the field of several complex variables in the setting of a very special but basic class of domains, the so-called Reinhardt domains. In this way the reader may learn much about this area without encountering too many technical difficulties. Chapter 1 describes the fundamental notions and the phenomenon of simultaneous holomorphic extension. Chapter 2 presents a fairly complete discussion of biholomorphisms of bounded (complete) Reinhardt domains in the two dimensional case. The third chapter gives a classification of Reinhardt domains of existence for the most important classes of holomorphic functions. The last chapter deals with invariant functions and gives explicit calculations of many of them on certain Reinhardt domains. Numerous exercises are included to help the readers with their understanding of the material. Further results and open problems are added which may be useful as seminar topics. The primary aim of this book is to introduce students or non-experts to some of the main research areas in several complex variables. The book provides a friendly invitation to this field as the only prerequisite is a basic knowledge of analysis. 3 3 2008 978-3-03719-049-4 978-3-03719-549-9 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/049 http://www.ems-ph.org/doi/10.4171/049 Corrected 2nd printing, 2010 Tammo tom Dieck University of Göttingen, Germany Algebraic topology Manifolds and cell complexes 55-01, 57-01 Covering spaces, fibrations, cofibrations, homotopy groups, cell complexes, fibre bundles, vector bundles, classifying spaces, singular and axiomatic homology and cohomology, smooth manifolds, duality, characteristic classes, bordism. This book is written as a textbook on algebraic topology. The first part covers the material for two introductory courses about homotopy and homology. The second part presents more advanced applications and concepts (duality, characteristic classes, homotopy groups of spheres, bordism). The author recommends to start an introductory course with homotopy theory. For this purpose, classical results are presented with new elementary proofs. Alternatively, one could start more traditionally with singular and axiomatic homology. Additional chapters are devoted to the geometry of manifolds, cell complexes and fibre bundles. A special feature is the rich supply of nearly 500 exercises and problems. Several sections include topics which have not appeared before in textbooks as well as simplified proofs for some important results. Prerequisites are standard point set topology (as recalled in the first chapter), elementary algebraic notions (modules, tensor product), and some terminology from category theory. The aim of the book is to introduce advanced undergraduate and graduate (masters) students to basic tools, concepts and results of algebraic topology. Sufficient background material from geometry and algebra is included. 9 1 2008 978-3-03719-048-7 978-3-03719-548-2 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/048 http://www.ems-ph.org/doi/10.4171/048 Benjamin Enriquez IRMA, Strasbourg, France Quantum theory 81R50; 81R12 Tensor categories, integrable systems, knot invariants The volume starts with a lecture course by P. Etingof on tensor categories (notes by D. Calaque). This course is an introduction to tensor categories, leading to topics of recent research such as realizability of fusion rings, Ocneanu rigidity, module categories, weak Hopf algebras, Morita theory for tensor categories, lifting theory, categorical dimensions, Frobenius–Perron dimensions, and the classification of tensor categories. The remainder of the book consists of three detailed expositions on associators and the Vassiliev invariants of knots, classical and quantum integrable systems and elliptic algebras, and the groups of algebra automorphisms of quantum groups. The preface sets the results presented in perspective. Directed at research mathematicians and theoretical physicists as well as graduate students, the volume gives an overview of the ongoing research in the domain of quantum groups, an important subject of current mathematical physics. 6 11 2008 978-3-03719-047-0 978-3-03719-547-5 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/047 http://www.ems-ph.org/doi/10.4171/047 12

Damien Calaque Université Montpellier 2, MONTPELLIER CEDEX 5, FRANCE Pavel Etingof Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES Category theory; homological algebra Nonassociative rings and algebras General We give a review of some recent developments in the theory of tensor categories. The topics include realizability of fusion rings, Ocneanu rigidity, module categories, weak Hopf algebras, Morita theory for tensor categories, lifting theory, categorical dimensions, Frobenius–Perron dimensions, and classification of tensor categories. 1 38 1 10.4171/047-1/1 http://www.ems-ph.org/doi/10.4171/047-1/1

Category theory; homological algebra Nonassociative rings and algebras General We determine explicitly a rational even Drinfeld associator Φ in a completion of the universal enveloping algebra of the Lie superalgebra gl(1 | 1)⊕3. More generally, we define a new algebra of trivalent diagrams that has a unique even horizontal group-like Drinfeld associator Φ. The associator Φ is mapped to Φ by a weight system. As a related result of independent interest, we show how O. Viro’s generalization Δ1 of the multi-variable Alexander polynomial can be obtained from the universal Vassiliev invariant of trivalent graphs. We determine Φ by using the invariant Δ1(tetrahedron) of a planar tetrahedron. 39 80 1 10.4171/047-1/2 http://www.ems-ph.org/doi/10.4171/047-1/2

Alexander Odesskii University of Manchester, MANCHESTER, UNITED KINGDOM Vladimir Rubtsov Université d'Angers, ANGERS CEDEX, FRANCE Category theory; homological algebra Nonassociative rings and algebras General We construct new integrable systems (IS), both classical and quantum, associated with elliptic algebras. Our constructions are based both on a construction of commuting families in skew fields and on properties of the elliptic algebras and their representations. We give some examples showing how these IS are related to previously studied systems. 81 105 1 10.4171/047-1/3 http://www.ems-ph.org/doi/10.4171/047-1/3

Nicolás Andruskiewitsch Universidad Nacional de Cordoba, CORDOBA, ARGENTINA Category theory; homological algebra Nonassociative rings and algebras General Let g be a simple complex finite dimensional Lie algebra and let Uq+(ℊ) be the positive part of the quantum enveloping algebra of g.If g is of type A2, the group of algebra automorphisms of Uq+(ℊ) is a semidirect product (k×)2 ⋊ Autdiagr(ℊ) any algebra automorphism is an automorphism of braided Hopf algebra and preserves the standard grading [2]. This intriguing smallness of the group of algebra automorphisms raises questions about the extent of these phenomena. We discuss some of them in the present paper. We introduce the notion of “algebra with few automorphisms” and establish some consequences. We prove some exploratory results concerning the group of algebra automorphisms for the type B2. We study the Hopf algebra automorphisms of Nichols algebras and their bosonizations and compute in particular the group of Hopf algebra automorphisms of Uq+(ℊ). 107 133 1 10.4171/047-1/4 http://www.ems-ph.org/doi/10.4171/047-1/4

Michel Weber IRMA, Strasbourg, France Dynamical systems and ergodic theory Probability theory and stochastic processes 37-02, 60-02 Dynamical systems, measure-preserving transformation, ergodic theorems, spectral theorems, convergence almost everywhere, central limit theorem, stochastic processes, gaussian processes, metric entropy method, majorizing measure method This book presents in a concise and accessible way, as well as in a common setting, various tools and methods arising from spectral theory, ergodic theory and stochastic processes theory, which form the basis of and contribute interactively a great deal to the current research on almost everywhere convergence problems. The text is divided into four parts. Part I is devoted to spectral results such as von Neumann’s theorem, spectral regularizations inequalities and their link with square functions and entropy numbers of ergodic averages. The representation of a weakly stationary process as Fourier transform of some random orthogonal measure, and a study of Gaposhkin’s spectral criterion conclude this part.  Classical results such as mixing in dynamical systems, Birkhoff's pointwise theorem, dominated ergodic theorems, oscillations functions of ergodic averages, transference principle, Wiener–Wintner theorem, Banach principle, continuity principle, Bourgain's entropy criteria, Burton–Denker’s central limit theorem are covered in part II.  The metric entropy method and the majorizing measure method, including a succinct study of Gaussian processes, are treated in part III, with applications to suprema of random polynomials.  Part IV contains a study of Riemann sums and of the convergence properties of the system {f(nkx), k ≥ 1}, as well as a probabilistic approach concerning divisors with applications. Researchers working in dynamical systems and at the crossroads of spectral theory, ergodic theory and stochastic processes will find the tools, methods and results presented in this book of great interest. It is written in a style accessible to graduate students throughout. 8 29 2009 978-3-03719-046-3 978-3-03719-546-8 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/046 http://www.ems-ph.org/doi/10.4171/046 14 Gennadiy Feldman B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences, Kharkov, Ukraine Probability theory and stochastic processes Abstract harmonic analysis Statistics 60-02, 60B15, 62E10; 43A05, 43A25, 43A35, 39B52 Locally compact Abelian group, Gaussian distribution, characterization problems of mathematical statistics, functional equation This book deals with the characterization of probability distributions. It is well known that both the sum and the difference of two Gaussian independent random variables with equal variance are independent as well. The converse statement was proved independently by M. Kac and S. N. Bernstein. This result is a famous example of a characterization theorem. In general, characterization problems in mathematical statistics are statements in which the description of possible distributions of random variables follows from properties of some functions in these variables. In recent years, a great deal of attention has been focused upon generalizing the classical characterization theorems to random variables with values in various algebraic structures such as locally compact Abelian groups, Lie groups, quantum groups, or symmetric spaces. The present book is aimed at the generalization of some well-known characterization theorems to the case of independent random variables taking values in a locally compact Abelian group X. The main attention is paid to the characterization of the Gaussian and the idempotent distribution (group analogs of the Kac–Bernstein, Skitovich–Darmois, and Heyde theorems). The solution of the corresponding problems is reduced to the solution of some functional equations in the class of continuous positive definite functions defined on the character group of X. Group analogs of the Cramér and Marcinkiewicz theorems are also studied. The author is an expert in algebraic probability theory. His comprehensive and self-contained monograph is addressed to mathematicians working in probability theory on algebraic structures, abstract harmonic analysis, and functional equations. The book concludes with comments and unsolved problems that provide further stimulation for future research in the theory. 4 29 2008 978-3-03719-045-6 978-3-03719-545-1 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/045 http://www.ems-ph.org/doi/10.4171/045 5 Camillo De Lellis University of Zurich Measure and integration Real functions Calculus of variations and optimal control; optimization 28A75; 26B15, 49Q15, 49Q20 The characterization of rectifiable sets through the existence of densities is a pearl of geometric measure theory. The difficult proof, due to Preiss, relies on many beautiful and deep ideas and novel techniques. Some of them have already proven useful in other contexts, whereas others have not yet been exploited. These notes give a simple and short presentation of the former, and provide some perspective of the latter. This text emerged from a course on rectifiability given at the University of Zürich. It is addressed both to researchers and students, the only prerequisite is a solid knowledge in standard measure theory. The first four chapters give an introduction to rectifiable sets and measures in euclidean spaces, covering classical topics such as the area formula, the theorem of Marstrand and the most elementary rectifiability criterions. The fifth chapter is dedicated to a subtle rectifiability criterion due to Marstrand and generalized by Mattila, and the last three focus on Preiss' result. The aim is to provide a self-contained reference for anyone interested in an overview of this fascinating topic. 2 12 2008 978-3-03719-044-9 978-3-03719-544-4 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/044 http://www.ems-ph.org/doi/10.4171/044 From Hopf Algebras to Multiplicative Unitaries and Beyond Thomas Timmermann University of Münster, Germany Functional analysis 16W30, 46-01, 46L55; 22A22, 22D25, 22D35, 46L10 Hopf algebras, duality, multiplicative unitaries, quantum groups, quantum groupoids This book provides an introduction to the theory of quantum groups with emphasis on their duality and on the setting of operator algebras. Part I of the text presents the basic theory of Hopf algebras, Van Daele’s duality theory of algebraic quantum groups, and Woronowicz’s compact quantum groups, staying in a purely algebraic setting. Part II focuses on quantum groups in the setting of operator algebras. Woronowicz’s compact quantum groups are treated in the setting of C*-algebras, and the fundamental multiplicative unitaries of Baaj and Skandalis are studied in detail. An outline of Kustermans’ and Vaes’ comprehensive theory of locally compact quantum groups completes this part. Part III leads to selected topics, such as coactions, Baaj–Skandalis-duality, and approaches to quantum groupoids in the setting of operator algebras. The book is addressed to graduate students and non-experts from other fields. Only basic knowledge of (multi-) linear algebra is required for the first part, while the second and third part assume some familiarity with Hilbert spaces, C*-algebras, and von Neumann algebras. 2 28 2008 978-3-03719-043-2 978-3-03719-543-7 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/043 http://www.ems-ph.org/doi/10.4171/043 Dorothee Haroske University of Jena, Germany Hans Triebel University of Jena, Germany Partial differential equations Functional analysis 35-01, 46-01; 35J25, 35P15, 42B35, 46E35, 46F05, 47B06, 47F05 It is the main aim of this book to develop at an accessible, moderate level an L2 theory for elliptic differential operators of second order on bounded smooth domains in Euclidean n-space, including a priori estimates for boundary-value problems in terms of (fractional) Sobolev spaces on domains and on their boundaries, together with a related spectral theory. The presentation is preceded by an introduction to the classical theory for the Laplace–Poisson equation, and some chapters providing required ingredients such as the theory of distributions, Sobolev spaces and the spectral theory in Hilbert spaces. The book grew out of two-semester courses the authors have given several times over a period of ten years at the Friedrich Schiller University of Jena. It is addressed to graduate students and mathematicians who have a working knowledge of calculus, measure theory and the basic elements of functional analysis (as usually covered by undergraduate courses) and who are seeking an accessible introduction to some aspects of the theory of function spaces and its applications to elliptic equations. 11 21 2007 978-3-03719-042-5 978-3-03719-542-0 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/042 http://www.ems-ph.org/doi/10.4171/042 Oleg Bogopolski TU Dortmund, Germany Group theory and generalizations 20-01; 20D08, 20E05, 20E06, 20E08, 20F28 Simple groups, sporadic groups, Steiner systems, codes, free products with amalgamation, HNN extensions, actions on trees, free groups and their automorphisms, train tracks This book quickly introduces beginners to general group theory and then focuses on three main themes: finite group theory, including sporadic groups; combinatorial and geometric group theory, including the Bass–Serre theory of groups acting on trees; the theory of train tracks by Bestvina and Handel for automorphisms of free groups. With its many examples, exercises, and full solutions to selected exercises, this text provides a gentle introduction that is ideal for self-study and an excellent preparation for applications. A distinguished feature of the presentation is that algebraic and geometric techniques are balanced. The beautiful theory of train tracks is illustrated by two nontrivial examples. Presupposing only a basic knowledge of algebra, the book is addressed to anyone interested in group theory: from advanced undergraduate and graduate students to specialists. 2 29 2008 978-3-03719-041-8 978-3-03719-541-3 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/041 http://www.ems-ph.org/doi/10.4171/041 Gohar Harutyunyan University of Oldenburg, Germany B.-Wolfgang Schulze University of Potsdam, Germany Partial differential equations 35-02; 35J25, 35S15 Ellipticity on manifolds with singular boundaries, edge and corner pseudo-differential calculus, problems with mixed and transmission conditions, problems of Zaremba type, crack problems, meromorphic symbolic structures, relative index Mixed, transmission, or crack problems belong to the analysis of boundary value problems on manifolds with singularities. The Zaremba problem with a jump between Dirichlet and Neumann conditions along an interface on the boundary is a classical example. The central theme of this book is to study mixed problems in standard Sobolev spaces as well as in weighted edge spaces where the interfaces are interpreted as edges. Parametrices and regularity of solutions are obtained within a systematic calculus of boundary value problems on manifolds with conical or edge singularities. This calculus allows singularities on the interface, and homotopies between mixed and crack problems. Additional edge conditions are computed in terms of relative index results. In a detailed final chapter, the intuitive ideas of the approach are illustrated, and there is a discussion of future challenges. A special feature of the text is the inclusion of many worked out examples which help the reader to appreciate the scope of the theory and to treat new cases of practical interest. This book is addressed to mathematicians and physicists interested in models with singularities, associated boundary value problems, and their solvability strategies based on pseudo-differential operators. The material is also useful for students in higher semesters and young researchers, as well as for experienced specialists working in analysis on manifolds with geometric singularities, the applications of index theory and spectral theory, operator algebras with symbolic structures, quantisation, and asymptotic analysis. 11 21 2007 978-3-03719-040-1 978-3-03719-540-6 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/040 http://www.ems-ph.org/doi/10.4171/040 4 Ralf Meyer University of Göttingen, Germany $K$-theory Functional analysis 19-02, 46-02; 46L80, 46A17, 46H30, 19D55, 19K35 Cyclic homology, bornology, Banach algebra, non-commutative geometry, functional calculus, K-theory Periodic cyclic homology is a homology theory for non-commutative algebras that plays a similar role in non-commutative geometry as de Rham cohomology for smooth manifolds. While it produces good results for algebras of smooth or polynomial functions, it fails for bigger algebras such as most Banach algebras or C*-algebras. Analytic and local cyclic homology are variants of periodic cyclic homology that work better for such algebras. In this book the author develops and compares these theories, emphasising their homological properties. This includes the excision theorem, invariance under passage to certain dense subalgebras, a Universal Coefficient Theorem that relates them to K-theory, and the Chern–Connes character for K-theory and K-homology. The cyclic homology theories studied in this text require a good deal of functional analysis in bornological vector spaces, which is supplied in the first chapters. The focal points here are the relationship with inductive systems and the functional calculus in non-commutative bornological algebras. The book is mainly intended for researchers and advanced graduate students interested in non-commutative geometry. Some chapters are more elementary and independent of the rest of the book, and will be of interest to researchers and students working in functional analysis and its applications. 8 6 2007 978-3-03719-039-5 978-3-03719-539-0 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/039 http://www.ems-ph.org/doi/10.4171/039 3 Volume I: Diophantine Problems and Polynomials Volume II: Elementary, Analytic and Geometric Number Theory Henryk Iwaniec Rutgers University, USA Władysław Narkiewicz University of Wroc&#322;aw, Poland Jerzy Urbanowicz IM PAN, Warsaw, Poland Number theory Field theory and polynomials 11, 12 Andrzej Schinzel, born in 1937, is a leading number theorist whose work has a lasting impact on modern mathematics. He is the author of over 200 research articles in various branches of arithmetics, including elementary, analytic and algebraic number theory. He has also been, for nearly 40 years, the editor of Acta Arithmetica, the first international journal devoted exclusively to number theory. These Selecta contain Schinzel's most important articles published between 1955 and 2006. The arrangement is by topic, with each major category introduced by an expert's comment. Many of the hundred selected papers deal with arithmetical and algebraic properties of polynomials in one or several variables, but there are also articles on Euler's totient function, the favorite subject of Schinzel's early research, on prime numbers (including the famous paper with Sierpiński on the Hypothesis “H”), algebraic number theory, diophantine equations, analytical number theory and geometry of numbers. Volume II concludes with some papers from outside number theory, as well as a list of unsolved problems and unproved conjectures, taken from the work of Schinzel. 4 4 2007 978-3-03719-038-8 978-3-03719-538-3 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/038 http://www.ems-ph.org/doi/10.4171/038 Christian Bär University of Potsdam, Germany Nicolas Ginoux University of Regensburg, Germany Frank Pfäffle University of Potsdam, Germany Global analysis, analysis on manifolds Partial differential equations Differential geometry Primary 58J45, 81T20; secondary 35L05, 35L15, 53C50, 81T05 This book provides a detailed introduction to linear wave equations on Lorentzian manifolds (for vector-bundle valued fields). After a collection of preliminary material in the first chapter one finds in the second chapter the construction of local fundamental solutions together with their Hadamard expansion. The third chapter establishes the existence and uniqueness of global fundamental solutions on globally hyperbolic spacetimes and discusses Green's operators and well-posedness of the Cauchy problem. The last chapter is devoted to field quantization in the sense of algebraic quantum field theory. The necessary basics on C*-algebras and CCR-representations are developed in full detail. The text provides a self-contained introduction to these topics addressed to graduate students in mathematics and physics. At the same time it is intended as a reference for researchers in global analysis, general relativity, and quantum field theory. 3 21 2007 978-3-03719-037-1 978-3-03719-537-6 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/037 http://www.ems-ph.org/doi/10.4171/037 Sergei Buyalo Steklov Institute of Mathematics, St. Petersburg, Russia Viktor Schroeder University of Zurich, Switzerland Geometry Differential geometry 51F99, 53C23, 55M10; 53C23, 54C20, 20F67, 20F69 Asymptotic geometry is the study of metric spaces from a large scale point of view, where the local geometry does not come into play. An important class of model spaces are the hyperbolic spaces (in the sense of Gromov), for which the asymptotic geometry is nicely encoded in the boundary at infinity. In the first part of this book, in analogy with the concepts of classical hyperbolic geometry, the authors provide a systematic account of the basic theory of Gromov hyperbolic spaces. These spaces have been studied extensively in the last twenty years, and have found applications in group theory, geometric topology, Kleinian groups, as well as dynamics and rigidity theory. In the second part of the book, various aspects of the asymptotic geometry of arbitrary metric spaces are considered. It turns out that the boundary at infinity approach is not appropriate in the general case, but dimension theory proves useful for finding interesting results and applications. The text leads concisely to some central aspects of the theory. Each chapter concludes with a separate section containing supplementary results and bibliographical notes. Here the theory is also illustrated with numerous examples as well as relations to the neighboring fields of comparison geometry and geometric group theory. The book is based on lectures the authors presented at the Steklov Institute in St. Petersburg and the University of Zurich. It addressed to graduate students and researchers working in geometry, topology, and geometric group theory. 5 24 2007 978-3-03719-036-4 978-3-03719-536-9 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/036 http://www.ems-ph.org/doi/10.4171/036 A geometric approach Guus Balkema University of Amsterdam, The Netherlands Paul Embrechts ETH Zurich, Switzerland Probability theory and stochastic processes Game theory, economics, social and behavioral sciences 60G70, 60F99, 91B30, 91B70, 62G32, 60G55 Quantitative Risk Management (QRM) has become a field of research of considerable importance to numerous areas of application, including insurance, banking, energy, medicine, reliability. Mainly motivated by examples from insurance and finance, the authors develop a theory for handling multivariate extremes. The approach borrows ideas from portfolio theory and aims at an intuitive approach in the spirit of the Peaks over Thresholds method. The point of view is geometric. It leads to a probabilistic description of what in QRM language may be referred to as a high risk scenario: the conditional behaviour of risk factors given that a large move on a linear combination (portfolio, say) has been observed. The theoretical models which describe such conditional extremal behaviour are characterized and their relation to the limit theory for coordinatewise maxima is explained. The first part is an elegant exposition of coordinatewise extreme value theory; the second half develops the more basic geometric theory. Besides a precise mathematical deduction of the main results, the text yields numerous discussions of a more applied nature. A twenty page preview introduces the key concepts; the extensive introduction provides links to financial mathematics and insurance theory. The book is based on a graduate course on point processes and extremes. It could form the basis for an advanced course on multivariate extreme value theory or a course on mathematical issues underlying risk. Students in statistics and finance with a mathematical, quantitative background are the prime audience. Actuaries and risk managers involved in data based risk analysis will find the models discussed in the book stimulating. The text contains many indications for further research. 9 6 2007 978-3-03719-035-7 978-3-03719-535-2 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/035 http://www.ems-ph.org/doi/10.4171/035 Pavel Etingof Massachusetts Institute of Technology, Cambridge, USA Associative rings and algebras Algebraic geometry Mechanics of particles and systems 16G30, 70H06, 16S80, 14A22 Calogero–Moser systems, which were originally discovered by specialists in integrable systems are currently at the crossroads of many areas of mathematics and within the scope of interests of many mathematicians. More specifically, these systems and their generalizations turned out to have intrinsic connections with such fields as algebraic geometry (Hilbert schemes of surfaces), representation theory (double affine Hecke algebras, Lie groups, quantum groups), deformation theory (symplectic reflection algebras), homological algebra (Koszul algebras), Poisson geometry, etc. The goal of the present lecture notes is to give an introduction to the theory of Calogero–Moser systems, highlighting their interplay with these fields. Since these lectures are designed for non-experts, we give short introductions to each of the subjects involved, and provide a number of exercises. The book will be suitable for mathematics graduate students and researchers in the areas of representation theory, noncommutative algebra, algebraic geometry, and related areas. 3 3 2007 978-3-03719-034-0 978-3-03719-534-5 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/034 http://www.ems-ph.org/doi/10.4171/034 Initial Value Problems and Local Regularity Theory Panagiota Daskalopoulos Columbia University, New York, USA Carlos Kenig University of Chicago, USA Partial differential equations 35-02, 35K55; 35K65 Degenerate diffusion, local regularity estimates, compactness for families of solutions, initial and boundary value problems for slow and fast diffusion The book deals with existence, uniqueness, regularity and asymptotic behavior of solutions to the initial value problem (Cauchy problem) and the initial-Dirichlet problem for a class of degenerate diffusions modeled on the porous medium type equation ut = Δum, m ≥ 0, u ≥ 0. Such models arise in plasma physics, diffusions through porous media, thin liquid film dynamics as well as in geometric flows such as the Ricci flow on surfaces and the Yamabe flow. The approach presented to these problems is through the use of local regularity estimates and Harnack type inequalities, which yield compactness for families of solutions. The theory is quite complete in the slow diffusion case (m > 1) and in the supercritical fast diffusion case (mc < m < 1, mc = (n – 2)+/n) while many problems remain in the range m ≤ mc. All of these aspects of the theory are discussed in the book. The book is addressed to both researchers and to graduate students with a good background in analysis and some previous exposure to partial differential equations. 5 31 2007 978-3-03719-033-3 978-3-03719-533-8 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/033 http://www.ems-ph.org/doi/10.4171/033 1 A Structure Theory for Pro-Lie Algebras, Pro-Lie Groups, and Connected Locally Compact Groups Karl Hofmann TU Darmstadt, Germany Sidney Morris University of Ballarat, Australia Topological groups, Lie groups Primary 22-02; secondary 17B65, 22D05, 22E20, 22E65, 44A13, 44M40, 58B25 Pro-Lie groups, pro-Lie algebras, Lie theory of connected pro-Lie groups, exponential function, structure theory of locally compact groups, completeness, quotient groups, open mapping theorem, Levi–Malcev splitting, local Iwasawa splitting Lie groups were introduced in 1870 by the Norwegian mathematician Sophus Lie. A century later Jean Dieudonné quipped that Lie groups had moved to the center of mathematics and that one cannot undertake anything without them. If a complete topological group G can be approximated by Lie groups in the sense that every identity neighborhood U of G contains a normal subgroup N such that G/N is a Lie group, then it is called a pro-Lie group. Every locally compact connected topological group and every compact group is a pro-Lie group. While the class of locally compact groups is not closed under the formation of arbitrary products, the class of pro-Lie groups is.   For half a century, locally compact pro-Lie groups have drifted through the literature, yet this is the first book which systematically treats the Lie and structure theory of pro-Lie groups irrespective of local compactness. This study fits very well into that current trend which addresses infinite dimensional Lie groups. The results of this text are based on a theory of pro-Lie algebras which parallels the structure theory of finite dimensional real Lie algebras to an astonishing degree even though it has to overcome greater technical obstacles. This book exposes a Lie theory of connected pro-Lie groups (and hence of connected locally compact groups) and illuminates the manifold ways in which their structure theory reduces to that of compact groups on the one hand and of finite dimensional Lie groups on the other. It is a continuation of the authors' fundamental monograph on the structure of compact groups (1998, 2006), and is an invaluable tool for researchers in topological groups, Lie theory, harmonic analysis and representation theory. It is written to be accessible to advanced graduate students wishing to study this fascinating and important area of current research, which has so many fruitful interactions with other fields of mathematics. 5 31 2007 978-3-03719-032-6 978-3-03719-532-1 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/032 http://www.ems-ph.org/doi/10.4171/032 2 Demetrios Christodoulou ETH Z&#252;rich, Switzerland Partial differential equations Global analysis, analysis on manifolds Fluid mechanics 35L67; 35L65, 35L70, 58J45, 76L05, 76N15, 76Y05 The equations describing the motion of a perfect fluid were first formulated by Euler in 1752. These equations were among the first partial differential equations to be written down, but, after a lapse of two and a half centuries, we are still far from adequately understanding the observed phenomena which are supposed to lie within their domain of validity. These phenomena include the formation and evolution of shocks in compressible fluids, the subject of the present monograph. The first work on shock formation was done by Riemann in 1858. However, his analysis was limited to the simplified case of one space dimension. Since then, several deep physical insights have been attained and new methods of mathematical analysis invented. Nevertheless, the theory of the formation and evolution of shocks in real three-dimensional fluids has remained up to this day fundamentally incomplete. This monograph considers the relativistic Euler equations in three space dimensions for a perfect fluid with an arbitrary equation of state. We consider initial data for these equations which outside a sphere coincide with the data corresponding to a constant state. Under suitable restriction on the size of the initial departure from the constant state, we establish theorems that give a complete description of the maximal classical development. In particular, it is shown that the boundary of the domain of the maximal classical development has a singular part where the inverse density of the wave fronts vanishes, signalling shock formation. The theorems give a detailed description of the geometry of this singular boundary and a detailed analysis of the behavior of the solution there. A complete picture of shock formation in three-dimensional fluids is thereby obtained. The approach is geometric, the central concept being that of the acoustical spacetime manifold. The monograph will be of interest to people working in partial differential equations in general and in fluid mechanics in particular. 1 10 2007 978-3-03719-031-9 978-3-03719-531-4 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/031 http://www.ems-ph.org/doi/10.4171/031 Reto Müller ETH Zürich Global analysis, analysis on manifolds Partial differential equations Differential geometry 58J35, 53C44; 53C21, 35K55, 58E11, 58E10 The classical Harnack inequalities play an important role in the study of parabolic partial differential equations. The idea of finding a differential version of such a classical Harnack inequality goes back to Peter Li and Shing Tung Yau, who introduced a pointwise gradient estimate for a solution of the linear heat equation on a manifold which leads to a classical Harnack type inequality if being integrated along a path. Their idea has been successfully adopted and generalized to (nonlinear) geometric heat flows such as mean curvature flow or Ricci flow; most of this work was done by Richard Hamilton. In 2002, Grisha Perelman presented a new kind of differential Harnack inequality which involves both the (adjoint) linear heat equation and the Ricci flow. This led to a completely new approach to the Ricci flow that allowed interpretation as a gradient flow which maximizes different entropy functionals. This approach forms the main analytic core of Perelman's attempt to prove the Poincaré conjecture. It is, however, of completely independent interest and may as well prove useful in various other areas, such as, for instance, the theory of Kähler manifolds. The goal of this book is to explain this analytic tool in full detail for the two examples of the linear heat equation and the Ricci flow. It begins with the original Li–Yau result, presents Hamilton's Harnack inequalities for the Ricci flow, and ends with Perelman's entropy formulas and space-time geodesics. The text is a self-contained, modern introduction to the Ricci flow and the analytic methods to study it. It is primarily addressed to students who have a basic introductory knowledge of analysis and of Riemannian geometry and who are attracted to further study in geometric analysis. No previous knowledge of differential Harnack inequalities or the Ricci flow is required. 8 30 2006 978-3-03719-030-2 978-3-03719-530-7 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/030 http://www.ems-ph.org/doi/10.4171/030 Athanase Papadopoulos IRMA, Strasbourg, France Functions of a complex variable Several complex variables and analytic spaces Primary 30-00, 32G15, 30F60; secondary 30C62, 57N16, 53A35, 20F65, 30F20, 30F25, 30F10, 30F30, 30F40, 30F45, 14H15, 20H10, 30F15, 53B35, 57M07, 57M60, 14H60, 14D20, 57M20, 20F38 The Teichmüller space of a surface was introduced by O. Teichmüller in the 1930s. It is a basic tool in the study of Riemann's moduli space and of the mapping class group. These objects are fundamental in several fields of mathematics including algebraic geometry, number theory, topology, geometry, and dynamics. The original setting of Teichmüller theory is complex analysis. The work of Thurston in the 1970s brought techniques of hyperbolic geometry in the study of Teichmüller space and of its asymptotic geometry. Teichmüller spaces are also studied from the point of view of the representation theory of the fundamental group of the surface in a Lie group G, most notably G = PSL(2,ℝ) and G = PSL(2,ℂ). In the 1980s, there evolved an essentially combinatorial treatment of the Teichmüller and moduli spaces involving techniques and ideas from high-energy physics, namely from string theory. The current research interests include the quantization of Teichmüller space, the Weil–Petersson symplectic and Poisson geometry of this space as well as gauge-theoretic extensions of these structures. The quantization theories can lead to new invariants of hyperbolic 3-manifolds. The purpose of this handbook is to give a panorama of some of the most important aspects of Teichmüller theory. The handbook should be useful to specialists in the field, to graduate students, and more generally to mathematicians who want to learn about the subject. All the chapters are self-contained and have a pedagogical character. They are written by leading experts in the subject. 5 25 2007 978-3-03719-029-6 978-3-03719-529-1 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/029 http://www.ems-ph.org/doi/10.4171/029 11

Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Functions of a complex variable General This article is an extensive introduction to the book. The first part is dedicated to an exposition of the structure of Teichmüller spaces. The second part is an overview of the material contained in this volume. 1 30 1 10.4171/029-1/1 http://www.ems-ph.org/doi/10.4171/029-1/1

Georgios Daskalopoulos Brown University, PROVIDENCE, UNITED STATES Richard Wentworth The Johns Hopkins University, BALTIMORE, UNITED STATES Functions of a complex variable Several complex variables and analytic spaces General Teichmüller theory is rich in applications to topology and physics. By way of the mapping class group the subject is closely related to knot theory and three-manifolds. From the uniformization theorem, Teichmüller theory is part of the more general study of Kleinian groups and character varieties. Conformal field theory and quantum cohomology make use of the algebraic and geometric properties of the Riemann moduli space. The goal of this exposition is to present some of the more recent activity using analysis, and in particular harmonic maps, in the context of Teichmüller theory, representations of surface groups, mapping class groups, and Weil–Petersson geometry. Topics have been selected in order to illustrate the theme that the analytic and topological points of view complement each other in a useful way. For example, we shall present four different proofs of the fact that Teichmüller space is a cell, and we shall discuss the recent completion of a harmonic maps approach to Teichmüller’s existence and uniqueness theorems on extremal quasiconformal maps. 33 109 1 10.4171/029-1/2 http://www.ems-ph.org/doi/10.4171/029-1/2

Athanase Papadopoulos Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Guillaume Théret Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Several complex variables and analytic spaces Manifolds and cell complexes General This is a survey on the Teichmüller metric and on Thurston’s asymmetric metric on Teichmüller space. The stress is on the similarities and the differences between the two metrics. The results on Teichmüller's metric that are described in this chapter are classical, but some of the results on Thurston’s asymmetric metric are new. The authors study in particular the global behaviour of the distance functions and on the limiting behaviour of geodesic rays. Stretch lines and anti-stretch lines are studied in some detail. The chapter ends with a commented list of open questions. 111 204 1 10.4171/029-1/3 http://www.ems-ph.org/doi/10.4171/029-1/3

Robert Penner Aarhus University, AARHUS C, DENMARK Several complex variables and analytic spaces General Previous work of the author has developed coordinates on bundles over the classical Teichmüller spaces of punctured surfaces and on the space of cosets of the Möbius group in the group of orientation-preserving homeomorphisms of the circle, and this work is surveyed here. Joint work with Dragomir Šarić is also sketched which extends these results to the setting of the Teichmüller space of the solenoid of a punctured surface, which is defined here in analogy to Dennis Sullivan’s original definition for the case of closed surfaces. Because of the connections with the classical modular group, the punctured solenoid and its Teichmüller theory have connections with number theory. 205 221 1 10.4171/029-1/4 http://www.ems-ph.org/doi/10.4171/029-1/4

Jean-Pierre Otal Université Paul Sabatier, TOULOUSE CEDEX 9, FRANCE Several complex variables and analytic spaces Functions of a complex variable General In the first section, we review the construction due to L. Bers of the complex structure on the Teichmüller space of a Fuchsian group. Then we recall the definition of the space of Hölder distributions on the space of geodesics of ℍ2, introduced by F. Bonahon for cocompact Fuchsian groups. This definition was recently extended to the case of general Fuchsian groups by D. Šarić who showed that the Liouville map embeds the Teichmüller space into the space of geodesic Hölder distributions. We show that this embedding is analytic. 223 248 1 10.4171/029-1/5 http://www.ems-ph.org/doi/10.4171/029-1/5

William Harvey King's College London, LONDON, UNITED KINGDOM Several complex variables and analytic spaces Algebraic geometry Functions of a complex variable General This chapter concerns Teichmüller theory from the point of view of the deformation theory of algebraic curves. The author surveys in particular the deformation theory of Veech’s discs and other Teichmüller discs, of families associated to triangle groups and arithmetically defined curves, with a section on Grothendieck’s dessins d’enfants. 249 292 1 10.4171/029-1/6 http://www.ems-ph.org/doi/10.4171/029-1/6

Frank Herrlich Karlsruhe Institute of Technology, KARLSRUHE, GERMANY Gabriela Schmithüsen Universität Karlsruhe, KARLSRUHE, GERMANY Functions of a complex variable Several complex variables and analytic spaces General We study the boundary of Teichmüller disks in Tg, a partial compactification of Teichmüller space, and their image in Schottky space.   We give a broad introduction to Teichmüller disks and explain the relation between Teichmüller curves and Veech groups.   Furthermore, we describe Braungardt’s construction of Tg and compare it with the Abikoff augmented Teichmüller space. Following Masur, we give a description of Strebel rays that makes it easy to understand their end points on the boundary of Tg. This prepares the description of boundary points that a Teichmüller disk has, with a particular emphasis to the case that it leads to a Teichmüller curve.   Further on we turn to Schottky space and describe two different approaches to obtain a partial compactification. We give an overview how the boundaries of Schottky space, Teichmüller space and moduli space match together and how the actions of the diverse groups on them are linked. Finally we consider the image of Teichmüller disks in Schottky space and show that one can choose the projection from Teichmüller space to Schottky space in such a manner that the image of the Teichmüller disk is a quotient by an infinite group. 293 349 1 10.4171/029-1/7 http://www.ems-ph.org/doi/10.4171/029-1/7

Shigeyuki Morita University of Tokyo, TOKYO, JAPAN Manifolds and cell complexes General In this chapter, first we consider the diffeomorphism group Diff M of a general C∞ manifold M and introduce the concept of the diffeotopy group of M, which is by definition the group of isotopy classes of elements of Diff M. We also consider certain subgroups of Diff M consisting of those diffeomorphisms which preserve a given volume form or a symplectic form on M. Then we specialize to the particular case where M is a closed oriented surface. In this case, the orientation preserving diffeotopy group is usually called the mapping class group. We also consider several groups related to the mapping class groups of surfaces such as the Torelli groups, the (outer) automorphism groups of free groups, the IA automorphism groups and the symplectomorphism groups of surfaces. We give some of the basic results concerning the algebraic structure of these groups. Then we would like to present a few problems for future research. 353 386 1 10.4171/029-1/8 http://www.ems-ph.org/doi/10.4171/029-1/8

John Loftin Rutgers University, NEWARK, UNITED STATES Functions of a complex variable Manifolds and cell complexes General This survey describes how subgroups of mapping class groups of a finite type surface can be studied via the geometric and dynamical properties of their action on the compactified Teichmüller space of the surface. 387 410 1 10.4171/029-1/9 http://www.ems-ph.org/doi/10.4171/029-1/9

Albert Marden University of Minnesota, MINNEAPOLIS, UNITED STATES Functions of a complex variable Manifolds and cell complexes General This report is an exposition of the theory of deformations of hyperbolic structures of 2- and 3-manifolds, and its relationship to Teichmüller theory of Riemann surfaces. 411 446 1 10.4171/029-1/10 http://www.ems-ph.org/doi/10.4171/029-1/10

Ursula Hamenstädt Universität Bonn, BONN, GERMANY Several complex variables and analytic spaces Manifolds and cell complexes General Using train tracks on a nonexceptional oriented surface S of finite type in a systematic way we give a proof that the complex of curves C(S) of S is a hyperbolic geodesic metric space. We also discuss the relation between the geometry of the complex of curves and the geometry of Teichmüller space. 447 467 1 10.4171/029-1/11 http://www.ems-ph.org/doi/10.4171/029-1/11

Charalampos Charitos Agricultural University of Athens, Athens, GREECE Ioannis Papadoperakis Agricultural University of Athens, ATHENS, GREECE Manifolds and cell complexes Several complex variables and analytic spaces General We give a finite number of real parameters for the Teichmüller space of a hyperbolic pair of pants with a conical singularity. We also give a finite number of real parameters for a space of hyperbolic structures on a class of 2-dimensional simplicial complexes. Such a space of hyperbolic structures associated to a simplicial complex can be regarded as a Teichmüller space for that complex. 471 506 1 10.4171/029-1/12 http://www.ems-ph.org/doi/10.4171/029-1/12

Marc Troyanov Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND Manifolds and cell complexes General The goal of this article is to develop some aspects of the deformation theory of piecewise flat structures on surfaces and use this theory to construct new geometric structures on the moduli space of Riemann surfaces. 507 540 1 10.4171/029-1/13 http://www.ems-ph.org/doi/10.4171/029-1/13

Christian Mercat Université de Montpellier II, MONTPELLIER CEDEX 5, FRANCE Functions of a complex variable General We survey the theory of discrete Riemann surfaces. The theory takes place on a cellular decomposition of a surface, together with its Poincaré dual, equipped with a discrete conformal structure. Many theorems of the continuous theory follow through to the discrete case; we will define the discrete analogs of period matrices, Riemann’s bilinear relations, exponential of constant argument and series. We present the notion of criticality and its relationship with integrability. 541 575 1 10.4171/029-1/14 http://www.ems-ph.org/doi/10.4171/029-1/14

Leonid Chekhov Steklov Mathematical Institute, MOSCOW, RUSSIAN FEDERATION Robert Penner Aarhus University, AARHUS C, DENMARK Several complex variables and analytic spaces General In earlier work, Chekhov and Fock have given a quantization of Teichmüller space as a Poisson manifold, and the current paper first surveys this material adding further mathematical and other detail, including the underlying geometric work by Penner on classical Teichmüller theory. In particular, the earlier quantum ordering solution is found to essentially agree with an “improved” operator ordering given by serially traversing general edge-paths on a graph in the underlying surface. Now, insofar as Thurston’s sphere of projectivized foliations of compact support provides a useful compactification for Teichmüller space in the classical case, it is natural to consider corresponding limits of appropriate operators to provide a framework for studying degenerations of quantum hyperbolic structures. After surveying the required background material on Thurston theory and “train tracks”, the current paper continues to give a quantization of Thurston’s boundary in the special case of the once-punctured torus, where there are already substantial analytical and combinatorial challenges. Indeed, an operatorial version of continued fractions as well as the improved quantum ordering are required to prove existence of these limits. Since Thurston’s boundary for the once-punctured torus is a topological circle, the main new result may be regarded as a quantization of this circle. There is a discussion of quantizing Thurston’s boundary spheres for higher genus surfaces in closing remarks. 579 645 1 10.4171/029-1/15 http://www.ems-ph.org/doi/10.4171/029-1/15

Vladimir Fock Institute for Theoretical and Experimental Physics, STRASBOURG CEDEX, FRANCE Alexander Goncharov Yale University, NEW HAVEN, UNITED STATES Several complex variables and analytic spaces Functions of a complex variable Manifolds and cell complexes General We survey explicit coordinate descriptions for two versions of Teichmüller and lamination spaces for open surfaces, and extend them to the more general set-up of surfaces with distinguished collections of points on the boundary. Main features, such as mapping class group action, Poisson and symplectic structures and others, are described in these terms. The lamination spaces are interpreted as the tropical limits of the Teichmüller ones. Canonical pairings between lamination and Teichmüller spaces are constructed. 647 684 1 10.4171/029-1/16 http://www.ems-ph.org/doi/10.4171/029-1/16

Jörg Teschner Freie Universität Berlin, BERLIN, GERMANY Several complex variables and analytic spaces Differential geometry General We review the quantization of Teichmüller spaces as initiated by Kashaev and Checkov–Fock. It is shown that the quantized Teichmüller spaces have factorization properties similar to those required in the definition of a modular functor. 685 760 1 10.4171/029-1/17 http://www.ems-ph.org/doi/10.4171/029-1/17

Rinat Kashaev Université de Genève, GENÈVE, SWITZERLAND General Penner’s decorated Teichmüller space of a punctured surface can be generalized to a decorated moduli space of flat irreducible PSL2(ℝ)-connections with parabolicity conditions around punctures. This space, similarly to the decorated Teichmüller space, admits a parametrization which is well suited for quantization. The quantum ‘ax + b’ group of Woronowicz and Zakrzewski leads to a viable quantum theory. 761 782 1 10.4171/029-1/18 http://www.ems-ph.org/doi/10.4171/029-1/18

Louise Nyssen Université Montpellier, France Number theory Convex and discrete geometry Computer science Quantum theory 11F; 11L, 11M, 81T, 52C, 68R15 There is a rich and historical relationship between theoretical physics and number theory. This volume presents a selection of problems which are currently in full development and inspire a lot of research going on. Each of the seven contributions starts with an introductory survey which makes it possible even for non-specialists to understand the results and to gain an idea of the great variety of subjects and techniques used. Topics covered are: phase locking in oscillating systems, crystallography, Hopf algebras and renormalisation theory, Zeta-function and random matrices, Kloosterman sums and the local Langlands correspondence. Intended for research mathematicians and theoretical physicists as well as graduate students, this volume gives an overview of recent developments in an exciting subject crossing several disciplines. 10 12 2006 978-3-03719-028-9 978-3-03719-528-4 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/028 http://www.ems-ph.org/doi/10.4171/028 10

Michel Planat Institut FEMTO-ST, BESANÇON, FRANCE Number theory Nonassociative rings and algebras Statistics Quantum theory An overview of the paradigm of phase-locking at the nonlinear, geometric, quantum and finite field level is attempted. It is applied to finite resolution measurements, the 1/f noise in a communication receiver, and to quantum information concepts. The Mangoldt function of prime number theory is found to be the generator of low frequency noise in the coupling coefficient between oscillators, in the scattering coefficient, and in quantum statistical states. On the other hand, phase operators defined on finite fields (and rings) control the quantum information primitives. Huyghens coupled pendulums, the Adler equation, the Arnold map, continued fraction expansions, discrete Möbius transformations, Ford circles, coherent and squeezed states, Ramanujan and Weil sums, the Riemann zeta function, Bost and Connes KMS states, Galois fields and rings and their additive characters are used successfully to unveil the phase dynamics. 1 38 1 10.4171/028-1/1 http://www.ems-ph.org/doi/10.4171/028-1/1

Jean-Louis Verger-Gaugry Université Grenoble I, SAINT MARTIN D'HERES CEDEX, FRANCE Number theory Nonassociative rings and algebras Statistics Quantum theory The first part of this paper is a survey on links between geometry of numbers and aperiodic crystals in physics, viewed from the mathematical side. In a second part, we prove the existence of a canonical cut-and-project scheme above a (SFU-set) self-similar finitely generated packing of (equal) spheres Λ in ℝn and investigate its consequences, in particular the role played by the Euclidean and inhomogeneous minima of the algebraic number field generated by the self-similarity on the Delone constant of the sphere packing. We discuss the isolation phenomenon. The degree d of this field divides the ℤ-rank of ℤ[Λ − Λ]. We give a lower bound of the Delone constant of a k-thin SFU-set (sphere packing) which arises from a model set or a Meyer set when d is large enough. 39 78 1 10.4171/028-1/2 http://www.ems-ph.org/doi/10.4171/028-1/2

Jean-Pierre Gazeau Université Denis Diderot - Paris 7, PARIS CEDEX 05, FRANCE Zuzana Masáková Czech Technical University, PRAHA 2, CZECH REPUBLIC Edita Pelantová Czech Technical University, PRAHA 2, CZECH REPUBLIC Number theory Nonassociative rings and algebras Statistics Quantum theory One-dimensional cut-and-project point sets obtained from the square lattice in the plane are considered from a unifying point of view and in the perspective of aperiodic wavelet constructions. We successively examine their geometrical aspects, combinatorial properties from the point of view of the theory of languages, and self-similarity with algebraic scaling factor θ. We explain the relation of the cut-and-project sets to non-standard numeration systems based on θ. We finally examine the substitutivity, a weakened version of substitution invariance, which provides us with an algorithm for symbolic generation of cut-and-project sequences. 79 131 1 10.4171/028-1/3 http://www.ems-ph.org/doi/10.4171/028-1/3

Christoph Bergbauer Freie Universität Berlin, BERLIN, GERMANY Dirk Kreimer Humboldt-Universität, BERLIN, GERMANY Number theory Nonassociative rings and algebras Statistics Quantum theory In this review we discuss the relevance of the Hochschild cohomology of renormalization Hopf algebras for local quantum field theories and their equations of motion. 133 164 1 10.4171/028-1/4 http://www.ems-ph.org/doi/10.4171/028-1/4

Emmanuel Royer Université de Montpellier II, MONTPELLIER CEDEX 5, FRANCE Number theory Nonassociative rings and algebras Statistics Quantum theory Since Euler and Riemann, various links have been established between the behaviour of prime numbers and the analytical properties of the Riemann ζ function. The zeroes of ζ have a great importance that justifies their fine study. Since seventies, a rich tool has appeared for the study of zeroes: the statistical spectral properties of the unitary matrices that are a model for ζ. The aim of this survey is to explain the link between unitary matrices and ζ, and its extension to the L-functions. 165 224 1 10.4171/028-1/5 http://www.ems-ph.org/doi/10.4171/028-1/5

Philippe Michel Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND Number theory Nonassociative rings and algebras Statistics Quantum theory 225 251 1 10.4171/028-1/6 http://www.ems-ph.org/doi/10.4171/028-1/6

Ariane Mézard Université de Versailles Saint-Quentin, VERSAILLES CEDEX, FRANCE Number theory Nonassociative rings and algebras Statistics Quantum theory This paper is an introduction to the local Langlands correspondence in dimension 2. We define some representations of GL2(ℚp) and some Galois representations with specified properties. We present strategies to connect them. We conclude with a discussion on the expected correspondence for p-adic representations. 253 265 1 10.4171/028-1/7 http://www.ems-ph.org/doi/10.4171/028-1/7

Theory and Statistical Applications Eustasio del Barrio Universidad de Valladolid, Spain Paul Deheuvels Université de Paris VI, France Sara van de Geer ETH Zürich, Switzerland Probability theory and stochastic processes Statistics 60F05, 60K35, 62C12, 62F05, 62F12, 62G30, 62Jxx The theory of empirical processes constitutes the mathematical toolbox of asymptotic statistics. Its growth was accelerated by the 1950s work on the Functional Central Limit Theorem and the Invariance Principle. The theory has developed in parallel with statistical methodologies, and has been successfully applied to a large diversity of problems related to the asymptotic behaviour of statistical procedures. The three sets of lecture notes in the book offer a wide panorama of contemporary empirical processes theory. Techniques are developed in the framework of probability in Banach spaces, Hungarian-style strong approximations, using tools from general stochastic process theory. Other tools appear in this text in connection with historical as well as modern applications, such as goodness-of-fit tests, density estimation or general M-estimators. This book gives an excellent overview of the broad scope of the theory of empirical processes. It will be an invaluable aid for students and researchers interested in an advanced and well-documented approach to the selected topics. 1 10 2007 978-3-03719-027-2 978-3-03719-527-7 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/027 http://www.ems-ph.org/doi/10.4171/027 Volume I: Linear Information Erich Novak University of Jena, Germany Henryk Woźniakowski Columbia University, New York, USA, and University of Warsaw, Poland Numerical analysis Computer science 65-02; 65Y20, 68Q17, 68Q25, 41A63, 46E22, 28C20, 46E30 Curse of dimensionality, tractability, high-dimensional numerical problems, worst case setting, average case setting, randomized algorithms, weighted norms, product weights, finite-order weights Multivariate problems occur in many applications. These problems are defined on spaces of d-variate functions and d can be huge – in the hundreds or even in the thousands. Some high-dimensional problems can be solved efficiently to within ε, i.e., the cost increases polynomially in ε−1 and d. However, there are many multivariate problems for which even the minimal cost increases exponentially in d. This exponential dependence on d is called intractability or the curse of dimensionality. This is the first of a three-volume set comprising a comprehensive study of the tractability of multivariate problems. It is devoted to algorithms using linear information consisting of arbitrary linear functionals. The theory for multivariate problems is developed in various settings: worst case, average case, randomized and probabilistic. A problem is tractable if its minimal cost is not exponential in ε−1 and d. There are various notions of tractability, depending on how we measure the lack of exponential dependence. For example, a problem is polynomially tractable if its minimal cost is polynomial in ε−1 and d. The study of tractability was initiated about 15 years ago. This is the first research monograph on this subject. Many multivariate problems suffer from the curse of dimensionality when they are defined over classical (unweighted) spaces. But many practically important problems are solved today for huge d in a reasonable time. One of the most intriguing challenges of theory is to understand why this is possible. Multivariate problems may become tractable if they are defined over weighted spaces with properly decaying weights. In this case, all variables and groups of variables are moderated by weights. The main purpose of this book is to study weighted spaces and to obtain conditions on the weights that are necessary and sufficient to achieve various notions of tractability. The book is of interest for researchers working in computational mathematics, especially in approximation of high-dimensional problems. It may be also suitable for graduate courses and seminars. The text concludes with a list of thirty open problems that can be good candidates for future tractability research. 9 2 2008 978-3-03719-026-5 978-3-03719-526-0 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/026 http://www.ems-ph.org/doi/10.4171/026 6 Werner Ballmann University of Bonn, Germany Differential geometry Several complex variables and analytic spaces Global analysis, analysis on manifolds 53C55, 32Q15, 32Qxx, 58Jxx, 53C35 These notes are based on lectures the author held at the University of Bonn and the Erwin-Schrödinger-Institute in Vienna. The aim is to give a thorough introduction to the theory of Kähler manifolds with special emphasis on the differential geometric side of Kähler geometry. The exposition starts with a short discussion of complex manifolds and holomorphic vector bundles and a detailed account of the basic differential geometric properties of Kähler manifolds. The more advanced topics are the cohomology of Kähler manifolds, Calabi conjecture, Gromov's Kähler hyperbolic spaces, and the Kodaira embedding theorem. Some familiarity with global analysis and partial differential equations is assumed, in particular in the part on the Calabi conjecture. There are appendices on Chern–Weil theory, symmetric spaces, and L2-cohomology. 7 10 2006 978-3-03719-025-8 978-3-03719-525-3 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/025 http://www.ems-ph.org/doi/10.4171/025 Joseph Várilly Universidad de Costa Rica Global analysis, analysis on manifolds Functional analysis Quantum theory 58B34; 46L87, 81T75 Noncommutative geometry, inspired by quantum physics, describes singular spaces by their noncommutative coordinate algebras, and metric structures by Dirac-like operators. Such metric geometries are described mathematically by Connes' theory of spectral triples. These lectures, delivered at an EMS Summer School on noncommutative geometry and its applications, provide an overview of spectral triples based on examples. This introduction is aimed at graduate students of both mathematics and theoretical physics. It deals with Dirac operators on spin manifolds, noncommutative tori, Moyal quantization and tangent groupoids, action functionals, and isospectral deformations. The structural framework is the concept of a noncommutative spin geometry; the condiditons on spectral triples which determine this concept are developed in detail. The emphasis throughout is on gaining understanding by computing the details of specific examples. The book provides a middle ground between a comprehensive text and a narrowly focused research monograph. It is intended for self-study, enabling the reader to gain access to the essentials of noncommutative geometry. New features since the original course are an expanded bibliography and a survey of more recent examples and applications of spectral triples. 6 30 2006 978-3-03719-024-1 978-3-03719-524-6 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/024 http://www.ems-ph.org/doi/10.4171/024 Sergey Matveev Chelyabinsk State University, Russia Algebraic topology 55-01 Algebraic topology is the study of the global properties of spaces by means of algebra. It is an important branch of modern mathematics with a wide degree of applicability to other fields, including geometric topology, differential geometry, functional analysis, differential equations, algebraic geometry, number theory, and theoretical physics. This book provides an introduction to the basic concepts and methods of algebraic topology for the beginner. It presents elements of both homology theory and homotopy theory, and includes various applications. The author's intention is to rely on the geometric approach by appealing to the reader's own intuition to help understanding. The numerous illustrations in the text also serve this purpose. Two features make the text different from the standard literature: first, special attention is given to providing explicit algorithms for calculating the homology groups and for manipulating the fundamental groups. Second, the book contains many exercises, all of which are supplied with hints or solutions. This makes the book suitable for both classroom use and for independent study. 5 10 2006 978-3-03719-023-4 978-3-03719-523-9 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/023 http://www.ems-ph.org/doi/10.4171/023 Marta Sanz-Solé Universitat de Barcelona, Spain Javier Soria Universitat de Barcelona, Spain Juan Luis Varona Universidad de La Rioja, Logroño, Spain Joan Verdera Universitat Autónoma de Barcelona, Spain General 00Bxx Note: The ICM2006 Proceedings are available only as a set, individual volumes cannot be ordered. The International Congress of Mathematicians (ICM) is held every four years. It is a major scientific event, bringing together mathematicians from all over the world, and demonstrating the vital role that mathematics play in our society. In particular, the Fields Medals are awarded to recognize outstanding mathematical achievement. At the same time, the International Mathematical Union awards the Nevanlinna Prize for work in the field of theoretical computer science. The proceedings of ICM 2006, published as a three-volume set, present an overview of current research in all areas of mathematics and provide a permanent record the congress. The first volume features the works of Fields Medallists and the Nevanlinna Prize winner, the plenary lectures, and the speeches and pictures of the opening and closing ceremonies and award sessions. The other two volumes present the invited lectures, arranged according to their mathematical subject. 5 15 2007 978-3-03719-022-7 978-3-03719-522-2 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/022 http://www.ems-ph.org/doi/10.4171/022

Giovanni Felder ETH Zürich, ZÜRICH, SWITZERLAND Group theory and generalizations History and biography General Andrei Okounkov’s initial area of research is group representation theory, with particular emphasis on combinatorial and asymptotic aspects. He used this subject as a starting point to obtain spectacular results in many different areas of mathematics and mathematical physics, from complex and real algebraic geometry to statistical mechanics, dynamical systems, probability theory and topological string theory. The research of Okounkov has its roots in very basic notions such as partitions, which form a recurrent theme in his work. A partition λ of a natural number n is a non-increasing sequence of integers λ1 ≥λ2 ≥···≥0 adding up to n. Partitions are a basic combinatorial notion at the heart of the representation theory. Okounkov started his career in this field in Moscow where he worked with G. Olshanski, through whom he came in contact with A. Vershik and his school in St. Petersburg, in particular S. Kerov. The research programme of these mathematicians, to which Okounkov made substantial contributions, has at its core the idea that partitions and other notions of representation theory should be considered as random objects with respect to natural probability measures. This idea was further developed by Okounkov, who showed that, together with insights from geometry and ideas of high energy physics, it can be applied to the most diverse areas of mathematics. This is an account of some of the highlights mostly of his recent research. I am grateful to Enrico Arbarello for explanations and for providing me with very useful notes on Okounkov’s work in algebraic geometry and its context. 55 64 1 10.4171/022-1/1 http://www.ems-ph.org/doi/10.4171/022-1/1

John Lott University of California, BERKELEY, UNITED STATES Differential geometry History and biography General Grigory Perelman has been awarded the Fields Medal for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow. Perelman was born in 1966 and received his doctorate from St. Petersburg State University. He quickly became renowned for his work in Riemannian geometry and Alexandrov geometry, the latter being a form of Riemannian geometry for metric spaces. Some of Perelman’s results in Alexandrov geometry are summarized in his 1994 ICM talk [20]. We state one of his results in Riemannian geometry. In a short and striking article, Perelman proved the so-called Soul Conjecture. 66 76 1 10.4171/022-1/2 http://www.ems-ph.org/doi/10.4171/022-1/2

Charles Fefferman Princeton University, PRINCETON, UNITED STATES Geometry History and biography General Mathematics at the highest level has several flavors. On seeing it, one might say: (A) What amazing technical power! (B) What a grand synthesis! (C) How could anyone not have seen this before? (D) Where on earth did this come from? The work of Terence Tao encompasses all of the above. One cannot hope to capture its extraordinary range in a few pages. My goal here is simply to exhibit a few contributions by Tao and his collaborators, sufficient to produce all the reactions (A)... (D). I shall discuss the Kakeya problem, nonlinear Schrödinger equations and arithmetic progressions of primes. 78 86 1 10.4171/022-1/3 http://www.ems-ph.org/doi/10.4171/022-1/3

Charles Newman New York University, NEW YORK, UNITED STATES Probability theory and stochastic processes History and biography General It is my great pleasure to briefly report on some of Wendelin Werner’s research accomplishments that have led to his being awarded a Fields Medal at this International Congress of Mathematicians of 2006. There are a number of aspects of Werner’s work that add to my pleasure in this event. One is that he was trained as a probabilist, receiving his Ph.D. in 1993 under the supervision of Jean-François Le Gall in Paris with a dissertation concerning planar Brownian Motion – which, as we shall see, plays a major role in his later work as well. Until now, Probability Theory had not been represented among Fields Medals and so I am enormously pleased to be here to witness a change in that history. I myself was originally trained, not in Probability Theory, but in Mathematical Physics. Werner’s work, together with his collaborators such as Greg Lawler, Oded Schramm and Stas Smirnov, involves applications of Probablity and Conformal Mapping Theory to fundamental issues in Statistical Physics, as we shall discuss. A second source of pleasure is my belief that this, together with other work of recent years, represents a watershed in the interaction between Mathematics and Physics generally. Namely, mathematicians such as Werner are not only providing rigorous proofs of already existing claims in the Physics literature, but beyond that are providing quite new conceptual understanding of basic phenomena – in this case, a direct geometric picture of the intrinsically random structure of physical systems at their critical points (at least in two dimensions). One simple but important example is percolation – see Figure 1. Permit me a somewhat more personal remark as director of the Courant Institute for the past four years. We have a scientific viewpoint, as did our predecessor institute in Göttingen – namely, that an important goal should be the elimination of artificial distinctions between the Mathematical Sciences and their applications in other Sciences – I believe Wendelin Werner’s work brilliantly lives up to that philosophy. Yet a third source of pleasure concerns the collaborative nature of much of Werner’s work. Beautiful and productive mathematics can be the result of many different personal workstyles. But the highly interactive style, of which Werner, together with Lawler, Schramm and his other collaborators, is a leading exemplar, appeals to many of us as simultaneously good for the soul while leading to work stronger than the sum of its parts. It is a promising sign to see Fields Medals awarded for this style of work. 88 95 1 10.4171/022-1/4 http://www.ems-ph.org/doi/10.4171/022-1/4

John Hopcroft Cornell University, ITHACA, UNITED STATES Computer science History and biography General Jon Kleinberg’s research has helped lay the theoretical foundations for the information age. He has developed the theory that underlies search engines, collaborative filtering, organizing and extracting information from sources such as the World Wide Web, news streams, and the large data collections that are becoming available in astronomy, bioinformatics and many other areas. The following is a brief overview of five of his major contributions. 97 108 1 10.4171/022-1/5 http://www.ems-ph.org/doi/10.4171/022-1/5

Hans Föllmer Humboldt-Universität zu Berlin, BERLIN, GERMANY Probability theory and stochastic processes History and biography General 109 123 1 10.4171/022-1/6 http://www.ems-ph.org/doi/10.4171/022-1/6

Percy Deift New York University, NEW YORK, UNITED STATES Probability theory and stochastic processes Operator theory General All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all governed by the same laws and formulae of thermodynamics. In this paper we describe some recent history of universality ideas in physics starting with Wigner’s model for the scattering of neutrons off large nuclei and show how these ideas have led mathematicians to investigate universal behavior for a variety of mathematical systems. This is true not only for systems which have a physical origin, but also for systems which arise in a purely mathematical context such as the Riemann hypothesis, and a version of the card game solitaire called patience sorting. 125 152 1 10.4171/022-1/7 http://www.ems-ph.org/doi/10.4171/022-1/7

Jean-Pierre Demailly Université Grenoble I, SAINT MARTIN D'HERES CEDEX, FRANCE Algebraic geometry Several complex variables and analytic spaces General Our goal is to survey some of the main advances which took place recently in the study of the geometry of projective or compact Kähler manifolds: very efficient new transcendental techniques, a better understanding of the geometric structure of cones of positive cohomology classes and of the deformation theory of Kähler manifolds, new results around the invariance of plurigenera and in the minimal model program. 153 186 1 10.4171/022-1/8 http://www.ems-ph.org/doi/10.4171/022-1/8

Ronald DeVore Texas A&M University, COLLEGE STATION, UNITED STATES Approximations and expansions Functional analysis Numerical analysis Computer science A large portion of computation is concerned with approximating a function u. Typically, there are many ways to proceed with such an approximation leading to a variety of algorithms. We address the question of how we should evaluate such algorithms and compare them. In particular, when can we say that a particular algorithm is optimal or near optimal? We shall base our analysis on the approximation error that is achieved with a given (computational or information) budget n. We shall see that the formulation of optimal algorithms depends to a large extent on the context of the problem. For example, numerically approximating the solution to a PDE is different from approximating a signal or image (for the purposes of compression). 187 215 1 10.4171/022-1/9 http://www.ems-ph.org/doi/10.4171/022-1/9

Yasha Eliashberg Stanford University, STANFORD, UNITED STATES Differential geometry General Symplectic field theory (SFT) attempts to approach the theory of holomorphic curves in symplectic manifolds (also called Gromov-Witten theory) in the spirit of a topological field theory. This naturally leads to new algebraic structures which seems to have interesting applications and connections not only in symplectic geometry but also in other areas of mathematics, e.g. topology and integrable PDE. In this talk we sketch out the formal algebraic structure of SFT and discuss some current work towards its applications. 217 246 1 10.4171/022-1/10 http://www.ems-ph.org/doi/10.4171/022-1/10

Étienne Ghys École Normale Supérieure de Lyon, LYON CEDEX 07, FRANCE Dynamical systems and ergodic theory Manifolds and cell complexes General The trajectories of a vector field in 3-space can be very entangled; the flow can swirl, spiral, create vortices etc. Periodic orbits define knots whose topology can sometimes be very complicated. In this talk, I will survey some advances in the qualitative and quantitative description of this kind of phenomenon. The first part will be devoted to vorticity, helicity, and asymptotic cycles for flows. The second part will deal with various notions of rotation and spin for surface diffeomorphisms. Finally, I will describe the important example of the geodesic flow on the modular surface, where the linking between geodesics turns out to be related to well-known arithmetical functions. 247 277 1 10.4171/022-1/11 http://www.ems-ph.org/doi/10.4171/022-1/11

Henryk Iwaniec Rutgers University, PISCATAWAY, UNITED STATES Number theory General The classical memoir by Riemann on the zeta function was motivated by questions about the distribution of prime numbers. But there are important problems concerning prime numbers which cannot be addressed along these lines, for example the representation of primes by polynomials. In this talk I will show a panorama of techniques, which modern analytic number theorists use in the study of prime numbers. Among these are sieve methods. I will explain how the primes are captured by adopting new axioms for sieve theory. I shall also discuss recent progress in traditional questions about primes, such as small gaps, and fundamental ones such as equidistribution in arithmetic progressions. However, my primary objective is to indicate the current directions in Prime Number Theory. 279 306 1 10.4171/022-1/12 http://www.ems-ph.org/doi/10.4171/022-1/12

Iain Johnstone Stanford University, STANFORD, UNITED STATES Statistics Linear and multilinear algebra; matrix theory General Multivariate statistical analysis is concerned with observations on several variables which are thought to possess some degree of inter-dependence. Driven by problems in genetics and the social sciences, it first flowered in the earlier half of the last century. Subsequently, random matrix theory (RMT) developed, initially within physics, and more recently widely in mathematics. While some of the central objects of study in RMT are identical to those of multivariate statistics, statistical theory was slow to exploit the connection. However, with vast data collection ever more common, data sets now often have as many or more variables than the number of individuals observed. In such contexts, the techniques and results of RMT have much to offer multivariate statistics. The paper reviews some of the progress to date. 307 333 1 10.4171/022-1/13 http://www.ems-ph.org/doi/10.4171/022-1/13

Kazuya Kato Kyoto University, KYOTO, JAPAN Number theory Algebraic geometry General This is an introduction to Iwasawa theory and its generalizations. We discuss some main conjectures and related subjects. 335 357 1 10.4171/022-1/14 http://www.ems-ph.org/doi/10.4171/022-1/14

Robert Kohn New York University, NEW YORK, UNITED STATES Calculus of variations and optimal control; optimization Mechanics of deformable solids Statistical mechanics, structure of matter General Many physical systems can be modelled by nonconvex variational problems regularized by higher-order terms. Examples include martensitic phase transformation, micromagnetics, and the Ginzburg–Landau model of nucleation. We are interested in the singular limit, when the coefficient of the higher-order term tends to zero. Our attention is on the internal structure of walls, and the character of microstructure when it forms. We also study the pathways of thermally-activated transitions, modeled via the minimization of action rather than energy. Our viewpoint is variational, focusing on matching upper and lower bounds. 359 383 1 10.4171/022-1/15 http://www.ems-ph.org/doi/10.4171/022-1/15

Jørgen Ellegaard Andersen University of Aarhus, AARHUS C, DENMARK $K$-theory Several complex variables and analytic spaces Algebraic topology Manifolds and cell complexes This text aims to explain what topology, at present, has to say about a few of the many moduli spaces that are currently under study in mathematics. The most prominent one is the moduli space Mg of all Riemann surfaces of genus g. Other examples include the Gromov–Witten moduli space of pseudo-holomorphic curves in a symplectic background, the moduli space of graphs and Waldhausen’s algebraic K-theory of spaces. 385 411 1 10.4171/022-1/16 http://www.ems-ph.org/doi/10.4171/022-1/16

Arkadi Nemirovski Georgia Institute of Technology, ATLANTA, UNITED STATES Operations research, mathematical programming Calculus of variations and optimal control; optimization General During the last two decades, major developments in convex optimization were focusing on conic programming, primarily, on linear, conic quadratic and semidefinite optimization. Conic programming allows to reveal rich structure which usually is possessed by a convex program and to exploit this structure in order to process the program efficiently. In the paper, we overview the major components of the resulting theory (conic duality and primal-dual interior point polynomial time algorithms), outline the extremely rich “expressive abilities” of conic quadratic and semidefinite programming and discuss a number of instructive applications. 413 444 1 10.4171/022-1/17 http://www.ems-ph.org/doi/10.4171/022-1/17

Sorin Popa University of California Los Angeles, LOS ANGELES, UNITED STATES Functional analysis Topological groups, Lie groups Measure and integration Dynamical systems and ergodic theory We present some recent rigidity results for von Neumann algebras (II1 factors) and equivalence relations arising from measure preserving actions of groups on probability spaces which satisfy a combination of deformation and rigidity properties. This includes strong rigidity results for factors with calculation of their fundamental group and cocycle superrigidity for actions with applications to orbit equivalence ergodic theory. 445 477 1 10.4171/022-1/18 http://www.ems-ph.org/doi/10.4171/022-1/18

Alfio Quarteroni EPFL, SB, IACS-CMCS, Bat. MA, LAUSANNE, SWITZERLAND Biology and other natural sciences Numerical analysis Fluid mechanics General In this paper we introduce some basic differential models for the description of blood flow in the circulatory system. We comment on their mathematical properties, their meaningfulness and their limitation to yield realistic and accurate numerical simulations, and their contribution for a better understanding of cardiovascular physio-pathology. 479 512 1 10.4171/022-1/19 http://www.ems-ph.org/doi/10.4171/022-1/19

Oded Schramm , REDMOND, UNITED STATES Probability theory and stochastic processes Functions of a complex variable Statistical mechanics, structure of matter General Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions become accessible to mathematical proof. One of the new developments is the discovery of a one-parameter family of random curves called stochastic Loewner evolution or SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical physics models as the mesh of the grid on which the model is defined tends to zero. The main purpose of this article is to list a collection of open problems. Some of the open problems indicate aspects of the physics knowledge that have not yet been understood mathematically. Other problems are questions about the nature of the SLE curves themselves. Before we present the open problems, the definition of SLE will be motivated and explained, and a brief sketch of recent results will be presented. 513 543 1 10.4171/022-1/20 http://www.ems-ph.org/doi/10.4171/022-1/20

Richard Stanley Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES Combinatorics Probability theory and stochastic processes General We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence of a permutation w of 1,2,...,n was obtained by Vershik–Kerov and (almost) by Logan–Shepp. The entire limiting distribution of is(w) was then determined by Baik, Deift, and Johansson. These techniques can be applied to other classes of permutations, such as involutions, and are related to the distribution of eigenvalues of elements of the classical groups. A number of generalizations and variations of increasing/decreasing subsequences are discussed, including the theory of pattern avoidance, unimodal and alternating subsequences, and crossings and nestings of matchings and set partitions. 545 579 1 10.4171/022-1/21 http://www.ems-ph.org/doi/10.4171/022-1/21

Terence Tao University of California Los Angeles, LOS ANGELES, UNITED STATES Number theory Combinatorics Dynamical systems and ergodic theory Fourier analysis A famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to a decomposition of any object into a structured (low-complexity) component and a random (discorrelated) component. Important examples of these types of decompositions include the Furstenberg structure theorem and the Szemerédi regularity lemma. One recent application of this dichotomy is the result of Green and Tao establishing that the prime numbers contain arbitrarily long arithmetic progressions (despite having density zero in the integers). The power of this dichotomy is evidenced by the fact that the Green–Tao theorem requires surprisingly little technology from analytic number theory, relying instead almost exclusively on manifestations of this dichotomy such as Szemerédi’s theorem. In this paper we survey various manifestations of this dichotomy in combinatorics, harmonic analysis, ergodic theory, and number theory. As we hope to emphasize here, the underlying themes in these arguments are remarkably similar even though the contexts are radically different. 581 608 1 10.4171/022-1/22 http://www.ems-ph.org/doi/10.4171/022-1/22

Juan Luis Vázquez Universidad Autónoma de Madrid, MADRID, SPAIN Partial differential equations General We review some topics in the mathematical theory of nonlinear diffusion. Attention is focused on the porous medium equation and the fast diffusion equation, including logarithmic diffusion. Special features are the existence of free boundaries, the limited regularity of the solutions and the peculiar asymptotic laws for porous medium flows, while for fast diffusions we find the phenomena of finite-time extinction, delayed regularization, nonuniqueness and instantaneous extinction. Logarithmic diffusion with its strong geometrical flavor is also discussed. Connections with functional analysis, semigroup theory, physics of continuous media, probability and differential geometry are underlined. 609 634 1 10.4171/022-1/23 http://www.ems-ph.org/doi/10.4171/022-1/23

Michèle Vergne École Polytechnique, PALAISEAU CEDEX, FRANCE $K$-theory Convex and discrete geometry General We will discuss the equivariant cohomology of a manifold endowed with the action of a Lie group. Localization formulae for equivariant integrals are explained by a vanishing theorem for equivariant cohomology with generalized coefficients. We then give applications to integration of characteristic classes on symplectic quotients and to indices of transversally elliptic operators. In particular, we state a conjecture for the index of a transversally elliptic operator linked to a Hamiltonian action. In the last part, we describe algorithms for numerical computations of values of multivariate spline functions and of vector-partition functions of classical root systems. 635 664 1 10.4171/022-1/24 http://www.ems-ph.org/doi/10.4171/022-1/24

Avi Wigderson Institute for Advanced Study, PRINCETON, UNITED STATES Computer science General The P versus NP question distinguished itself as the central question of theoretical computer science nearly four decades ago. The quest to resolve it, and more generally, to understand the power and limits of efficient computation, has led to the development of computational complexity theory. While this mathematical discipline in general, and the P vs. NP problem in particular, have gained prominence within the mathematics community in the past decade, it is still largely viewed as a problem of computer science. In this paper I shall try to explain why this problem, and others in computational complexity, are not only mathematical problems but also problems about mathematics, faced by the working mathematician. I shall describe the underlying concepts and problems, the attempts to understand and solve them, and some of the research directions this led us to. I shall explain some of the important results, as well as the major goals and conjectures which still elude us. All this will hopefully give a taste of the motivations, richness and interconnectedness of our field. I shall conclude with a few non computational problems, which capture P vs. NP and related computational complexity problems, hopefully inviting more mathematicians to attack them as well. I believe it important to give many examples, and to underlie the intuition (and sometimes, philosophy) behind definitions and results. This may slow the pace of this article for some, in the hope to make it clearer to others. 665 712 1 10.4171/022-1/25 http://www.ems-ph.org/doi/10.4171/022-1/25

John Morgan Columbia University, NEW YORK, UNITED STATES Manifolds and cell complexes General 713 736 1 10.4171/022-1/26 http://www.ems-ph.org/doi/10.4171/022-1/26

General 737 756 1 10.4171/022-1/27 http://www.ems-ph.org/doi/10.4171/022-1/27

General 757 776 1 10.4171/022-1/28 http://www.ems-ph.org/doi/10.4171/022-1/28

José Sánchez-Ron Universidad Autónoma de Madrid, MADRID, SPAIN General We review the history of the International Congresses of Mathematicians, from Zurich 1897 to Madrid 2006, mentioning some of the most significant personages and events (scientific as well as political) of such meetings (as a matter of fact, that history did not begin 1897 in Zurich, but in 1893 at the Chicago World’s Columbian Exposition). We report on Felix Klein, HenriPoincaré,DavidHilbert,VitoVolterra,EmmyNoether,LudwigPrandtl, LaurentSchwartz, Klein’s Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, and the role played by the Mathematical Tripos in the education at Cambridge University during the 19th and the first decades of the 20th century (and its influence in the design of the Cambridge 1912 Congress). Some of the further topics discussed here are how the two world wars affected the congresses, the creation of the Fields medals, as well as some of the main changes that mathematics has experienced, internally (i.e., in what it refers to problems, ideas and theories) as well as institutionally (including its manifold connexions with society) during the period covered by the existence of the International Congresses of Mathematicians. 777 793 1 10.4171/022-1/29 http://www.ems-ph.org/doi/10.4171/022-1/29

Rodney Downey Victoria University, WELLINGTON, NEW ZEALAND Computer science Mathematical logic and foundations General We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability. 1 26 2 10.4171/022-2/1 http://www.ems-ph.org/doi/10.4171/022-2/1

Itay Neeman UCLA, LOS ANGELES, UNITED STATES Mathematical logic and foundations General The principle of determinacy has been crucial to the study of definable sets of real numbers. This paper surveys some of the uses of determinacy, concentrating specifically on the connection between determinacy and large cardinals, and takes this connection further, to the level of games of length ω1. 27 43 2 10.4171/022-2/2 http://www.ems-ph.org/doi/10.4171/022-2/2

Michael Rathjen University of Leeds, LEEDS, UNITED KINGDOM Mathematical logic and foundations General Ordinal analysis of theories is a core area of proof theory whose origins can be traced back to Hilbert’s programme – the aim of which was to lay to rest all worries about the foundations of mathematics once and for all by securing mathematics via an absolute proof of consistency. Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen’s head in the course of his consistency proof of arithmetic. The central theme of ordinal analysis is the classification of theories by means of transfinite ordinals that measure their ‘consistency strength’ and ‘computational power’. The so-called proof-theoretic ordinal of a theory also serves to characterize its provably recursive functions and can yield both conservation and combinatorial independence results. This paper intends to survey the development of “ordinally informative” proof theory from the work of Gentzen up to more recent advances in determining the proof-theoretic ordinals of strong subsystems of second order arithmetic. 45 69 2 10.4171/022-2/3 http://www.ems-ph.org/doi/10.4171/022-2/3

Thomas Scanlon University of California, BERKELEY, UNITED STATES Mathematical logic and foundations Number theory Field theory and polynomials Commutative rings and algebras Generalizing and synthesizing earlier work on the model theory of valued difference fields and on the model theory of valued fields with analytic structure, we prove Ax–Kochen– Eršov style relative completeness and relative quantifier elimination theorems for a theory of valuation rings with analytic and difference structure. Specializing our results to the case of W[Fpalg], the ring of Witt vectors of the algebraic closure of the field with p elements, given together with the relative Frobenius and the Tate algebras as analytic structure, we develop a model theoretic account of Buium’s p-differential functions. In so doing, we derive a uniform p-adic version of the Manin–Mumford conjecture. 71 92 2 10.4171/022-2/4 http://www.ems-ph.org/doi/10.4171/022-2/4

Simon Thomas Rutgers University, PISCATAWAY, UNITED STATES Mathematical logic and foundations Group theory and generalizations Dynamical systems and ergodic theory General In 1937, Baer solved the classification problem for the torsion-free abelian groups of rank 1. Since then, despite the efforts of many mathematicians, no satisfactory solution has been found of the classification problem for the torsion-free abelian groups of rank n ≥ 2. So it is natural to ask whether the classification problem for the higher rank groups is genuinely difficult. In this article, I will explain how this question can be partially answered, using ideas from descriptive set theory and Zimmer’s superrigidity theory. 93 116 2 10.4171/022-2/5 http://www.ems-ph.org/doi/10.4171/022-2/5

William Crawley-Boevey University of Leeds, LEEDS, UNITED KINGDOM Associative rings and algebras Algebraic geometry Linear and multilinear algebra; matrix theory General We describe recent work on preprojective algebras and moduli spaces of their representations. We give an analogue of Kac’s Theorem, characterizing the dimension types of indecomposable coherent sheaves over weighted projective lines in terms of loop algebras of Kac–Moody Lie algebras, and explain how it is proved using Hall algebras. We discuss applications to the problem of describing the possible conjugacy classes of sums and products of matrices in known conjugacy classes. 117 129 2 10.4171/022-2/6 http://www.ems-ph.org/doi/10.4171/022-2/6

Marcus du Sautoy Oxford University, OXFORD, UNITED KINGDOM Fritz Grunewald Heinrich-Heine-Universität, DÜSSELDORF, GERMANY Group theory and generalizations Number theory Nonassociative rings and algebras General We report on progress and problems concerning the analytical behaviour of the zeta functions of groups and rings. We also describe how these generating functions are special cases of adelic cone integrals for which our results hold. 131 149 2 10.4171/022-2/7 http://www.ems-ph.org/doi/10.4171/022-2/7

Bernhard Keller Université Paris Diderot - Paris 7, PARIS CEDEX 13, FRANCE Category theory; homological algebra Associative rings and algebras General Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, Dugger–Shipley, Toën, Toën–Vaquié, and others. 151 190 2 10.4171/022-2/8 http://www.ems-ph.org/doi/10.4171/022-2/8

Raphaël Rouquier Oxford University, OXFORD, UNITED KINGDOM Group theory and generalizations Associative rings and algebras Category theory; homological algebra General We discuss the homological algebra of representation theory of finite dimensional algebras and finite groups. We present various methods for the construction and the study of equivalences of derived categories: local group theory, geometry and categorifications. 191 221 2 10.4171/022-2/9 http://www.ems-ph.org/doi/10.4171/022-2/9

Mark Sapir Vanderbilt University, NASHVILLE, UNITED STATES Group theory and generalizations General This is a survey of the recent work in algorithmic and asymptotic properties of groups. I discuss Dehn functions of groups, complexity of the word problem, Higman embeddings, and constructions of finitely presented groups with extreme properties (monsters). 223 244 2 10.4171/022-2/10 http://www.ems-ph.org/doi/10.4171/022-2/10

Ákos Seress Ohio State University, COLUMBUS, UNITED STATES Group theory and generalizations General We survey algorithms to compute with large finite permutation and matrix groups. Particular attention will be given to handling both types of groups with similar methods, using structural properties to answer even basic questions such as the order of the input group. 245 258 2 10.4171/022-2/11 http://www.ems-ph.org/doi/10.4171/022-2/11

Agata Smoktunowicz University of Edinburgh, EDINBURGH, UNITED KINGDOM Associative rings and algebras General In this paper we survey some results on the structure of noncommutative rings. We focus particularly on nil rings, Jacobson radical rings and rings with finite Gelfand–Kirillov dimension. 259 269 2 10.4171/022-2/12 http://www.ems-ph.org/doi/10.4171/022-2/12

Manjul Bhargava Princeton University, PRINCETON, UNITED STATES Number theory General In 1801 Gauss laid down a remarkable law of composition on integral binary quadratic forms. This discovery, known as Gauss composition, not only had a profound influence on elementary number theory but also laid the foundations for ideal theory and modern algebraic number theory. Even today, Gauss composition remains one of the best ways of understanding ideal class groups of quadratic fields. The question arises as to whether there might exist similar laws of composition on other spaces of forms that could shed light on the structure of other algebraic number rings and fields. In this article we present several such higher analogues of Gauss composition, and we describe how each of these composition laws can be interpreted in terms of ideal classes in appropriate rings of algebraic integers. We also discuss several applications of these composition laws, including the resolution of a critical case of the Cohen–Lenstra–Martinet heuristics, and a solution of the long-standing problem of counting the number of quartic and quintic fields of bounded discriminant. In addition, we describe the mysterious relationship between these various composition laws and the exceptional Lie groups. Finally, we discuss prospects for future work and conclude with several open questions. 271 294 2 10.4171/022-2/13 http://www.ems-ph.org/doi/10.4171/022-2/13

Ching-Li Chai University of Pennsylvania, PHILADELPHIA, UNITED STATES Algebraic geometry Number theory General Let p be a prime number, and let ℳ be a modular variety of PEL type over Fp which classifies abelian varieties in characteristic p with extra symmetries of a fixed PEL type. Consider the p-divisible group with extra symmetries consisting of all p-power torsions of the universal abelian scheme over ℳ. The locus in ℳ corresponding to a fixed isomorphism type of a p-divisible group with extra symmetry is called a leaf by F. Oort. Each leaf is a smooth locally closed subvariety of the modular variety ℳ which is stable under all prime-to-p Hecke correspondences on ℳ. Oort conjectured that every Hecke orbit is dense in the leaf containing it. Tools fashioned for this conjecture include (a) rigidity, (b) global monodromy, and (c) canonical coordinates. The theory of canonical coordinates generalizes the classical Serre–Tate coordinates; it asserts that locally at the level of jet-spaces, every leaf is built up from p-divisible formal groups through a finite family of fibrations in a canonical way. The Hecke orbit conjecture is affirmed when ℳ is a Siegel modular variety classifying principally polarized abelian varieties of a fixed dimension, and also when ℳ is a Hilbert modular variety classifying abelian varieties with real multiplications. The proof of the Siegel case, joint with F. Oort, uses the irreducibility of non-supersingular leaves in Hilbert modular varieties due to C.-F. Yu. That proof relies heavily on a special property of Siegel modular varieties: The set of Fp-rational points of a Siegel modular variety Ag,n is filled up by Fp-rational points of Hilbert modular varieties contained in Ag,n. Possible directions for further progress include Tate-linear subvarieties and p-adic monodromy. The title of this article suggests that each leaf deserves to be viewed as a Shimura variety in characteristic p in its own right. 295 312 2 10.4171/022-2/14 http://www.ems-ph.org/doi/10.4171/022-2/14

Henri Darmon McGill University, MONTREAL, CANADA Number theory General Elliptic curves over ℂ are equipped with a systematic collection of Heegner points arising from the theory of complex multiplication and defined over abelian extensions of imaginary quadratic fields. These points are the key to the most decisive progress in the last decades on the Birch and Swinnerton-Dyer conjecture: an essentially complete proof for elliptic curves over ℂ of analytic rank ≤ 1, arising from the work of Gross–Zagier and Kolyvagin. In [Da2], it is suggested that Heegner points admit a host of conjectural generalisations, referred to as Stark–Heegner points because they occupy relative to their classical counterparts a position somewhat analogous to Stark units relative to elliptic or circular units. A better understanding of Stark–Heegner points would lead to progress on two related arithmetic questions: the explicit construction of global points on elliptic curves (a key issue arising in the Birch and Swinnerton-Dyer conjecture) and the analytic construction of class fields sought for in Kronecker’s Jugendtraum and Hilbert’s twelfth problem. The goal of this article is to survey Heegner points, Stark–Heegner points, their arithmetic applications and their relations (both proved, and conjectured) with special values of L-series attached to modular forms. 313 345 2 10.4171/022-2/15 http://www.ems-ph.org/doi/10.4171/022-2/15

Kazuhiro Fujiwara Nagoya University, NAGOYA, CHIKUSA-KU, JAPAN Number theory General Shimura varieties are arithmetic quotients of locally symmetric spaces which are canonically defined over number fields. In this article, we discuss recent developments on the reciprocity law realized on cohomology groups of Shimura varieties which relate Galois representations and automorphic representations. Focus is put on the control of ℓ-adic families of Galois representations by ℓ-adic families of automorphic representations. Arithmetic geometrical ideas and methods on Shimura varieties are used for this purpose. A geometrical realization of the Jacquet–Langlands correspondence is discussed as an example. 347 371 2 10.4171/022-2/16 http://www.ems-ph.org/doi/10.4171/022-2/16

Ben Green University of Oxford, OXFORD, UNITED KINGDOM Number theory General The Hardy–Littlewood method is a well-known technique in analytic number theory. Among its spectacular applications are Vinogradov’s 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the number of 3-term progressions of primes, all less than N. This article surveys recent developments of the author and T. Tao, in which the Hardy–Littlewood method has been generalised to obtain, for example, an asymptotic for the number of 4-term arithmetic progressions of primes less than N. 373 399 2 10.4171/022-2/17 http://www.ems-ph.org/doi/10.4171/022-2/17

Gérard Laumon Université Paris-Sud, ORSAY CEDEX, FRANCE Algebraic geometry Number theory Topological groups, Lie groups General In order to compute the Hasse–Weil zeta functions of the Shimura varieties or to establish some cases of the Langlands functoriality one first needs to stabilize the Arthur–Selberg trace formula. This stabilization can be done only if some combinatorial identities between orbital integrals over p-adic reductive groups are satisfied. The series of these conjectural identities form the so-called “Fundamental Lemma”. We present here some key points of the geometric approaches which have been used by Goresky, Kottwitz and MacPherson on the one hand and Ngô Bao Châu and myself one the other hand, to prove some cases of the Fundamental Lemma. 401 419 2 10.4171/022-2/18 http://www.ems-ph.org/doi/10.4171/022-2/18

Philippe Michel Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND Akshay Venkatesh Stanford University, STANFORD, UNITED STATES Number theory General An old question of Linnik asks about the equidistribution of integral points on a large sphere. This question proved to be very rich: it is intimately linked to modular forms, to subconvex estimates for L-functions, and to dynamics of torus actions on homogeneous spaces. Indeed, Linnik gave a partial answer using ergodic methods, and his question was completely answered by Duke using harmonic analysis and modular forms. We survey the context of these ideas and their developments over the last decades. 421 457 2 10.4171/022-2/19 http://www.ems-ph.org/doi/10.4171/022-2/19

Wiesława Nizioł University of Utah, SALT LAKE CITY, UNITED STATES Algebraic geometry Number theory General We will discuss recent progress by many people in the program of representing motivic cohomology with torsion coefficients of arithmetic schemes by various arithmetic p-adic cohomologies: étale, logarithmic de Rham–Witt, and syntomic. To illustrate possible applications in arithmetic geometry we will sketch proofs of the absolute purity conjecture in étale cohomology and comparison theorems of p-adic Hodge theory. 459 472 2 10.4171/022-2/20 http://www.ems-ph.org/doi/10.4171/022-2/20

Christopher Skinner University of Michigan, ANN ARBOR, UNITED STATES Eric Urban Columbia University, NEW YORK, UNITED STATES Number theory General This paper connects the vanishing at the central critical value of the L-functions of certain polarized regular motives with the positivity of the rank of the associated p-adic (Bloch–Kato) Selmer groups. For the motives studied it is shown that vanishing of the L-value implies positivity of the rank of the Selmer group. It is further shown that if the order of vanishing is positive and even then the Selmer group has rank at least two. The proofs make extensive use of families of p-adic modular forms. Additionally, the proofs assume the existence of Galois representations associated to holomorphic eigenforms on unitary groups over an imaginary quadratic field. 473 500 2 10.4171/022-2/21 http://www.ems-ph.org/doi/10.4171/022-2/21

Vinayak Vatsal University of British Columbia, VANCOUVER, CANADA Algebraic geometry Dynamical systems and ergodic theory General This article surveys the various known results on non-vanishing of special values of L-functions in p-adic families, with an emphasis on the rigidity theorems that underlie the proof in each case. 501 514 2 10.4171/022-2/22 http://www.ems-ph.org/doi/10.4171/022-2/22

Valery Alexeev University of Georgia, ATHENS, UNITED STATES Algebraic geometry General The Minimal Model Program offers natural higher-dimensional analogues of stable n-pointed curves and maps: stable pairs consisting of a projective variety X of dimension ≥ 2 and a divisor B, that should satisfy a few simple conditions, and stable maps f : (X, B) → Y . Although MMP remains conjectural in higher dimensions, in several important situations the moduli spaces of stable pairs, generalizing those of Deligne–Mumford, Knudsen and Kontsevich, can be constructed more directly, and in considerable generality. We review these constructions, with particular attention paid to varieties with group action, and list some open problems. 515 536 2 10.4171/022-2/23 http://www.ems-ph.org/doi/10.4171/022-2/23

Jean-Benoît Bost Université Paris-Sud, ORSAY CEDEX, FRANCE Number theory Algebraic geometry Dynamical systems and ergodic theory General We discuss criteria for the algebraicity of a formal subscheme V^ in the completion X^P at some rational point P of an algebraic variety X over some field K. In particular we consider the case where K is a function field or a number field, and we discuss applications concerning the algebraicity of leaves of algebraic foliations, algebraic groups, absolute Tate cycles, and the rationality of germs of formal functions on a curve over a number field. 537 562 2 10.4171/022-2/24 http://www.ems-ph.org/doi/10.4171/022-2/24

Tom Bridgeland University of Sheffield, SHEFFIELD, UNITED KINGDOM Category theory; homological algebra Algebraic geometry General We discuss derived categories of coherent sheaves on algebraic varieties. We focus on the case of non-singular Calabi–Yau varieties and consider two unsolved problems: proving that birational varieties have equivalent derived categories, and computing the group of derived autoequivalences. We also introduce the space of stability conditions on a triangulated category and explain its relevance to these two problems. 563 582 2 10.4171/022-2/25 http://www.ems-ph.org/doi/10.4171/022-2/25

Lawrence Ein University of Illinois at Chicago, CHICAGO, UNITED STATES Mircea Mustaţă University of Michigan, ANN ARBOR, UNITED STATES Algebraic geometry General Let X be a smooth complex variety and Y be a closed subvariety of X, or more generally, a closed subscheme of X. We are interested in invariants attached to the singularities of the pair (X,Y). We discuss various methods to construct such invariants, coming from the theory of multiplier ideals, D-modules, the geometry of the space of arcs and characteristic p techniques. We present several applications of these invariants to algebra, higher dimensional birational geometry and to singularities. 583 602 2 10.4171/022-2/26 http://www.ems-ph.org/doi/10.4171/022-2/26

Tom Graber California Institute of Technology, PASADENA, UNITED STATES Algebraic geometry General We survey some recent results proving the existence of rational points over one dimensional function fields and finite fields on varieties containing many rational curves. We also consider conjectural extensions of these theorems to higher dimensional function fields. 603 611 2 10.4171/022-2/27 http://www.ems-ph.org/doi/10.4171/022-2/27

Jun-Muk Hwang Cheongryangri 2-Dong, SEOUL, SOUTH KOREA Algebraic geometry Several complex variables and analytic spaces General Rigidity questions on rational homogeneous spaces arise naturally as higher dimensional generalizations of Riemann’s uniformization theorem in one complex variable. We will give an overview of some results obtained in this area by the study of minimal rational curves and geometric structures defined by the their tangent directions. 613 626 2 10.4171/022-2/28 http://www.ems-ph.org/doi/10.4171/022-2/28

Tomohide Terasoma University of Tokyo, MEGURO, TOKYO, JAPAN Algebraic geometry Several complex variables and analytic spaces General Many relations are known between multiple zeta values ζ(k1,...,kn). A relation coming from the associator condition for the Drinfeld associator, the generating function of multiple zeta values, is a geometric relation. By the theory of mixed motives, we can control the dimension of the rational linear hull of multiple zeta values. The harmonic shuffle relation also comes from geometry, and more strongly, this is implied by the associator relation. 627 635 2 10.4171/022-2/29 http://www.ems-ph.org/doi/10.4171/022-2/29

Yuri Tschinkel New York University, NEW YORK, UNITED STATES Algebraic geometry General I discuss some arithmetic aspects of higher-dimensional algebraic geometry. I focus on varieties with many rational points and on connections with classification theory and the minimal model program. 637 651 2 10.4171/022-2/30 http://www.ems-ph.org/doi/10.4171/022-2/30

Jarosław Włodarczyk Purdue University, WEST LAFAYETTE, UNITED STATES Algebraic geometry General We develop a Morse-like theory for complex algebraic varieties. In this theory a Morse function is replaced by a ℂ*-action. The critical points of the Morse function correspond to connected fixed point components. “Passing through the fixed points” induces some simple birational transformations called blow-ups, blow-downs and flips which are analogous to spherical modifications. In classical Morse theory by means of a Morse function we can decompose the manifold into elementary pieces – “handles”. In algebraic Morse theory we decompose a birational map between two smooth complex algebraic varieties into a sequence of blow-ups and blow-downs with smooth centers. 653 682 2 10.4171/022-2/31 http://www.ems-ph.org/doi/10.4171/022-2/31

Christoph Böhm Universität Münster, MÜNSTER, GERMANY Burkhard Wilking Universität Münster, MÜNSTER, GERMANY Differential geometry General We show that the normalized Ricci flow evolves metrics with positive curvature operator on compact manifolds to limit metrics of constant positive sectional curvature. In this note we only indicate the proof, the details will be published somewhere else. 683 690 2 10.4171/022-2/32 http://www.ems-ph.org/doi/10.4171/022-2/32

Simon Brendle Stanford University, STANFORD, UNITED STATES Differential geometry General I will review recent results regarding two questions that arise in connection with the Yamabe problem. The first problem is concerned with the conformal deformation of Riemannian metrics by their scalar curvature. This leads to a parabolic evolution equation, which can be interpreted as the flow of steepest descent for the Yamabe functional. I will provide conditions which guarantee that the flow converges to a metric of constant scalar curvature as t→∞. The second problem is concerned with the set of constant scalar curvature metrics in a given conformal class. I will discuss under what conditions this set is compact. A recurring theme in the study of both problems is that blow-up can be ruled out by means of the positive mass theorem provided that the dimension is less than 6, whereas the Weyl tensor plays a crucial role in higher dimensions. 691 704 2 10.4171/022-2/33 http://www.ems-ph.org/doi/10.4171/022-2/33

Ko Honda University of Southern California, LOS ANGELES, UNITED STATES Manifolds and cell complexes Differential geometry General The goal of this article is to survey recent developments in the theory of contact structures in dimension three. 705 717 2 10.4171/022-2/34 http://www.ems-ph.org/doi/10.4171/022-2/34

Michael Kapovich University of California at Davis, DAVIS, UNITED STATES Topological groups, Lie groups Algebraic geometry Differential geometry General We present a survey of our recent work on generalized triangle inequalities in infinitesimal symmetric spaces, nonpositively curved symmetric spaces and Euclidean buildings. We also explain how these results can be used to analyze some basic problems of algebraic group theory including the problem of decomposition of tensor products of irreducible representations of complex reductive Lie groups. Among the applications is a generalization of the Saturation Theorem of Knutson and Tao to Lie groups other than SL(n,ℂ). 719 741 2 10.4171/022-2/35 http://www.ems-ph.org/doi/10.4171/022-2/35

Bruce Kleiner Courant Institute of Mathematical Sciences, NEW YORK, UNITED STATES Functions of a complex variable Group theory and generalizations General This is a survey of recent developments at the interface between quasiconformal analysis and the asymptotic geometry of Gromov hyperbolic groups. The main theme is the extension of Mostow rigidity and related theorems to a broader class of hyperbolic groups, using recently developed analytic structure of the boundary. 743 768 2 10.4171/022-2/36 http://www.ems-ph.org/doi/10.4171/022-2/36

Philippe Charron Université de Montréal, Montréal, QC, CANADA Differential geometry General In these notes, I will present collaborative works on Lagrangian submanifolds that I realised mainly with Octav Cornea (the cluster complex, which naturally leads to a universal Lagrangian Floer theory), but also, at an earlier stage, with Jean-Claude Sikorav. To cover the subject in a more complete and adequate way, I will also mention very recent works by Barraud–Cornea and by Welschinger, closely related to the subject of these notes. The aim of the cluster machinery is to resolve the well known problem of real codimension 1 bubbling off of disks in the Gromov–Floer theory; see Fukaya–Oh–Ohta–Ono (especially the two lectures by Oh and Ono in these proceedings) for a different, earlier, approach. 769 789 2 10.4171/022-2/37 http://www.ems-ph.org/doi/10.4171/022-2/37

Xiaobo Liu University of Notre Dame, NOTRE DAME, UNITED STATES Differential geometry Algebraic geometry General The purpose of this note is to explain how much information of Gromov–Witten invariants of compact symplectic manifolds are determined by the geometry of moduli spaces of curves. In the case when a manifold has semisimple quantum cohomology, we believe that the information obtained this way might determine all higher genus Gromov–Witten invariants and could be used to study the Virasoro conjecture of Eguchi–Hori–Xiong and S. Katz. 791 812 2 10.4171/022-2/38 http://www.ems-ph.org/doi/10.4171/022-2/38

Toshiki Mabuchi Osaka University, OSAKA, JAPAN Differential geometry Several complex variables and analytic spaces General The Hitchin–Kobayashi correspondence for vector bundles, established by Donaldson, Kobayashi, Lübke, Uhlenbeck andYau, states that an indecomposable holomorphic vector bundle over a compact Kähler manifold is stable in the sense of Takemoto–Mumford if and only if the vector bundle admits a Hermitian-Einstein metric. Its manifold analogue known as Yau’s conjecture, which originated from Calabi’s conjecture, asks whether “stability” and “existence of extremal metrics” for polarized manifolds are equivalent. In this note the recent progress of this subject, by Donaldson, Tian and our group, together with its relationship to algebraic geometry will be discussed. 813 826 2 10.4171/022-2/39 http://www.ems-ph.org/doi/10.4171/022-2/39

Grigory Mikhalkin Université de Genève, GENÈVE 4, SWITZERLAND Algebraic geometry Convex and discrete geometry General From a formal perspective tropical geometry can be viewed as a branch of geometry manipulating with certain piecewise-linear objects that take over the rôle of classical algebraic varieties. This talk outlines some basic notions of this area and surveys some of its applications for the problems in classical (real and complex) geometry. 827 852 2 10.4171/022-2/40 http://www.ems-ph.org/doi/10.4171/022-2/40

William Minicozzi II The Johns Hopkins University, BALTIMORE, UNITED STATES Differential geometry General The study of embedded minimal surfaces in ℝ3 is a classical problem, dating to the mid 1700s, and many people have made key contributions. We will survey a few recent advances, focusing on joint work with Tobias H. Colding of MIT and Courant Institute, and taking the opportunity to focus on results that have not been highlighted elsewhere. 853 877 2 10.4171/022-2/41 http://www.ems-ph.org/doi/10.4171/022-2/41

Yong-Geun Oh University of Wisconsin-Madison, MADISON CITY, UNITED STATES Kenji Fukaya Kyoto University, KYOTO, JAPAN Differential geometry Algebraic geometry General In this article the authors review what the Floer homology is and what it does in symplectic geometry both in the closed string and in the open string context. In the first case, the authors will explain how the chain level Floer theory leads to the C0 symplectic invariants of Hamiltonian flows and to the study of topological Hamiltonian dynamics. In the second case, the authors explain how Floer’s original construction of Lagrangian intersection Floer homology is obstructed in general as soon as one leaves the category of exact Lagrangian submanifolds. They will survey the construction of the Floer complex and describe its obstruction in terms of the filtered A∞-algebras. This can be promoted to the level of A∞-category (Fukaya category) of symplectic manifolds. Some applications of this general machinery to the study of the topology of Lagrangian embeddings in relation to symplectic topology and to mirror symmetry are also reviewed. 879 905 2 10.4171/022-2/42 http://www.ems-ph.org/doi/10.4171/022-2/42

Antonio Ros Universidad de Granada, GRANADA, SPAIN Differential geometry General We present a synthesis of the situation as it now stands about the various moduli spaces of properly embedded minimal surfaces of finite topology in flat 3-manifolds. This family includes the case of minimal surfaces with finite total curvature in ℝ3 as well as singly, doubly and triply periodic minimal surfaces. 907 926 2 10.4171/022-2/43 http://www.ems-ph.org/doi/10.4171/022-2/43

Chuu-Lian Terng University of California, Irvine, IRVINE, UNITED STATES Differential geometry Partial differential equations Dynamical systems and ergodic theory General The 1-d Schrödinger flow on S2, the Gauss–Codazzi equation for flat Lagrangian submanifolds in ℝ2n, and the space-time monopole equation are all examples of geometric soliton equations. The linear systems with a spectral parameter (Lax pair) associated to these equations satisfy the reality condition associated to SU(n). In this article, we explain the method developed jointly with K. Uhlenbeck that uses various loop group factorizations to construct inverse scattering transforms, Bäcklund transformations, and solutions to Cauchy problems for these equations. 927 950 2 10.4171/022-2/44 http://www.ems-ph.org/doi/10.4171/022-2/44

Ian Agol University of California, BERKELEY, UNITED STATES Functions of a complex variable Manifolds and cell complexes General We prove that there are only finitely many arithmetic Kleinian maximal reflection groups. 951 960 2 10.4171/022-2/45 http://www.ems-ph.org/doi/10.4171/022-2/45

Martin Bridson University of Oxford, OXFORD, UNITED KINGDOM Group theory and generalizations General A universe of finitely presented groups is sketched and explained, leading to a discussion of the fundamental role that manifestations of non-positive curvature play in group theory. The geometry of the word problem and associated filling invariants are discussed. The subgroup structure of direct products of hyperbolic groups is analysed and a process for encoding diverse phenomena into finitely presented subdirect products is explained. Such an encoding is used to solve problems of Grothendieck concerning profinite completions and representations of groups. In each context, explicit groups are crafted to solve problems of a geometric nature. 961 987 2 10.4171/022-2/46 http://www.ems-ph.org/doi/10.4171/022-2/46

Aaron Lauda University of Southern California, LOS ANGELES, UNITED STATES Manifolds and cell complexes General This is a short survey of algebro-combinatorial link homology theories which have the Jones polynomial and other link polynomials as their Euler characteristics. 989 999 2 10.4171/022-2/47 http://www.ems-ph.org/doi/10.4171/022-2/47

Yair Minsky Yale University, NEW HAVEN, UNITED STATES Manifolds and cell complexes Functions of a complex variable General A survey of the role of the complex of curves in recent work on 3-manifolds and mapping class groups. 1001 1033 2 10.4171/022-2/48 http://www.ems-ph.org/doi/10.4171/022-2/48

Fabien Morel Universität München, MÜNCHEN, GERMANY Algebraic geometry $K$-theory Algebraic topology General We present some recent results in A1-algebraic topology, which means both in A1-homotopy theory of schemes and its relationship with algebraic geometry. This refers to the classical relationship between homotopy theory and (differential) topology. We explain several examples of “motivic” versions of classical results: the theory of the Brouwer degree, the classification of A1-coverings through the A1-fundamental group, the Hurewicz Theorem and the A1-homotopy of algebraic spheres, and the A1-homotopy classification of vector bundles. We also give some applications and perspectives. 1035 1059 2 10.4171/022-2/49 http://www.ems-ph.org/doi/10.4171/022-2/49

Kaoru Ono Hokkaido University, SAPPORO, JAPAN Differential geometry General In the middle of the 1980s, Floer initiated a new theory, which is now called the Floer theory. Since then the theory has been developed in various ways. In this article we report some recent progress in Floer theory in symplectic geometry. For example, we give an outline of a proof of the flux conjecture, which states that the Hamiltonian diffeomorphism group is C1-closed in the group of symplectomorphisms for closed symplectic manifolds. We also give a brief survey on the obstruction–deformation theory for Floer theory of Lagrangian submanifolds and explain some of its applications. 1061 1082 2 10.4171/022-2/50 http://www.ems-ph.org/doi/10.4171/022-2/50

Peter Ozsváth Princeton University, PRINCETON, UNITED STATES Zoltán Szabó Princeton University, PRINCETON, UNITED STATES Differential geometry Manifolds and cell complexes General We review the construction of Heegaard–Floer homology for closed three-manifolds and also for knots and links in the three-sphere. We also discuss three applications of this invariant to knot theory: studying the Thurston norm of a link complement, the slice genus of a knot, and the unknotting number of a knot. We emphasize the application to the Thurston norm, and illustrate the theory in the case of the Conway link. 1083 1099 2 10.4171/022-2/51 http://www.ems-ph.org/doi/10.4171/022-2/51

Karen Vogtmann University of Warwick, COVENTRY, UNITED KINGDOM Group theory and generalizations General There are intriguing analogies between automorphism groups of finitely generated free groups and mapping class groups of surfaces on the one hand, and arithmetic groups such as GL(n, ℤ) on the other. We explore aspects of these analogies, focusing on cohomological properties. Each cohomological feature is studied with the aid of topological and geometric constructions closely related to the groups. These constructions often reveal unexpected connections with other areas of mathematics. 1101 1117 2 10.4171/022-2/52 http://www.ems-ph.org/doi/10.4171/022-2/52

Roman Bezrukavnikov Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES Nonassociative rings and algebras Category theory; homological algebra Group theory and generalizations General Springer resolution of the set of nilpotent elements in a semisimple Lie algebra plays a central role in geometric representation theory. A new structure on this variety has arisen in several representation theoretic constructions, such as the (local) geometric Langlands duality and modular representation theory. It is also related to some algebro-geometric problems, such as the derived equivalence conjecture and description of T. Bridgeland’s space of stability conditions. The structure can be described as a noncommutative counterpart of the resolution, or as a t-structure on the derived category of the resolution. The intriguing fact that the same t-structure appears in these seemingly disparate subjects has strong technical consequences for modular representation theory. 1119 1144 2 10.4171/022-2/53 http://www.ems-ph.org/doi/10.4171/022-2/53

Alexander Braverman Brown University, PROVIDENCE, UNITED STATES Topological groups, Lie groups Algebraic geometry General Given a projective variety X and a smooth projective curve C one may consider the moduli space of maps C → X. This space admits certain compactification whose points are called quasi-maps. In the last decade it has been discovered that in the case when X is a (partial) flag variety of a semi-simple algebraic group G (or, more generally, of any symmetrizable Kac–Moody Lie algebra) these compactifications play an important role in such fields as geometric representation theory, geometric Langlands correspondence, geometry and topology of moduli spaces of G-bundles on algebraic surfaces, 4-dimensional super-symmetric gauge theory (and probably many others). This paper is a survey of the recent results about quasi-maps as well as their applications in different branches of representation theory and algebraic geometry. 1145 1170 2 10.4171/022-2/54 http://www.ems-ph.org/doi/10.4171/022-2/54

Guy Henniart Université Paris-Sud, ORSAY CX, FRANCE Topological groups, Lie groups General Let F be a locally compact non Archimedean field, and D a division algebra with centre F and finite dimension d2 over F. Fix an integer r ≥ 1, and let G = GLn(F), G' = GLr(D), where n = rd. Smooth irreducible representations of G' are related to those of G via the Jacquet–Langlands correspondence, whereas the Langlands correspondence relates such representations of G to degree n representations of the absolute Galois group of F . We review some recent results on those correspondences, in particular on their explicit description. 1171 1182 2 10.4171/022-2/55 http://www.ems-ph.org/doi/10.4171/022-2/55

Nicolas Monod Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND Algebraic topology Group theory and generalizations General A selection of aspects of the theory of bounded cohomology is presented. The emphasis is on questions motivating the use of that theory as well as on some concrete issues suggested by its study. Specific topics include rigidity, bounds on characteristic classes, quasification, orbit equivalence, amenability. 1183 1211 2 10.4171/022-2/56 http://www.ems-ph.org/doi/10.4171/022-2/56

Bao-Châu Ngô Université Paris-Sud, ORSAY CEDEX, FRANCE Algebraic geometry Number theory Topological groups, Lie groups General The Hitchin fibration is a well suited tool to study the geometric side of the trace formula for Lie algebra from the point of view of moduli spaces of vector bundles over a curve. The endoscopy groups appear naturally when we decompose the cohomology of the Hitchin fibration by its natural symmetries. Following this dictionary, we can formulate a global and geometric version of Langlands–Shelstad’s fundamental lemma. This conjecture has been proved in the case of unitary groups in a joint work with G. Laumon. 1213 1225 2 10.4171/022-2/57 http://www.ems-ph.org/doi/10.4171/022-2/57

Eric Opdam University of Amsterdam, AMSTERDAM, NETHERLANDS Group theory and generalizations Topological groups, Lie groups Abstract harmonic analysis General Iwahori–Hecke algebras are ubiquitous. One encounters these algebras in subjects as diverse as harmonic analysis, equivariant K-theory, orthogonal polynomials, quantum groups, knot theory, algebraic combinatorics, and integrable models in statistical physics. In this exposition we will mostly concentrate on the analytic aspects of affine Hecke algebras and study them from the perspective of operator algebras. We will discuss the Plancherel theorem for these type of algebras, and based on a conjectural invariance property of their (operator algebraic) K-theory, study the structure of the tempered dual. 1227 1259 2 10.4171/022-2/58 http://www.ems-ph.org/doi/10.4171/022-2/58

Peter Schneider Universität Münster, MÜNSTER, GERMANY Number theory Topological groups, Lie groups General In this paper we give an overview over the basic features of the continuous representation theory of p-adic Lie groups as it has emerged during the last five years. The main motivation for developing such a theory is a possible extension of the local Langlands program to p-adic Galois representations. This is still very much in its infancy. But in the last section we will describe a first approximation to an extended Langlands functoriality principle for crystalline Galois representations. 1261 1282 2 10.4171/022-2/59 http://www.ems-ph.org/doi/10.4171/022-2/59

Yehuda Shalom University of California, Los Angeles, LOS ANGELES, UNITED STATES Group theory and generalizations General We present the surge of activity since 2005, around what we call the algebraic (as contrasted with the geometric) approach to Kazhdan’s property (T). The discussion includes also an announcement of a recent result (March 2006) regarding property (T) for linear groups over arbitrary finitely generated rings. 1283 1310 2 10.4171/022-2/60 http://www.ems-ph.org/doi/10.4171/022-2/60

David Soudry Tel Aviv University, RAMAT AVIV, TEL AVIV, ISRAEL Number theory General In this article I survey the descent method of Ginzburg, Rallis and Soudry and its main applications to the Langlands functorial lift of automorphic, cuspidal, generic representations on a classical group to (appropriate) GLn, and to establishing a local Langlands reciprocity law for (split) SO2n+1 (joint work with D. Jiang). The descent method arises when we consider certain residues of special cases of a family of global integrals, attached to pairs of automorphic, cuspidal representations, one on a classical group G and one on GLn. The last part of this article focuses on the case G = SOm (split), and the progress made in a joint work with S. Rallis, towards establishing, via the converse theorem, the functorial lift from any automorphic, cuspidal representation on G to GL2[m/2]. 1311 1325 2 10.4171/022-2/61 http://www.ems-ph.org/doi/10.4171/022-2/61

Birgit Speh Cornell University, ITHACA, UNITED STATES Topological groups, Lie groups General Let G be a semisimple Lie group with finitely many connected components and Lie algebra g, K a maximal compact subgroup of G, and X = G/K a symmetric space. A torsion free discrete subgroup Γ of G and a finite dimensional real or complex linear representation (ρ, E) of G define a locally symmetric space XΓ = Γ\G/K with a local system Ẽ. Then H∗ (Γ, E) = H∗ (Γ\X, Ẽ) is isomorphic to H∗ (g,K, C∞( Γ\G) ⊗ E). If Γ is an arithmetic group, then H∗ (g,K, C∞(Γ\G) ⊗ E) is isomorphic to the (g,K)-cohomology with coefficients in A(Γ\G) ⊗ E where A(Γ\G) is the space of automorphic forms. Using representation theory and the theory of automorphic forms a large amount of information about H∗ (Γ, E) can be deduced. 1327 1335 2 10.4171/022-2/62 http://www.ems-ph.org/doi/10.4171/022-2/62

Tonny Springer Utrecht University, UTRECHT, NETHERLANDS Algebraic geometry General This paper deals with recent results involving a compactification X of a semisimple group G. The emphasis is on the case that G is adjoint and X is its wonderful compactification. Group theoretical constructions in G have repercussions in X. The paper describes a number of them. 1337 1348 2 10.4171/022-2/63 http://www.ems-ph.org/doi/10.4171/022-2/63

Mario Bonk University of Michigan, ANN ARBOR, UNITED STATES Functions of a complex variable Group theory and generalizations General Many questions in analysis and geometry lead to problems of quasiconformal geometry on non-smooth or fractal spaces. For example, there is a close relation of this subject to the problem of characterizing fundamental groups of hyperbolic 3-orbifolds or to Thurston’s characterization of rational functions with finite post-critical set. In recent years, the classical theory of quasiconformal maps between Euclidean spaces has been successfully extended to more general settings and powerful tools have become available. Fractal 2-spheres or Sierpi´nski carpets are typical spaces for which this deeper understanding of their quasiconformal geometry is particularly relevant and interesting. 1349 1373 2 10.4171/022-2/64 http://www.ems-ph.org/doi/10.4171/022-2/64

Dorina Mitrea University of Missouri, COLUMBIA, UNITED STATES Fourier analysis Partial differential equations Operator theory General A Tb theorem is a boundedness criterion for singular integrals, which allows the L2 boundedness of a singular integral operator T to be deduced from sufficiently good behavior of T on some suitable non-degenerate test function b. However, in some PDE applications, including, for example, the solution of the Kato problem for square roots of divergence form elliptic operators, it may be easier to test the operator T locally (say on any given dyadic cube Q), on a test function bQ that depends upon Q, rather than on a single, globally defined b. Or to be more precise, in the applications, it may be easier to find a family of bQ’s for which TbQ is locally well behaved, than it is to find a single b for which Tb is nice globally. In this lecture, we shall discuss some versions of local Tb theorems, as well as some applications to PDE. 1375 1392 2 10.4171/022-2/65 http://www.ems-ph.org/doi/10.4171/022-2/65

Sergey Konyagin Moscow State University, MOSCOW, RUSSIAN FEDERATION Fourier analysis General The aim of this expository paper is to demonstrate that there are several challenging problems concerning the behavior of trigonometric Fourier series almost everywhere. 1393 1403 2 10.4171/022-2/66 http://www.ems-ph.org/doi/10.4171/022-2/66

Linda Preiss Rothschild University of California, San Diego, LA JOLLA, UNITED STATES Several complex variables and analytic spaces Algebraic geometry General This article is a survey of various applications of the method of iterated Segre mappings obtained by a number of mathematicians, including the author, over the past decade. This method is applied to various problems involving real submanifolds in complex space and their mappings. The article begins with a description of the iterated Segre mappings associated to generic submanifolds. The problems addressed concern transversality of holomorphic mappings, finite jet determination, local stability groups, and algebraicity of holomorphic mappings between real-algebraic manifolds. 1405 1419 2 10.4171/022-2/67 http://www.ems-ph.org/doi/10.4171/022-2/67

Stanislav Smirnov Université de Genève, GENÈVE 4, SWITZERLAND Statistical mechanics, structure of matter Functions of a complex variable Probability theory and stochastic processes Quantum theory Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising model, self-avoiding polymers, etc. This has led to numerous exact (but non-rigorous) predictions of their scaling exponents and dimensions. We will discuss how to prove the conformal invariance conjectures, especially in relation to Schramm–Loewner evolution. 1421 1451 2 10.4171/022-2/68 http://www.ems-ph.org/doi/10.4171/022-2/68

Emil Straube Texas A&M University, COLLEGE STATION, UNITED STATES Several complex variables and analytic spaces Partial differential equations General The ∂-Neumann problem is the fundamental boundary value problem in several complex variables. It features an elliptic operator coupled with non-coercive boundary conditions. The problem is globally regular on many, but not all, pseudoconvex domains. We discuss several recent developments in the L2-Sobolev theory of the ∂-Neumann problem that concern compactness and global regularity. 1453 1478 2 10.4171/022-2/69 http://www.ems-ph.org/doi/10.4171/022-2/69

Vladimir Temlyakov University of South Carolina, COLUMBIA, UNITED STATES Approximations and expansions Partial differential equations General This paper is a survey of recent results on greedy approximations with regard to bases. The theory of greedy approximations is a part of nonlinear approximations. The standard problem in this regard is the problem of m-term approximation where one fixes a basis and seeks to approximate a target function by a linear combination of m terms of the basis. When the basis is a wavelet basis or a basis of other waveforms, then this type of approximation is the starting point for compression algorithms. We are interested in the quantitative aspects of this type of approximation. Introducing the concept of best m-term approximation we obtain a lower bound for the accuracy of any method providing m-term approximation. It is known that a problem of simultaneous optimization over many parameters (like in best m-term approximation) is a very difficult problem. We would like to have an algorithm for constructing m-term approximants that adds at each step only one new element from the basis and keeps elements of the basis obtained at the previous steps. The primary object of our discussion is the Thresholding Greedy Algorithm (TGA) with regard to a given basis. The TGA, applied to a function f, picks at the m-th step an element with the m-th biggest coefficient (in absolute value) of the expansion of f in the series with respect to the basis. We show that this algorithm is very good for a wavelet basis and is not that good for the trigonometric system. We discuss in detail the behavior of the TGA with regard to the trigonometric system. We also discuss one example of an algorithm from a family of very general greedy algorithms that works in the case of a redundant system instead of a basis. It turns out that this general greedy algorithm is very good for the trigonometric system. 1479 1504 2 10.4171/022-2/70 http://www.ems-ph.org/doi/10.4171/022-2/70

Xavier Tolsa Universitat Autónoma de Barcelona, BARCELONA, SPAIN Functions of a complex variable Measure and integration Fourier analysis General A compact set E ⊂ ℂ is said to be removable for bounded analytic functions if for any open set Ω containing E, every bounded function analytic on Ω \ E has an analytic extension to Ω. Analytic capacity is a notion that, in a sense, measures the size of a set as a non removable singularity. In particular, a compact set is removable if and only if its analytic capacity vanishes. The so-called Painlevé problem consists in characterizing removable sets in geometric terms. Recently many results in connection with this very old and challenging problem have been obtained. Moreover, it has also been proved that analytic capacity is semiadditive. We review these results and other related questions dealing with rectifiability, the Cauchy transform, and the Riesz transforms. 1505 1527 2 10.4171/022-2/71 http://www.ems-ph.org/doi/10.4171/022-2/71

Franck Barthe Université Toulouse III, TOULOUSE CEDEX, FRANCE Difference and functional equations Real functions Convex and discrete geometry Information and communication, circuits The Brunn–Minkowski inequality gives a lower bound of the Lebesgue measure of a sum-set in terms of the measures of the individual sets. It has played a crucial role in the theory of convex bodies. This topic has many interactions with isoperimetry or functional analysis. Our aim here is to report some recent aspects of these interactions involving optimal mass transport or the Heat equation. Among other things, we will present Brunn–Minkowski inequalities for flat sets, or in Gauss space, as well as local versions of the theorem which apply to the study of entropy production in the central limit theorem. 1529 1546 2 10.4171/022-2/72 http://www.ems-ph.org/doi/10.4171/022-2/72

Bo'az Klartag Tel-Aviv University, TEL-AVIV, ISRAEL Convex and discrete geometry Functional analysis General The classical theorems of high-dimensional convex geometry exhibit a surprising level of regularity and order in arbitrary high-dimensional convex sets. These theorems are mainly concerned with the rough geometric features of general convex sets; the so-called “isomorphic” features. Recent results indicate that, perhaps, high-dimensional convex sets are also very regular on the almost-isometric scale. We review some related research directions in high-dimensional convex geometry, focusing in particular on the problem of geometric symmetrization. 1547 1562 2 10.4171/022-2/73 http://www.ems-ph.org/doi/10.4171/022-2/73

Narutaka Ozawa Kyoto University, KYOTO, JAPAN Functional analysis Group theory and generalizations Dynamical systems and ergodic theory Abstract harmonic analysis We will give a brief account of (topological) amenable actions and exactness for countable discrete groups. The class of exact groups contains most of the familiar groups and yet is manageable enough to provide interesting applications in geometric topology, von Neumann algebras and ergodic theory. 1563 1580 2 10.4171/022-2/74 http://www.ems-ph.org/doi/10.4171/022-2/74

Mikael Rørdam Københavns Universitetet, COPENHAGEN, DENMARK Functional analysis Group theory and generalizations Dynamical systems and ergodic theory Abstract harmonic analysis We give an overview of the development over the last 15 years of the theory of simple C*-algebras, in particular in regards to their classification and structure. We discuss dimension theory for (simple) C*-algebras, in particular the so-called stable and real ranks, and we explain how properties of C*-algebras of low dimension (stable rank one and real rank zero) was used by the author and P. Friis to give a new and simple proof of a theorem of H. Lin that almost commuting self-adjoint matrices are close to exactly commuting self-adjoint matrices. Elliott’s classification program is explained and is contrasted with recent examples of C*-algebras of “high dimension”, including an example of a simple C*-algebras with a finite and an infinite projection. 1581 1598 2 10.4171/022-2/75 http://www.ems-ph.org/doi/10.4171/022-2/75

Stanislaw Szarek Case Western Reserve University, CLEVELAND, UNITED STATES Functional analysis Linear and multilinear algebra; matrix theory Operator theory Convex and discrete geometry We discuss metric, algorithmic and geometric issues related to broadly understood complexity of high dimensional convex sets. The specific topics we bring up include metric entropy and its duality, derandomization of constructions of normed spaces or of convex bodies, and different fundamental questions related to geometric diversity of such bodies, as measured by various isomorphic (as opposed to isometric) invariants. 1599 1621 2 10.4171/022-2/76 http://www.ems-ph.org/doi/10.4171/022-2/76

Guoliang Yu Texas A&M University, COLLEGE STATION, UNITED STATES Functional analysis Group theory and generalizations Global analysis, analysis on manifolds General The Atiyah–Singer index theorem has been vastly generalized to higher index theory for elliptic operators in the context of noncommutative geometry. Higher index theory has important applications to problems in differential topology and differential geometry such as the Novikov Conjecture on homotopy invariance of higher signatures and the existence problem of Riemannian metrics with positive scalar curvature. In this article, I will give a survey on recent development of higher index theory, its applications, and its fascinating connection to the geometry of groups and metric spaces. 1623 1639 2 10.4171/022-2/77 http://www.ems-ph.org/doi/10.4171/022-2/77

Oleg Ageev N.E. Bauman State Technical University, MOSCOW, RUSSIAN FEDERATION Dynamical systems and ergodic theory Measure and integration Operator theory General This is a short survey of recent developments in one of the oldest areas of ergodic theory, sometimes called the spectral theory of dynamical systems. We mainly discuss the spectral realization problem in the rich class of all invertible measure preserving dynamical systems, a “behavior” of different spectral invariants in natural subclasses of dynamical systems, and a complete solution of Rokhlin’s problem on homogeneous spectrum in ergodic theory. 1641 1653 2 10.4171/022-2/78 http://www.ems-ph.org/doi/10.4171/022-2/78

Vitaly Bergelson Ohio State University, COLUMBUS, UNITED STATES Measure and integration Combinatorics Dynamical systems and ergodic theory General We discuss classical results of Ramsey theory together with their dynamical counterparts, survey recent developments and formulate some natural open questions and conjectures. 1655 1678 2 10.4171/022-2/79 http://www.ems-ph.org/doi/10.4171/022-2/79

Nikolai Chernov University of Alabama at Birmingham, BIRMINGHAM, UNITED STATES Dmitry Dolgopyat University of Maryland, COLLEGE PARK, UNITED STATES Dynamical systems and ergodic theory Order, lattices, ordered algebraic structures Ordinary differential equations General Mathematical theory of billiards is a fascinating subject providing a fertile source of new problems as well as conjecture testing in dynamics, geometry, mathematical physics and spectral theory. This survey is devoted to planar hyperbolic billiards with emphasis on their applications in statistical physics, where they provide many physically interesting and mathematically tractable models. 1679 1704 2 10.4171/022-2/80 http://www.ems-ph.org/doi/10.4171/022-2/80

Rafael de la Llave Georgia Institute of Technology, ATLANTA, UNITED STATES Dynamical systems and ergodic theory Ordinary differential equations Mechanics of particles and systems General We discuss some geometric structures that lead to instability in Hamiltonian systems arbitrarily close to integrable. The structures covered in this report are joint work with A. Delshams, T. M. Seara and M. Gidea. 1705 1729 2 10.4171/022-2/81 http://www.ems-ph.org/doi/10.4171/022-2/81

Manfred Einsiedler ETH Zürich, HG G 64.2, ZÜRICH, SWITZERLAND Elon Lindenstrauss The Hebrew University of Jerusalem, JERUSALEM, ISRAEL Dynamical systems and ergodic theory Number theory Quantum theory General We discuss dynamical properties of actions of diagonalizable groups on locally homogeneous spaces, particularly their invariant measures, and present some number theoretic and spectral applications. Entropy plays a key role in the study of theses invariant measures and in the applications. 1731 1759 2 10.4171/022-2/82 http://www.ems-ph.org/doi/10.4171/022-2/82

Robert Ghrist University of Illinois, URBANA, UNITED STATES Dynamical systems and ergodic theory Ordinary differential equations Partial differential equations Manifolds and cell complexes Forcing theorems based on topological features of invariant sets have played a fundamental role in dynamics and differential equations. This talk focuses on the recent work of Vandervorst, Van den Berg, and the author using braids to construct a forcing theory for scalar parabolic PDEs, second-order Lagrangian ODEs, and one-dimensional lattice dynamics. 1 26 3 10.4171/022-3/1 http://www.ems-ph.org/doi/10.4171/022-3/1

Anton Gorodetski University of California, Irvine, IRVINE, UNITED STATES Brian Hunt University of Maryland, COLLEGE PARK, UNITED STATES Vadim Kaloshin University of Maryland, COLLEGE PARK, UNITED STATES Dynamical systems and ergodic theory General We describe a general method of studying prevalent properties of diffeomorphisms of a compact manifold M, where by prevalent we mean true for Lebesgue almost every parameter ε in a generic finite-parameter family {fε} of diffeomorphisms on M. Usually a dynamical property P can be formulated in terms of properties Pn of trajectories of finite length n. Let P be such a dynamical property that can be expressed in terms of only periodic trajectories. The first idea of the method is to discretize M and split the set of all possible periodic trajectories of length n for the entire family {fε} into a finite number of approximating periodic pseudotrajectories. Then for each such pseudotrajectory, we estimate the measure of parameters for which it fails Pn. This bounds the total parameter measure for which Pn fails by a finite sum over the periodic pseudotrajectories of length n. Application of Newton interpolation polynomials to estimate the measure of parameters that fail Pn for a given periodic pseudotrajectory of length n is the second idea. We outline application of these ideas to two quite different problems: Growth of number of periodic points for prevalent diffeomorphisms (Kaloshin–Hunt). Palis’ conjecture on finititude of number of “localized” sinks for prevalent surface diffeomorphisms (Gorodetski–Kaloshin). 27 55 3 10.4171/022-3/2 http://www.ems-ph.org/doi/10.4171/022-3/2

Bryna Kra Northwestern University, EVANSTON, UNITED STATES Dynamical systems and ergodic theory General Multiple ergodic averages, such as the average of expressions like f1(Tn x) f2(T2n x) ... fk(Tkn x), were first studied in the ergodic theoretic proof of Szemerédi’s Theorem on arithmetic progressions. It turns out that all constraints on such averages (in a sense that we describe) have an algebraic character, arising from identities in nilpotent groups. We discuss these averages, several generalizations, and combinatorial implications of the results. 57 76 3 10.4171/022-3/3 http://www.ems-ph.org/doi/10.4171/022-3/3

Patrice Le Calvez Université de Paris XIII/CNRS, VILLETANEUSE, FRANCE Dynamical systems and ergodic theory General We will state an equivariant foliated version of the classical Brouwer Plane Translation Theorem and will explain how to apply this result to the study of homeomorphisms of surfaces. In particular we will explain why a diffeomorphism of a closed oriented surface of genus ≥ 1 that is the time-one map of a time dependent Hamiltonian vector field has infinitely many periodic orbits. This gives a positive answer in the case of surfaces to a more general question stated by C. Conley. We will give a survey of some recent results on homeomorphisms and diffeomorphisms of surfaces and will explain the links with the improved version of Brouwer’s theorem. 77 98 3 10.4171/022-3/4 http://www.ems-ph.org/doi/10.4171/022-3/4

Michael Shub University of Toronto, TORONTO, ONTARIO, CANADA Dynamical systems and ergodic theory General In the first part of this paper we study dynamical systems from the point of view of algebraic topology. What features of all dynamical systems are reflected by their actions on the homology of the phase space? In the second part we study recent progress on the conjecture that most partially hyperbolic dynamical systems which preserve a smooth invariant measure are ergodic, and we survey the known examples. Then we speculate on ways these results may be extended to the statistical study of more general dynamical systems. Finally, in the third part, we study two special classes of dynamical systems, the structurally stable and the affine. In the first case we study the relation of structural stability to entropy, and in the second we study stable ergodicity in the homogeneous space context. 99 120 3 10.4171/022-3/5 http://www.ems-ph.org/doi/10.4171/022-3/5

Anton Zorich Université de Rennes I, RENNES CEDEX, FRANCE Manifolds and cell complexes Functions of a complex variable Several complex variables and analytic spaces Dynamical systems and ergodic theory Various problems of geometry, topology and dynamical systems on surfaces as well as some questions concerning one-dimensional dynamical systems lead to the study of closed surfaces endowed with a flat metric with several cone-type singularities. In an important particular case, when the flat metric has trivial holonomy, the corresponding flat surfaces are naturally organized into families which appear to be isomorphic to moduli spaces of holomorphic oneforms. One can obtain much information about the geometry and dynamics of an individual flat surface by studying both its orbit under the Teichmüller geodesic flow and under the linear group action on the corresponding moduli space. We apply this general principle to the study of generic geodesics and to counting of closed geodesics on a flat surface. 121 146 3 10.4171/022-3/6 http://www.ems-ph.org/doi/10.4171/022-3/6

Stefano Bianchini SISSA-ISAS, TRIESTE, ITALY Partial differential equations General We study two problems related to hyperbolic systems with a dissipative source. In the first part, we consider the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in m space dimensions, under a coupling condition among hyperbolic and dissipative part known as the Shizuta–Kawashima condition. Under the assumption of small initial data, these solutions approach constant equilibrium state in the Lp-norm at a rate O(t-m/2(1-1/p)), as t → ∞, for p ∈ [min {m, 2}, ∞]. The main tool is given by a detailed analysis of the Green function for the linearized problem. If the space dimension m = 1 or the system is rotational invariant, it is possible to give an explicit form to the main terms in the Green kernel. In the second part, we consider the hyperbolic limit of special systems of balance laws: this means to study the limit of the solution to a system of balance laws under the rescaling (t, x) ↦ (t/ε, x/ε), as ε → 0. For some special dissipative systems in one space dimension, it is possible to prove the existence of the limit and to identify it as a solution to a system of conservation laws. 147 155 3 10.4171/022-3/7 http://www.ems-ph.org/doi/10.4171/022-3/7

Patrick Gérard Université Paris-Sud, ORSAY CX, FRANCE Partial differential equations General We survey recent wellposedness and illposedness results for the Cauchy problem for nonlinear Schrödinger equations in inhomogeneous media such as Riemannian manifolds or domains of the Euclidean space, trying to emphasize the influence of the geometry. The main tools are multilinear Strichartz estimates for the Schrödinger group. 156 182 3 10.4171/022-3/8 http://www.ems-ph.org/doi/10.4171/022-3/8

François Golse École Polytechnique, PARIS, FRANCE Statistical mechanics, structure of matter Dynamical systems and ergodic theory General Consider the motion of a single point particle bouncing in a fixed system of spherical obstacles. It is assumed that collisions are perfectly elastic, and that the particle is subject to no external force between collisions, so that the particle moves at constant speed. This type of dynamical system belongs to the class of dispersing billiards, and is referred to as a “Lorentz gas”. A Lorentz gas is called periodic when the obstacle centers form a lattice. Assuming that the initial position and direction of the particle are distributed under some smooth density with respect to the uniform measure, one seeks the evolution of that density under the dynamics defined by the particle motion in some large scale limit for which the number of collisions per unit of time is of the order of unity. This scaling limit is known as “the Boltzmann-Grad limit”, and is the regime of validity for the Boltzmann equation in the kinetic theory of gases. Whether this evolution is governed in such a limit by a PDE analogous to the Boltzmann equation is a natural question, and the topic of this paper. 183 201 3 10.4171/022-3/9 http://www.ems-ph.org/doi/10.4171/022-3/9

Matthew Gursky University of Notre Dame, NOTRE DAME, UNITED STATES Global analysis, analysis on manifolds Differential geometry General We describe several “uniformizing” problems in conformal geometry, all of which can be formulated as problems of existence for solutions of certain elliptic partial differential equations. For the sake of exposition we divide the discussion according to the type of PDE, beginning with semilinear equations related to the scalar curvature, then higher order equations arising from the Q-curvature, and finally fully nonlinear equations. 203 212 3 10.4171/022-3/10 http://www.ems-ph.org/doi/10.4171/022-3/10

Kenji Nishihara Waseda University, TOKYO, JAPAN Partial differential equations Calculus of variations and optimal control; optimization General In this article we discuss some recent results on the large-time behavior of solutions of Hamilton–Jacobi equations as well as some ideas and observations behind them and historical remarks concerning them. 213 227 3 10.4171/022-3/11 http://www.ems-ph.org/doi/10.4171/022-3/11

Mario Pulvirenti Università di Roma La Sapienza, ROMA, ITALY Partial differential equations Statistical mechanics, structure of matter General In this contribution we illustrate the delicate transition from the microscopic description of a particle system, given in terms of fundamental equations as the Newton or the Schrödinger equation, to the reduced kinetic picture, given in terms of the Boltzmann and Landau equations which are obtained under suitable scaling limits. Special emphasis is given to the so called weak-coupling limit. The content of the lecture is mostly devoted to the very many open problems, rather than to the few known results. 229 256 3 10.4171/022-3/12 http://www.ems-ph.org/doi/10.4171/022-3/12

Ovidiu Savin Columbia University, NEW YORK, UNITED STATES Partial differential equations General We discuss a conjecture of De Giorgi concerning the one dimensional symmetry of bounded, monotone in one direction, solutions of semilinear elliptic equations of the form ∆u = W'(u) in all ℝn. 257 265 3 10.4171/022-3/13 http://www.ems-ph.org/doi/10.4171/022-3/13

Sylvia Serfaty Université Pierre et Marie Curie (Paris VI), PARIS CEDEX 05, FRANCE Partial differential equations General We review some mathematical results on the Ginzburg–Landau model with and without magnetic field. The Ginzburg–Landau energy is the standard model for superconductivity, able to predict the existence of vortices (which are quantized, topological defects) in certain regimes of the applied magnetic field. We focus particularly on deriving limiting (or reduced) energies for the Ginzburg–Landau energy functional, depending on the various parameter regimes, in the spirit of Γ-convergence. These passages to the limit allow to perform a sort of dimension-reduction and to deduce a rather complete characterization of the behavior of vortices for energy-minimizers, in agreement with the physics results. We also describe the behavior of energy critical points, the stability of the solutions, the motion of vortices for solutions of the gradient-flow of the Ginzburg–Landau energy, and show how they are also governed by those of the limiting energies. 267 290 3 10.4171/022-3/14 http://www.ems-ph.org/doi/10.4171/022-3/14

Neil Trudinger Australian National University, CANBERRA, AUSTRALIA Partial differential equations Differential geometry General In conjunction with applications to optimal transportation and conformal geometry, there has been considerable research activity in recent years devoted to fully nonlinear, elliptic second order partial differential equations of a particular form, given by functions of the Hessian plus a lower order matrix function. Regularity is determined through the behaviour of this function with respect to the gradient variables. We present a selection of second derivative estimates and indicate briefly their application to optimal transportation and conformal deformation of Riemannian manifolds. 291 301 3 10.4171/022-3/15 http://www.ems-ph.org/doi/10.4171/022-3/15

Luis Vega Universidad del Pais Vasco, BILBAO, SPAIN Partial differential equations General I will review some recent work done in collaboration with C. E. Kenig, G. Ponce and C. Rolvung on a general method to solve locally in time the initial value problem for non-linear Schrödinger equations under some natural hypotheses of decay and regularity of the coefficients. Also some non-trapping conditions of the solutions of the hamiltonian flow associated to the initial data is needed. We will not assume ellipticity on the matrix of the leading order coefficients but just a non-degeneracy condition. The method is based on energy estimates which can be performed thanks to the construction of an integrating factor. This construction is of independent interest and relies on the analysis of some new pseudo-differential operators. 303 319 3 10.4171/022-3/16 http://www.ems-ph.org/doi/10.4171/022-3/16

Juan José Velazques Universidad Complutense de Madrid, MADRID, SPAIN Partial differential equations Biology and other natural sciences General This paper reviews several mathematical results for partial differential equations modelling chemotaxis. In particular, questions like singularity formation for the Keller–Segel model and continuation of the solutions beyond the blow-up time will be discussed. Some of the open problems that remain for the Keller–Segel model as well as some new mathematical problems arising in the study of chemotaxis problems will be discussed. 321 338 3 10.4171/022-3/17 http://www.ems-ph.org/doi/10.4171/022-3/17

Alberto Cattaneo Universität Zürich, ZÜRICH, SWITZERLAND Quantum theory Geometry Differential geometry Global analysis, analysis on manifolds This note describes the functional-integral quantization of two-dimensional topological field theories together with applications to problems in deformation quantization of Poisson manifolds and reduction of certain submanifolds. A brief introduction to smooth graded manifolds and to the Batalin–Vilkovisky formalism is included. 339 365 3 10.4171/022-3/18 http://www.ems-ph.org/doi/10.4171/022-3/18

Bernard Derrida Ecole Normale Superieure, PARIS CEDEX 05, FRANCE Probability theory and stochastic processes Statistical mechanics, structure of matter General Exclusion processes describe a gas of particles on a lattice with hard core repulsion. When such a lattice gas is maintained in contact with two reservoirs at unequal densities, or driven by an external field, it exhibits a non-equilibrium steady state. In one dimension, a number of properties of this steady state can be calculated exactly using a matrix ansatz. This talk gives a short review on results obtained recently by this matrix ansatz approach. 367 382 3 10.4171/022-3/19 http://www.ems-ph.org/doi/10.4171/022-3/19

Jean Michel Maillet École Normale Supérieure de Lyon, LYON CEDEX 07, FRANCE Quantum theory Statistical mechanics, structure of matter Group theory and generalizations Dynamical systems and ergodic theory We review recent progress in the computation of correlation functions of the XXZ spin-1/2 chain. We describe both finite and infinite chain results. Long distance asymptotic behavior is discussed. Our method is based on the resolution of the quantum inverse scattering problem in the framework of the algebraic Bethe ansatz. 383 408 3 10.4171/022-3/20 http://www.ems-ph.org/doi/10.4171/022-3/20

Marcos Mariño Instituto Superior Técnico, LISBOA, PORTUGAL Quantum theory General In this talk I summarize recent progress in the theory of Gromov–Witten invariants from topological string theory and string dualities. On the one hand, large N dualities have led to the theory of the topological vertex, which solves Gromov–Witten theory to all genera on toric, noncompact Calabi–Yau threefolds. On the other hand, heterotic/type II duality and the holomorphic anomaly equations can be used to analyze Gromov–Witten theory in some simple compact examples. I sketch the physical ideas behind these results and connect the results obtained in physics with the results obtained in algebraic geometry. 409 419 3 10.4171/022-3/21 http://www.ems-ph.org/doi/10.4171/022-3/21

Igor Rodnianski Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES Relativity and gravitational theory Partial differential equations General The paper revisits some of the classical and recent results on the Cauchy problem in General Relativity. Special emphasis is put on the problems concerning existence of a Cauchy development, break-down criteria and stability. The author would like to make a disclaimer that despite its general title the paper is not intended as a comprehensible survey. Due to the space-time constraints many remarkable results and developments are either mentioned briefly or not discussed at all. Most notably this concerns various work on the Einstein equations with matter and symmetry reduced problems. 421 442 3 10.4171/022-3/22 http://www.ems-ph.org/doi/10.4171/022-3/22

Christoph Schweigert Universität Hamburg, HAMBURG, GERMANY Jürgen Fuchs Karlstads Universitet, KARLSTAD, SWEDEN Quantum theory Category theory; homological algebra General A modular tensor category provides the appropriate data for the construction of a threedimensional topological field theory. We describe the following analogue for two-dimensional conformal field theories: a 2-category whose objects are symmetric special Frobenius algebras in a modular tensor category and whose morphisms are categories of bimodules. This 2-category provides sufficient ingredients for constructing all correlation functions of a two-dimensional rational conformal field theory. The bimodules have the physical interpretation of chiral data, boundary conditions, and topological defect lines of this theory. 443 458 3 10.4171/022-3/23 http://www.ems-ph.org/doi/10.4171/022-3/23

Avy Soffer Rutgers University, PISCATAWAY, UNITED STATES Partial differential equations General A survey of results and problems of soliton dynamics in dispersive and hyperbolic nonlinear PDE’s and the related spectral and scattering theory. I focus on the problem of large time behavior of the nonlinear Schrödinger equation, with both solitary and radiative waves appearing in the solution. The equations are nonintegrable in general and in arbitrary dimension. I will formulate the main conjectures relevant to soliton dynamics. 459 471 3 10.4171/022-3/24 http://www.ems-ph.org/doi/10.4171/022-3/24

Cédric Villani École Normale Supérieure de Lyon, LYON CEDEX 07, FRANCE Partial differential equations Fluid mechanics General In many problems coming from mathematical physics, the association of a degenerate diffusion operator with a conservative operator may lead to dissipation in all variables and convergence to equilibrium. One can draw an analogy with the well-studied phenomenon of hypoellipticity in regularity theory, and actually both phenomena have been studied together. Now a distinctive theory of “hypocoercivity” is starting to emerge, with already some striking results, and several challenging open problems. 473 498 3 10.4171/022-3/25 http://www.ems-ph.org/doi/10.4171/022-3/25

Anton Bovier Angewandte Analysis und Stochastik, BERLIN, GERMANY Probability theory and stochastic processes Statistical mechanics, structure of matter General Metastability is an ubiquitous phenomenon of the dynamical behaviour of complex systems. In this talk, I describe recent attempts towards a model-independent approach to metastability in the context of reversible Markov processes. I will present an outline of a general theory, based on careful use of potential theoretic ideas and indicate a number of concrete examples where this theory was used very successfully. I will also indicate some challenges for future work. 499 518 3 10.4171/022-3/26 http://www.ems-ph.org/doi/10.4171/022-3/26

Raphaël Cerf Université Paris-Sud, ORSAY CEDEX, FRANCE Statistical mechanics, structure of matter General One of the fundamental goals of statistical mechanics is to understand the macroscopic effects induced by the random forces acting at the microscopic level. Some satisfactory results are now available for the Ising model at equilibrium in the phase coexistence regime in any dimension: it is rigorously proved that the most likely shapes of the macroscopic droplets of one pure phase floating in the other pure phase are close to the Wulff crystal of the model. However, the dynamical processes leading to the emergence of a droplet are far from being understood. We formulate a classical conjecture: the scaling limit of the Glauber microscopic dynamics should be an anisotropic motion by mean curvature. 519 533 3 10.4171/022-3/27 http://www.ems-ph.org/doi/10.4171/022-3/27

Amir Dembo Stanford University, STANFORD, UNITED STATES Probability theory and stochastic processes Measure and integration Statistical mechanics, structure of matter General We review recent advances in the study of the fractal nature of certain random sets, the key to which is a multi-scale truncated second moment method. We focus on some of the fine properties of the sample path of the most basic stochastic processes such as the simple random walk and the Brownian motion. As we shall see, probability on trees inspires many of our proofs, with trees used to model the relevant correlation structure. Along the way we also mention a few open problems. 535 558 3 10.4171/022-3/28 http://www.ems-ph.org/doi/10.4171/022-3/28

Peter Donnelly University of Oxford, OXFORD, UNITED KINGDOM Biology and other natural sciences Numerical analysis General The completion of the human and other genome projects, and the ongoing development of high-throughput experimental methods for measuring genetic variation, have dramatically changed the scale of information available and the nature of the questions which can now be asked in modern biomedical genetics. Although there is a long history of mathematical modelling in genetics, these developments offer exciting new opportunities and challenges for the mathematical sciences. We focus here on the challenges within human population genetics, in which data document molecular genetic variation between different people. The explosion of data on human variation allows us to study aspects of the underlying evolutionary processes and the molecular mechanisms behind them; the patterns of genetic variation in different geographical regions and the ancestral histories of human populations; and the genetic basis of common human diseases. In each case, sophisticated mathematical, statistical, and computational tools are needed to unravel much of the information in the data, with many of the best methods combining complex stochastic modelling and modern computationally-intensive statistical methods. But the rewards are great: key pieces of scientific knowledge simply would not have been available by other means. 559 574 3 10.4171/022-3/29 http://www.ems-ph.org/doi/10.4171/022-3/29

K. David Elworthy University of Warwick, COVENTRY, UNITED KINGDOM Xue-Mei Li Loughborough University, LOUGHBOROUGH, UNITED KINGDOM Global analysis, analysis on manifolds Differential geometry Probability theory and stochastic processes General An approach to analysis on path spaces of Riemannian manifolds is described. The spaces are furnished with ‘Brownian motion’ measure which lies on continuous paths, though differentiation is restricted to directions given by tangent paths of finite energy. An introduction describes the background for paths on ℝm and Malliavin calculus. For manifold valued paths the approach is to use ‘Itô’ maps of suitable stochastic differential equations as charts. ‘Suitability’ involves the connection determined by the stochastic differential equation. Some fundamental open problems concerning the calculus and the resulting ‘Laplacian’ are described. A theory for more general diffusion measures is also briefly indicated. The same method is applied as an approach to getting over the fundamental difficulty of defining exterior differentiation as a closed operator, with success for one and two forms leading to a Hodge–Kodaira operator and decomposition for such forms. Finally there is a brief description of some related results for loop spaces. 575 594 3 10.4171/022-3/30 http://www.ems-ph.org/doi/10.4171/022-3/30

Jianqing Fan and Financial Engineering, PRINCETON, UNITED STATES Runze Li The Pennsylvania State University, UNIVERSITY PARK, UNITED STATES Statistics General Technological innovations have revolutionized the process of scientific research and knowledge discovery. The availability of massive data and challenges from frontiers of research and development have reshaped statistical thinking, data analysis and theoretical studies. The challenges of high-dimensionality arise in diverse fields of sciences and the humanities, ranging from computational biology and health studies to financial engineering and risk management. In all of these fields, variable selection and feature extraction are crucial for knowledge discovery. We first give a comprehensive overview of statistical challenges with high dimensionality in these diverse disciplines. We then approach the problem of variable selection and feature extraction using a unified framework: penalized likelihood methods. Issues relevant to the choice of penalty functions are addressed. We demonstrate that for a host of statistical problems, as long as the dimensionality is not excessively large, we can estimate the model parameters as well as if the best model is known in advance. The persistence property in risk minimization is also addressed. The applicability of such a theory and method to diverse statistical problems is demonstrated. Other related problems with high-dimensionality are also discussed. 595 622 3 10.4171/022-3/31 http://www.ems-ph.org/doi/10.4171/022-3/31

Alice Guionnet Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES Linear and multilinear algebra; matrix theory Combinatorics General We review recent developments in random matrix theory related with the enumeration of connected oriented graphs called maps. In particular, we show that the long standing use of matrix integrals in physics to tackle such issues can be made rigorous and discuss some applications. This talk is based on joint works with E. Maurel-Segala and O. Zeitouni. 623 636 3 10.4171/022-3/32 http://www.ems-ph.org/doi/10.4171/022-3/32

Steven Lalley University of Chicago, CHICAGO, UNITED STATES Combinatorics Group theory and generalizations General Various stochastic processes on nonamenable graphs and manifolds of exponential volume growth exhibit phases that do not occur in the corresponding processes on amenable graphs. Examples include: (1) branching diffusion and random walk on hyperbolic space, which for intermediate branching rates may survive globally but not locally; (2) contact processes on homogeneous trees, which likewise can survive globally while dying out locally; and (3) percolation on Cayley graphs of nonamenable groups, where for certain parameter values infinitely many infinite percolation clusters may coincide. This article surveys some of what is known about the intermediate phases and the upper phase transitions for these processes. 637 647 3 10.4171/022-3/33 http://www.ems-ph.org/doi/10.4171/022-3/33

Yves Le Jan Université Paris Sud, ORSAY CEDEX, FRANCE Probability theory and stochastic processes Fluid mechanics General Flows of random coalescing maps or flows of random transition probabilities can arise from simple stochastic differential equations when Ito’s theory of strong solutions ceases to apply. 649 667 3 10.4171/022-3/34 http://www.ems-ph.org/doi/10.4171/022-3/34

Peter McCullagh University of Chicago, CHICAGO, UNITED STATES Jie Yang University of Chicago, CHICAGO, UNITED STATES Statistics Computer science General Two families of stochastic processes are constructed that are intended for use in classification problems where the aim is to classify units or specimens or species on the basis of measured features. The first model is an exchangeable cluster process generated by a standard Dirichlet allocation scheme. The set of classes is not pre-specified, so a new unit may be assigned to a previously unobserved class. The second model, which is more flexible, uses a marked point process as the mechanism generating the units or events, each with its associated class and feature. The conditional distribution given the superposition process is obtained in closed form for one particular marked point process. This distribution determines the conditional class probabilities, and thus the prediction rule for subsequent units. 669 686 3 10.4171/022-3/35 http://www.ems-ph.org/doi/10.4171/022-3/35

Andrei Okounkov Princeton University, PRINCETON, UNITED STATES Quantum theory Algebraic geometry General Two families of stochastic processes are constructed that are intended for We summarize the connection between random partitions and N = 2 supersymmetric gauge theories in 4 dimensions and indicate how this relation extends to higher dimensions. 687 711 3 10.4171/022-3/36 http://www.ems-ph.org/doi/10.4171/022-3/36

Dominique Picard Université Paris 7, Denis Diderot, PARIS CEDEX 05, FRANCE Gérard Kerkyacharian Université Pierre et Marie Curie, PARIS, FRANCE Statistics General We consider the problem of recovering a function f when we receive a blurred (by a linear operator) and noisy version: Yε = Kf + εẆ . We will have as guides 2 famous examples of such inverse problems: the deconvolution and the Wicksell problem. The direct problem (K is the identity) isolates the denoising operation. It cannot be solved unless accepting to estimate a smoothed version of f: for instance, if f has an expansion on a basis, this smoothing might correspond to stopping the expansion at some stage m. Then a crucial problem lies in finding an equilibrium for m, considering the fact that for m large, the difference between f and its smoothed version is small, whereas the random effect introduces an error which is increasing with m. In the true inverse problem, in addition to denoising, we have to ‘inverse the operator’ K, an operation which not only creates the usual difficulties, but also introduces the necessity to control the additional instability due to the inversion of the random noise. Our purpose here is to emphasize the fact that in such a problem there generally exists a basis which is fully adapted to the problem, where for instance the inversion remains very stable: this is the singular value decomposition basis. On the other hand, the SVD basis might be difficult to determine and to numerically manipulate. It also might not be appropriate for the accurate description of the solution with a small number of parameters. Moreover, in many practical situations the signal provides inhomogeneous regularity, and its local features are especially interesting to recover. In such cases, other bases (in particular, localised bases such as wavelet bases) may be much more appropriate to give a good representation of the object at hand. Our approach here will be to produce estimation procedures keeping the advantages of a localisation properly without loosing the stability and computability of SVD decompositions. We will especially consider two cases. In the first one (which is the case of the deconvolution example) we show that a fairly simple algorithm (WAVE-VD), using an appropriate thresholding technique performed on a standard wavelet system, enables us to estimate the object with rates which are almost optimal up to logarithmic factors for any Lp loss function and on the whole range of Besov spaces. In the second case (which is the case of the Wicksell example where the SVD basis lies in the range of Jacobi polynomials) we prove that a similar algorithm (NEED-VD) can be performed provided one replaces the standard wavelet system by a second generation wavelet-type basis: the needlets. We use here the construction (essentially following the work of Petrushev and co-authors) of a localised frame linked with a prescribed basis (here Jacobi polynomials) using a Littlewood–Paley decomposition combined with a cubature formula. Section 5 describes the direct case (K = I). It has its own interest and will act as a guide for understanding the ‘true’ inverse models for a reader who is not familiar with nonparametric statistical estimation. It can be read first. Section 1 introduces the general inverse problem and describes the examples of deconvolution and Wicksell’s problem. A review of standard methods is given with a special focus on SVD methods. Section 2 describes the WAVE-VD procedure. Section 3 and 4 give a description of the needlets constructions and the performances of the NEED-VD procedure. 713 739 3 10.4171/022-3/37 http://www.ems-ph.org/doi/10.4171/022-3/37

Wendelin Werner ETH Zürich, ZÜRICH, SWITZERLAND Probability theory and stochastic processes Functions of a complex variable Statistical mechanics, structure of matter General We give an introduction to some aspects of recent results concerning conformally invariant measures. We focus in this note on the conformal restriction properties of some measures on curves and loops in the plane, and see that these properties in fact almost characterize the measures and allow to classify them. For example, there basically exists a unique measure μ on the set of self-avoiding loops in the plane, such that for any two conformally equivalent domains D and D', the restrictions of μ to the set of loops remaining in D and D' are conformally equivalent. This enables to show that a priori different discrete models define the same curves in the scaling limit and exhibit some surprising symmetries. It gives also a way to tie links between these concrete measures on curves and conformal field theory. Important roles in this theory are played by Brownian loops and by the Schramm–Loewner Evolutions (SLE). Most of the results described in this paper were derived in joint work with Greg Lawler, and Oded Schramm. 741 762 3 10.4171/022-3/38 http://www.ems-ph.org/doi/10.4171/022-3/38

Alexander Barvinok University of Michigan, ANN ARBOR, UNITED STATES Combinatorics Number theory Convex and discrete geometry Computer science Motivated by the formula for the sum of the geometric series, we consider various classes of sets S ⊂ ℤd of integer points for which an a priori “long” Laurent series or polynomial Σm∈S xm can be written as a “short” rational function f(S; x). Examples include the sets of integer points in rational polyhedra, integer semigroups, and Hilbert bases of rational cones, among others. We discuss applications to efficient counting and optimization and open questions. This enables to show that a priori different discrete models define the same curves in the scaling limit and exhibit some surprising symmetries. It gives also a way to tie links between these concrete measures on curves and conformal field theory. Important roles in this theory are played by Brownian loops and by the Schramm–Loewner Evolutions (SLE). Most of the results described in this paper were derived in joint work with Greg Lawler, and Oded Schramm. 763 787 3 10.4171/022-3/39 http://www.ems-ph.org/doi/10.4171/022-3/39

Mireille Bousquet-Mélou Université de Bordeaux I, TALENCE CEDEX, FRANCE Combinatorics Computer science General Let A be a class of objects, equipped with an integer size such that for all n the number an of objects of size n is finite. We are interested in the case where the generating function Σn an tn is rational, or more generally algebraic. This property has a practical interest, since one can usually say a lot on the numbers an, but also a combinatorial one: the rational or algebraic nature of the generating function suggests that the objects have a (possibly hidden) structure, similar to the linear structure of words in the rational case, and to the branching structure of trees in the algebraic case. We describe and illustrate this combinatorial intuition, and discuss its validity. While it seems to be satisfactory in the rational case, it is probably incomplete in the algebraic one. We conclude with open questions. This enables to show that a priori different discrete models define the same curves in the scaling limit and exhibit some surprising symmetries. It gives also a way to tie links between these concrete measures on curves and conformal field theory. Important roles in this theory are played by Brownian loops and by the Schramm–Loewner Evolutions (SLE). Most of the results described in this paper were derived in joint work with Greg Lawler, and Oded Schramm. 789 826 3 10.4171/022-3/40 http://www.ems-ph.org/doi/10.4171/022-3/40

Jim Geelen Faculty of Mathematics, WATERLOO, ONTARIO, CANADA Bert Gerards Centrum voor Wiskunde en Informatica, AMSTERDAM, NETHERLANDS Geoff Whittle Victoria University, WELLINGTON, NEW ZEALAND Combinatorics General We survey recent work that is aimed at generalizing the results and techniques of the Graph Minors Project of Robertson and Seymour to matrices and matroids. 827 842 3 10.4171/022-3/41 http://www.ems-ph.org/doi/10.4171/022-3/41

Mark Haiman University of California, BERKELEY, UNITED STATES Special functions Combinatorics General In the first part of this article we review the general theory of Cherednik algebras and non-symmetric Macdonald polynomials, including a formulation and proof of the fundamental duality theorem in its proper general context. In the last section we summarize some of the combinatorial results in this area which we have recently obtained in collaboration with J. Haglund and N. Loehr. 843 872 3 10.4171/022-3/42 http://www.ems-ph.org/doi/10.4171/022-3/42

Jeong Han Kim Theory Group, REDMOND, UNITED STATES Combinatorics General In the random graph G(n, p) with pn bounded, the degrees of the vertices are almost i.i.d. Poisson random variables with mean λ := p(n − 1). Motivated by this fact, we introduce the Poisson cloning model GPC(n, p) for random graphs in which the degrees are i.i.d. Poisson random variables with mean λ. We first establish a theorem that shows that the new model is equivalent to the classical model G(n, p) in an asymptotic sense. Next, we introduce a useful algorithm to generate the random graph GPC(n, p), called the cut-off line algorithm. Then GPC(n, p) equipped with the cut-off line algorithm enables us to very precisely analyze the sizes of the largest component and the t-core of G(n, p). This new approach for the problems yields not only elegant proofs but also improved bounds that are essentially best possible. We also consider the Poisson cloning model for random hypergraphs and the G(n, p)-core problem for random hypergraphs. 873 897 3 10.4171/022-3/43 http://www.ems-ph.org/doi/10.4171/022-3/43

Tomasz Łuczak Adam Mickiewicz University, POZNAN, POLAND Combinatorics General For the last ten years the theory of random structures has been one of the most rapidly evolving fields of discrete mathematics. The existence of sparse structures with good ‘global’ properties has been vital for many applications in mathematics and computer science, and studying properties of such objects led to many challenging mathematical problems. In the paper we report on recent progress on this subject related to some variants of Szemerédi’s Regularity Lemma. 899 909 3 10.4171/022-3/44 http://www.ems-ph.org/doi/10.4171/022-3/44

Imre Ruzsa Hungarian Academy of Sciences, BUDAPEST, HUNGARY Number theory Combinatorics General We meditate on the following questions. What are the best analogs of measure and dimension for discrete sets? How should a discrete analogue of the Brunn–Minkowski inequality look like? And back to the continuous case, are we happy with the usual concepts of measure and dimension for studying the addition of sets? 911 930 3 10.4171/022-3/45 http://www.ems-ph.org/doi/10.4171/022-3/45

Domingo Gómez-Pérez Universidad de Cantabria, SANTANDER, SPAIN Convex and discrete geometry General We give a self-contained introduction to the theory of secondary polytopes and geometric bistellar flips in triangulations of polytopes and point sets, as well as a review of some of the known results and connections to algebraic geometry, topological combinatorics, and other areas. As a new result, we announce the construction of a point set in general position with a disconnected space of triangulations. This shows, for the first time, that the poset of strict polyhedral subdivisions of a point set is not always connected. 931 962 3 10.4171/022-3/46 http://www.ems-ph.org/doi/10.4171/022-3/46

Robin Thomas Georgia Institute of Technology, ATLANTA, UNITED STATES Combinatorics Computer science Statistical mechanics, structure of matter General An orientation of a graph G is Pfaffian if every even cycle C such that G\V(C) has a perfect matching has an odd number of edges directed in either direction of the cycle. The significance of Pfaffian orientations is that if a graph has one, then the number of perfect matchings (a.k.a. the dimer problem) can be computed in polynomial time. The question of which bipartite graphs have Pfaffian orientations is equivalent to many other problems of interest, such as a permanent problem of Pólya, the even directed cycle problem, or the sign-nonsingular matrix problem for square matrices. These problems are now reasonably well-understood. On the other hand, it is not known how to efficiently test if a general graph is Pfaffian, but there are some interesting connections with crossing numbers and signs of edgecolorings of regular graphs. 963 984 3 10.4171/022-3/47 http://www.ems-ph.org/doi/10.4171/022-3/47

Manindra Agrawal Indian Institute of Technology, KANPUR, INDIA Computer science General We study the problem of expressing permanents of matrices as determinants of (possibly larger) matrices. This problem has close connections to the complexity of arithmetic computations: the complexities of computing permanent and determinant roughly correspond to arithmetic versions of the classes NP and P respectively. We survey known results about their relative complexity and describe two recently developed approaches that might lead to a proof of the conjecture that the permanent can only be expressed as the determinant of exponential-sized matrices. 985 997 3 10.4171/022-3/48 http://www.ems-ph.org/doi/10.4171/022-3/48

Alexander Holevo Steklov Mathematical Institute, MOSCOW, GSP-1, RUSSIAN FEDERATION Information and communication, circuits Quantum theory General In this lecture we survey the present status of the additivity problem for the classical capacity and related characteristics of quantum channels – one of the most profound mathematical problems of quantum information theory. 999 1018 3 10.4171/022-3/49 http://www.ems-ph.org/doi/10.4171/022-3/49

Jon Kleinberg Cornell University, ITHACA, UNITED STATES Computer science Combinatorics Game theory, economics, social and behavioral sciences General The study of complex networks has emerged over the past several years as a theme spanning many disciplines, ranging from mathematics and computer science to the social and biological sciences. A significant amount of recent work in this area has focused on the development of random graph models that capture some of the qualitative properties observed in large-scale network data; such models have the potential to help us reason, at a general level, about the ways in which real-world networks are organized. We survey one particular line of network research, concerned with small-world phenomena and decentralized search algorithms, that illustrates this style of analysis. We begin by describing a well-known experiment that provided the first empirical basis for the “six degrees of separation” phenomenon in social networks; we then discuss some probabilistic network models motivated by this work, illustrating how these models lead to novel algorithmic and graph-theoretic questions, and how they are supported by recent empirical studies of large social networks. 1019 1044 3 10.4171/022-3/50 http://www.ems-ph.org/doi/10.4171/022-3/50

Omer Reingold Weizmann Institute of Science, REHOVOT, ISRAEL Combinatorics Computer science General This presentation is aimed to communicate a recently found deterministic algorithm for determining connectivity in undirected graphs [40]. This algorithm uses the minimal amount of memory possible, up to a constant factor. Specifically, the algorithm’s memory is comparable to that needed to store the name of a single vertex of the graph (i.e., it is logarithmic in the size of the graph). Our algorithm also implies a deterministic, short (i.e. of polynomial length), universal sequence of steps which explores all the edges of every regular undirected graph. Such a sequence will get one out of every maze, and through the streets of every city. More formally we give universal exploration sequences for arbitrary graphs and universal traversal sequences for graphs with some natural restriction on their labelling. Both sequences are constructible with logarithmic memory and are thus only polynomially long. To obtain this algorithm, we give a method to transform (using small memory), an arbitrary connected undirected graph into an expander graph (which is a sparse but highly connected graph). 1045 1069 3 10.4171/022-3/51 http://www.ems-ph.org/doi/10.4171/022-3/51

Tim Roughgarden Stanford University, STANFORD, UNITED STATES Computer science Operations research, mathematical programming Game theory, economics, social and behavioral sciences General We survey one area of the emerging field of algorithmic game theory: the use of approximation measures to quantify the inefficiency of game-theoretic equilibria. Potential functions, which enable the application of optimization theory to the study of equilibria, have been a versatile and powerful tool in this area. We use potential functions to bound the inefficiency of equilibria in three diverse, natural classes of games: selfish routing networks, resource allocation games, and Shapley network design games. 1071 1094 3 10.4171/022-3/52 http://www.ems-ph.org/doi/10.4171/022-3/52

Ronitt Rubinfeld Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES Computer science General Sublinear time algorithms represent a new paradigm in computing, where an algorithm must give some sort of an answer after inspecting only a very small portion of the input. We discuss the sorts of answers that one might be able to achieve in this new setting. 1095 1110 3 10.4171/022-3/53 http://www.ems-ph.org/doi/10.4171/022-3/53

Luca Trevisan University of California, BERKELEY, UNITED STATES Computer science Combinatorics General In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or unknown. In computer science, probabilistic algorithms are sometimes simpler and more efficient than the best known deterministic algorithms for the same problem. Despite this evidence for the power of random choices, the computational theory of pseudorandomness shows that, under certain complexity-theoretic assumptions, every probabilistic algorithm has an efficient deterministic simulation and a large class of applications of the probabilistic method can be converted into explicit constructions. In this survey paper we describe connections between the conditional “derandomization” results of the computational theory of pseudorandomness and unconditional explicit constructions of certain combinatorial objects such as error-correcting codes and “randomness extractors.” 1111 1136 3 10.4171/022-3/54 http://www.ems-ph.org/doi/10.4171/022-3/54

Pavel Bochev Sandia National Laboratories, ALBUQUERQUE, UNITED STATES Max Gunzburger Florida State University, TALLAHASSEE, UNITED STATES Numerical analysis General Least-squares finite element methods are an attractive class of methods for the numerical solution of partial differential equations. They are motivated by the desire to recover, in general settings, the advantageous features of Rayleigh–Ritz methods such as the avoidance of discrete compatibility conditions and the production of symmetric and positive definite discrete systems. The methods are based on the minimization of convex functionals that are constructed from equation residuals. This paper focuses on theoretical and practical aspects of least-square finite element methods and includes discussions of what issues enter into their construction, analysis, and performance. It also includes a discussion of some open problems. 1137 1162 3 10.4171/022-3/55 http://www.ems-ph.org/doi/10.4171/022-3/55

Zhiming Chen Chinese Academy of Sciences, BEIJING, CHINA Numerical analysis General The adaptive finite element method based on a posteriori error estimates provides a systematic way to refine or coarsen the meshes according to the local a posteriori error estimator on the elements. One of the remarkable properties of the method is that for appropriately designed adaptive finite element procedures, the meshes and the associated numerical complexity are quasi-optimal in the sense that in two space dimensions, the finite element discretization error is proportional to N−1/2 in terms of the energy norm, where N is the number of elements of the underlying mesh. The purpose of this paper is to report some of the recent advances in the a posteriori error analysis and adaptive finite element methods for partial differential equations. Emphases will be paid on an adaptive perfectly matched layer technique for scattering problems and a sharp L1 a posteriori error analysis for nonlinear convection-diffusion problems. 1163 1180 3 10.4171/022-3/56 http://www.ems-ph.org/doi/10.4171/022-3/56

Ricardo Durán Universidad de Buenos Aires, BUENOS AIRES, ARGENTINA Numerical analysis General The finite element method is one of the most frequently used techniques to approximate the solution of partial differential equations. It consists in approximating the unknown solution by functions which are polynomials on each element of a given partition of the domain, made of triangles or quadrilaterals (or their generalizations to higher dimensions). A fundamental problem is to estimate the error between the exact solution u and its computable finite element approximation. In many situations this error can be bounded in terms of the best approximation of u by functions in the finite element space of piecewise polynomial functions. A natural way to estimate this best approximation is by means of the Lagrange interpolation or other similar procedures. Many works have considered the problem of interpolation error estimates. The classical error analysis for interpolations is based on the so-called regularity assumption, which excludes elements with different sizes in each direction (called anisotropic). The goal of this paper is to present a different approach which has been developed by many authors and can be applied to obtain error estimates for several interpolations under more general hypotheses. An important case in which anisotropic elements arise naturally is in the approximation of convection-diffusion problems which present boundary layers. We present some applications to these problems. Finally we consider the finite element approximation of the Stokes equations and present some results for non-conforming methods. 1181 1200 3 10.4171/022-3/57 http://www.ems-ph.org/doi/10.4171/022-3/57

Nira Dyn Tel Aviv University, TEL AVIV, ISRAEL Numerical analysis Approximations and expansions Computer science General Subdivision schemes are efficient computational methods for the design, representation and approximation of surfaces of arbitrary topology in ℝ3. Subdivision schemes generate curves/surfaces from discrete data by repeated refinements. This paper reviews some of the theory of linear stationary subdivision schemes and their applications in geometric modelling. The first part is concerned with “classical” schemes refining control points. The second part reviews linear subdivision schemes refining other objects, such as vectors of Hermite-type data, compact sets in ℝn and nets of curves in ℝ3. Examples of various schemes are presented. 1201 1226 3 10.4171/022-3/58 http://www.ems-ph.org/doi/10.4171/022-3/58

Ido Bright University of Washington, SEATTLE, UNITED STATES Numerical analysis Mechanics of deformable solids General Wave propagation algorithms are a class of high-resolution finite volume methods for solving hyperbolic partial differential equations arising in diverse applications. The development and use of the CLAWPACK software implementing these methods serves as a case study for a more general discussion of mathematical aspects of software development and the need for more reproducibility in computational research. Sample applications discussed include medical applications of shock waves and geophysical fluid dynamics modeling volcanoes and tsunamis. 1227 1253 3 10.4171/022-3/59 http://www.ems-ph.org/doi/10.4171/022-3/59

Yvon Maday Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE Numerical analysis General Numerical approximation of the solution of partial differential equations plays an important role in many areas such as engineering, mechanics, physics, chemistry, biology – for computer-aided design-analysis, computer-aided decision-making or simply better understanding. The fidelity of the simulations with respect to reality is achieved through the combined efforts to derive: (i) better models, (ii) faster numerical algorithm, (iii) more accurate discretization methods and (iv) improved large scale computing resources. In many situations, including optimization and control, the same model, depending on a parameter that is changing, has to be simulated over and over, multiplying by a large factor (up to 100 or 1000) the solution procedure cost of one single simulation. The reduced basis method allows to define a surrogate solution procedure, that, thanks to the complementary design of fidelity certificates on outputs, allows to speed up the computations by two to three orders of magnitude while maintaining a sufficient accuracy. We present here the basics of this approach for linear and non linear elliptic and parabolic PDE’s. 1255 1270 3 10.4171/022-3/60 http://www.ems-ph.org/doi/10.4171/022-3/60

Endre Süli Oxford University, OXFORD, UNITED KINGDOM Numerical analysis General Partial differential equations with nonnegative characteristic form arise in numerous mathematical models of physical phenomena: stochastic analysis, in particular, is a fertile source of equations of this kind. We survey recent developments concerning the finite element approximation of these equations, focusing on three relevant aspects: (a) stability and stabilisation; (b) hp-adaptive algorithms driven by residual-based a posteriori error bounds, capable of automatic variation of the granularity h of the finite element partition and of the local polynomial degree p; (c) complexity-reduction for high-dimensional transport-diffusion problems by stabilised sparse finite element methods. 1271 1297 3 10.4171/022-3/61 http://www.ems-ph.org/doi/10.4171/022-3/61

Vivek Borkar Tata Institute of Fundamental Research, BOMBAY, INDIA Systems theory; control Probability theory and stochastic processes General Results concerning existence and characterization of optimal controls for ergodic control of nondegenerate diffusion processes are described. Extensions to the general ‘controlled martingale problem’ are indicated, which cover in particular degenerate diffusions and some infinite dimensional problems. In conclusion, some related problems and open issues are discussed. 1299 1309 3 10.4171/022-3/62 http://www.ems-ph.org/doi/10.4171/022-3/62

Stephen Boyd Stanford University, STANFORD, UNITED STATES Combinatorics Operations research, mathematical programming General We consider the problem of choosing the edge weights of an undirected graph so as to maximize or minimize some function of the eigenvalues of the associated Laplacian matrix, subject to some constraints on the weights, such as nonnegativity, or a given total value. In many interesting cases this problem is convex, i.e., it involves minimizing a convex function (or maximizing a concave function) over a convex set. This allows us to give simple necessary and sufficient optimality conditions, derive interesting dual problems, find analytical solutions in some cases, and efficiently compute numerical solutions in all cases. In this overview we briefly describe some more specific cases of this general problem, which have been addressed in a series of recent papers. Fastest mixing Markov chain. Find edge transition probabilities that give the fastest mixing (symmetric, discrete-time) Markov chain on the graph. Fastest mixing Markov process. Find the edge transition rates that give the fastest mixing (symmetric, continuous-time) Markov process on the graph. Absolute algebraic connectivity. Find edge weights that maximize the algebraic connectivity of the graph (i.e., the smallest positive eigenvalue of its Laplacian matrix). The optimal value is called the absolute algebraic connectivity by Fiedler. Minimum total effective resistance. Find edge weights that minimize the total effective resistance of the graph. This is same as minimizing the average commute time from any node to any other, in the associated Markov chain. Fastest linear averaging. Find weights in a distributed averaging network that yield fastest convergence. Least steady-state mean-square deviation. Find weights in a distributed averaging network, driven by random noise, that minimizes the steady-state mean-square deviation of the node values. 1311 1319 3 10.4171/022-3/63 http://www.ems-ph.org/doi/10.4171/022-3/63

Oleg Emanouilov (Imanuvilov) Colorado State Univeristy, FORT COLLINS, UNITED STATES Systems theory; control Partial differential equations General In this paper we will discuss recent developments in controllability of evolution equations of fluid mechanics. The control is assumed to be distributed either on a part of the boundary or locally distributed in some subdomain. We will present some ideas of proof of main theorems. Special attention will be paid to the technique based on Carleman estimates. 1321 1338 3 10.4171/022-3/64 http://www.ems-ph.org/doi/10.4171/022-3/64

Arjan van der Schaft Rijksuniversiteit Groningen, GRONINGEN, NETHERLANDS Systems theory; control Dynamical systems and ergodic theory Mechanics of particles and systems General The theory of port-Hamiltonian systems provides a framework for the geometric description of network models of physical systems. It turns out that port-based network models of physical systems immediately lend themselves to a Hamiltonian description. While the usual geometric approach to Hamiltonian systems is based on the canonical symplectic structure of the phase space or on a Poisson structure that is obtained by (symmetry) reduction of the phase space, in the case of a port-Hamiltonian system the geometric structure derives from the interconnection of its sub-systems. This motivates to consider Dirac structures instead of Poisson structures, since this notion enables one to define Hamiltonian systems with algebraic constraints. As a result, any power-conserving interconnection of port-Hamiltonian systems again defines a port-Hamiltonian system. The port-Hamiltonian description offers a systematic framework for analysis, control and simulation of complex physical systems, for lumped-parameter as well as for distributed-parameter models. 1339 1365 3 10.4171/022-3/65 http://www.ems-ph.org/doi/10.4171/022-3/65

Olof Staffans Åbo Akademi University, ÅBO, FINLAND Systems theory; control Operator theory General We begin by discussing linear discrete time-invariant i/s/o (input/state/output) systems that satisfy certain ‘energy’ inequalities. These inequalities involve a quadratic storage function in the state space induced by a positive self-adjoint operator H that may be unbounded and have an unbounded inverse, and also a quadratic supply rate in the combined i/o (input/output) space. The three most commonly studied classes of supply rates are called scattering, impedance, and transmission. Although these three classes resemble each other, we show that there are still significant differences. We then present a new class of s/s (state/signal) systems which have a Hilbert state space and a Kreĭn signal space. The state space is used to store relevant information ı about the past evolution of the system, and the signal space is used to describe interactions with the surrounding world. A s/s system resembles an i/s/o system apart from the fact that inputs and outputs are not separated from each other. By decomposing the signal space into a direct sum of an input space and an output space one gets a standard i/s/o system, provided the decomposition is admissible, and different i/o decompositions lead to different i/o supply rates (for example of scattering, impedance, or transmission type). In the case of non-admissible decompositions we obtain right and left affine representations, both of the s/s system itself, and of the corresponding transfer function. In particular, in the case of a passive system we obtain right and left coprime representations of the generalized transfer functions corresponding to nonadmissible decompositions of the signal space, and we end up with transfer functions which are, e.g., generalized Potapov or Nevanlinna class functions. 1367 1388 3 10.4171/022-3/66 http://www.ems-ph.org/doi/10.4171/022-3/66

Enrique Zuazua Universidad Autónoma de Madrid, MADRID, SPAIN Systems theory; control Partial differential equations General In recent years important progress have been done in the context of numerical approximation of controllability problems for PDEs. It is by now well known that, often, numerical approximation schemes that are stable for solving initial-boundary value problems, develop instabilities when applied to controllability problems. This is due to the presence of spurious high frequency numerical solutions that the control mechanisms are not able to control uniformly as the mesh-size tends to zero. However, the theory is far from being complete. In this article we present some new results in this framework for the wave and the heat equations, which also raise a number of open questions and future directions of research. We first prove that a two-grid method, introduced by R. Glowinski, that is by now well-known to guarantee convergence for the 1 − d wave equation, also converges in the semilinear setting for globally Lipschitz nonlinearities. This result provides a further evidence of the robustness of the two-grid method. We then show that boundary controls for finite-difference space semi-discretizations of the heat equation converge when applied all along the boundary of the domain, a fact that does not hold for wave-like equations. This confirms that the strong irreversibility of the heat equation enhances the control properties of its numerical approximation schemes. This result fails when the control is restricted to some subsets of the boundary because of the lack of unique continuation of some high frequency eigenvectors of the underlying discrete eigenvalue problem. 1389 1417 3 10.4171/022-3/67 http://www.ems-ph.org/doi/10.4171/022-3/67

Russel Caflisch UCLA, LOS ANGELES, UNITED STATES Statistical mechanics, structure of matter General Epitaxy is the growth of a thin film on a substrate in which the crystal properties of the film are inherited from those of the substrate. Because of the wide range of relevant length and time scales, multiscale mathematical models have been developed to describe epitaxial growth. This presentation describes atomistic, island dynamics and continuum models. Island dynamics models are multiscale models that use continuum coarse-graining in the lateral direction, but retain atomistic discreteness in the growth direction. Establishing connections between the various length and time scales in these models is a principal goal of mathematical materials science. Progress towards this goal is described here, including the derivation of surface diffusion, line tension and continuum equations from atomistic, kinetic models. 1419 1432 3 10.4171/022-3/68 http://www.ems-ph.org/doi/10.4171/022-3/68

Emmanuel Candès California Institute of Technology, PASADENA, UNITED STATES Approximations and expansions Statistics Computer science General Conventional wisdom and common practice in acquisition and reconstruction of images from frequency data follow the basic principle of the Nyquist density sampling theory. This principle states that to reconstruct an image, the number of Fourier samples we need to acquire must match the desired resolution of the image, i.e. the number of pixels in the image. This paper surveys an emerging theory which goes by the name of “compressive sampling” or “compressed sensing,” and which says that this conventional wisdom is inaccurate. Perhaps surprisingly, it is possible to reconstruct images or signals of scientific interest accurately and sometimes even exactly from a number of samples which is far smaller than the desired resolution of the image/signal, e.g. the number of pixels in the image. It is believed that compressive sampling has far reaching implications. For example, it suggests the possibility of new data acquisition protocols that translate analog information into digital form with fewer sensors than what was considered necessary. This new sampling theory may come to underlie procedures for sampling and compressing data simultaneously. In this short survey, we provide some of the key mathematical insights underlying this new theory, and explain some of the interactions between compressive sampling and other fields such as statistics, information theory, coding theory, and theoretical computer science. 1433 1452 3 10.4171/022-3/69 http://www.ems-ph.org/doi/10.4171/022-3/69

Vicent Caselles Universitat Pompeu-Fabra, BARCELONA, SPAIN Computer science Numerical analysis Information and communication, circuits General This paper is devoted to the total variation (TV) based approach to image denoising and restoration. The restored image minimizes total variation in the class of images which satisfy the contraints given by the image acquisition model. We compute some explicit solutions of the denoising model which explain some of the features observed in numerical experiments. We also comment on some alternatives recently proposed by Y. Meyer which lead to u + v image decompositions. Finally we propose a total variation approach to image restoration, i.e., deconvolution and denoising, in which the image acquisition model is incorporated as a set of local constraints. 1453 1472 3 10.4171/022-3/70 http://www.ems-ph.org/doi/10.4171/022-3/70

Michael Griebel Universität Bonn, BONN, GERMANY Jan Hamaekers Universität Bonn, BONN, GERMANY Partial differential equations Numerical analysis General We present a direct discretization of the electronic Schrödinger equation. It is based on one-dimensional Meyer wavelets from which we build an anisotropic multiresolution analysis for general particle spaces by a tensor product construction. We restrict these spaces to the case of antisymmetric functions. To obtain finite-dimensional subspaces we first discuss semidiscretization with respect to the scale parameter by means of sparse grids which relies on mixed regularity and decay properties of the electronic wave functions. We then propose different techniques for a discretization with respect to the position parameter. Furthermore we present the results of our numerical experiments using this new generalized sparse grid methods for Schrödinger’s equation. 1473 1506 3 10.4171/022-3/71 http://www.ems-ph.org/doi/10.4171/022-3/71

Claude Le Bris Cité Descartes - Champs sur Marne, MARNE LA VALLÉE CEDEX 2, FRANCE Partial differential equations Calculus of variations and optimal control; optimization Quantum theory Statistical mechanics, structure of matter Molecular simulation is explored from the mathematical viewpoint. The field comprises computational chemistry and molecular dynamics. A variety of mathematical and numerical questions raised is reviewed. Placing the models and the techniques employed for simulation on a firm mathematical ground is a difficult task, which has begun decades ago. The time is right for assessing the field, and the issues and challenges ahead. 1507 1522 3 10.4171/022-3/72 http://www.ems-ph.org/doi/10.4171/022-3/72

Martin Nowak Harvard University, CAMBRIDGE, UNITED STATES Biology and other natural sciences General Cooperation means a donor pays a cost, c, for a recipient to get a benefit b. In evolutionary biology, cost and benefit are measured in terms of fitness. While mutation and selection represent the main ‘forces’ of evolutionary dynamics, cooperation is a fundamental principle that is required for every level of biological organization. Individual cells rely on cooperation among their components. Multi-cellular organisms exist because of cooperation among their cells. Social insects are masters of cooperation. Most aspects of human society are based on mechanisms that promote cooperation. Whenever evolution constructs something entirely new (such as multi-cellularity or human language), cooperation is needed. Evolutionary construction is based on cooperation. I will present five basic principles for the evolution of cooperation, which arise in the theories of kin selection, direct reciprocity, indirect reciprocity, graph selection and group selection. 1523 1540 3 10.4171/022-3/73 http://www.ems-ph.org/doi/10.4171/022-3/73

David Nualart University of Kansas, LAWRENCE, UNITED STATES Probability theory and stochastic processes General Fractional Brownian motion (fBm) is a centered self-similar Gaussian process with stationary increments, which depends on a parameter H ∈ (0, 1) called the Hurst index. In this note we will survey some facts about the stochastic calculus with respect to fBm using a pathwise approach and the techniques of the Malliavin calculus. Some applications in turbulence and finance will be discussed. 1541 1562 3 10.4171/022-3/74 http://www.ems-ph.org/doi/10.4171/022-3/74

Anders Szepessy Kungl. Tekniska Högskolan, STOCKHOLM, SWEDEN Statistical mechanics, structure of matter Numerical analysis General The dynamics of dendritic growth of a crystal in an undercooled melt is determined by macroscopic diffusion-convection of heat and capillary forces acting on length scales compared to the nanometer width of the solid-liquid interface. Its modeling is useful for instance in processing techniques based on casting. The phase field method is widely used to study evolution of such microstructures of phase transformations on a continuum level; it couples the energy equation to a phenomenological Allen–Cahn/Ginzburg–Landau equation modeling the dynamics of an order parameter determining the solid and liquid phases, including also stochastic fluctuations to obtain the qualitative correct result of dendritic side branching. This lecture presents some ideas to derive stochastic phase field models from atomistic formulations by coarse-graining molecular dynamics and kinetic Monte Carlo methods. 1563 1582 3 10.4171/022-3/75 http://www.ems-ph.org/doi/10.4171/022-3/75

Petar Kenderov Bulgarian Academy of Sciences, SOFIA, BULGARIA Mathematics education General Mathematics competitions, together with the people and organizations engaged with them, form an immense and vibrant global network today. This network has many roles. Competitions help identify students with higher abilities in mathematics. They motivate these students to develop their talents and to seek professional realization in science. Competitions have positive impact on education and on educational institutions. Last but not least, a significant part of the classical mathematical heritage known as “Elementary Mathematics” is preserved, kept alive and developed through the network of competitions and competition-related activities. Nevertheless, competitions need to evolve in order to meet the demands of the new century. These and many other items are outlined and discussed in the paper. 1583 1598 3 10.4171/022-3/76 http://www.ems-ph.org/doi/10.4171/022-3/76

Alan Siegel New York University, NEW YORK, UNITED STATES Mathematics education General The technical portion of this paper concerns a videotape classroom study of eighth grade mathematics lessons in Japan, and how methodological design errors led to conclusions that are refuted by the actual video data. We document these errors, and trace their distillation into one- and two-sentence education policy recommendations articulated in U.S. government position papers, implemented in classrooms across the U.S. and imported by countries around the world. We also present the historical context needed to understand the misrepresentations cited in support of questionable education policy. 1599 1630 3 10.4171/022-3/77 http://www.ems-ph.org/doi/10.4171/022-3/77

Ian Stewart University of Warwick, COVENTRY, UNITED KINGDOM General It is becoming increasingly necessary, and important, for mathematicians to engage with the general public. Our subject is widely misunderstood, and its vital role in today’s society goes mostly unobserved. Most people are unaware that any mathematics exists beyond what they did at school. So our prime objective must be to make people aware that new mathematics is constantly being invented, and that the applications of mathematics are essential in a technological world. The mass media can play a significant role in encouraging such understanding, but the world of the media is very different from the academic world. I describe what it is like to engage with the media, concentrating on my own experiences of the past 40 years. 1631 1644 3 10.4171/022-3/78 http://www.ems-ph.org/doi/10.4171/022-3/78

Mathematics education General This article sets the background for the panel session at the ICM on controversial issues in K-12 mathematics education. Three specific issues have been selected: Technology, skill building and the role of test and assessment. For each of these, a list of questions has been prepared. After introducing the three themes and the associated questions, this article presents the positions on these of the two panelists: Professor Anthony Ralston, from the State University of New York at Buffalo in the US, and Professor Ehud de Shalit from the Hebrew University of Jerusalem in Israel. The article ends with some personal comments from the coordinator of the panel: Professor Michèle Artigue from the University Paris 7 in France. 1645 1661 3 10.4171/022-3/79 http://www.ems-ph.org/doi/10.4171/022-3/79

Mathematics education General This is a panel discussion on PISA and TIMSS, two international comparative studies in educational achievement. The panelists are Jan de Lange of the Freudenthal Institute, the Netherlands, for PISA, and William Schmidt of Michigan State University, the United States, for TIMSS, with Lee Peng Yee of National Institute of Education, Singapore, as a moderator. They are to explain the nature, the aims, and the conclusions of the two studies, and to argue over their relative merits. This document contains three initial statements from the above-mentioned participants respectively. 1663 1672 3 10.4171/022-3/80 http://www.ems-ph.org/doi/10.4171/022-3/80

Mathematics education General The need for mathematics educators, schoolteachers and mathematicians to work together to improve K-12 mathematics education continues to be a great concern throughout the world. The main paper for this panel discussion proposes a paradigm or perspective within which to organize this working together of the different groups. This is to see mathematics education as mathematical engineering. From this perspective, the challenge of the mathematics educator and the schoolteacher is to customize mathematics to students’ needs. The role, in turn, of the university mathematician is to customize mathematics courses for teachers so that they in turn may be able to customize the mathematics for the different needs of their students. This paradigm is then discussed in different contexts, in the United States, Hong Kong, Switzerland, and the Philippines. The paradigm is seen to be fruitful in these different contexts. 1673 1696 3 10.4171/022-3/81 http://www.ems-ph.org/doi/10.4171/022-3/81

Leo Corry Tel Aviv University, TEL AVIV, ISRAEL History and biography Mathematical logic and foundations Mechanics of particles and systems Relativity and gravitational theory The sixth of Hilbert’s famous 1900 list of twenty-three problems is a programmatic call for the axiomatization of physical sciences. Contrary to a prevalent view this problem was naturally rooted at the core of Hilbert’s conception of what axiomatization is all about. The axiomatic method embodied in his work on geometry at the turn of the twentieth-century originated in a preoccupation with foundational questions related with empirical science, including geometry and other physical disciplines at a similar level. From all the problems in the list, the sixth is the only one that continually engaged his efforts over a very long period, at least between 1894 and 1932. 1697 1718 3 10.4171/022-3/82 http://www.ems-ph.org/doi/10.4171/022-3/82

Niccolo Guicciardini Universita di Siena, SIENA, ITALY History and biography General In my talk I will consider Newton’s views on mathematical method. Newton never wrote extensively about this issue. However, in his polemic writings addressed against Descartes and Leibniz he expressed the idea that his method was superior to the ones proposed by the French and the German. Considering these writings can help us in understanding the role attributed to algebra and calculus in Newton’s mathematical thought. 1719 1742 3 10.4171/022-3/83 http://www.ems-ph.org/doi/10.4171/022-3/83

Mathematics education General In addition to the current state of knowledge about the learning of mathematics and its aims in today’s society, the main purpose of this paper is discussing ways of improving the process of learning, and especially, in that regard, the role of e-learning technologies. We chart the situation of e-learning mathematics as of December, 2005, including distance-learning or open university courses, and then we consider a number of areas where e-learning is likely to develop. Finally, we assess the impact of e-learning on the role of the new educators in mathematics. 1743 1768 3 10.4171/022-3/84 http://www.ems-ph.org/doi/10.4171/022-3/84

Sergei Kuksin Heriot-Watt University, Edinburgh, UK Partial differential equations Fluid mechanics 35Q30, 76F05 The book gives an account of recent achievements in the mathematical theory of two-dimensional turbulence, described by the 2D Navier–Stokes equation, perturbed by a random force. The main results presented here were obtained during the last five to ten years and, up to now, have been available only in papers in the primary literature. Their summary and synthesis here, beginning with some preliminaries on partial differential equations and stochastics, make the book a self-contained account that will appeal to readers with a general background in analysis. After laying the groundwork, the author goes on to recent results on ergodicity of random dynamical systems, which the randomly forced Navier-Stokes equation defines in the function space of divergence-free vector fields, including a Central Limit Theorem. The physical meaning of these results is discussed as well as their relations with the theory of attractors. Next, the author studies the behaviour of solutions when the viscosity goes to zero. In the final section these dynamical methods are used to derive the so-called balance relations - the infinitely many algebraical relations satisfied by the solutions. 4 30 2006 978-3-03719-021-0 978-3-03719-521-5 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/021 http://www.ems-ph.org/doi/10.4171/021 Andrey A. Bolibrukh Memorial Volume Daniel Bertrand Université Pierre et Marie Curie, Paris, France Benjamin Enriquez IRMA, Strasbourg, France Claude Mitschi IRMA, Strasbourg, France Claude Sabbah Ecole Polytechnique, Palaiseau, France Reinhard Schäfke IRMA, Strasbourg, France Algebraic geometry Group theory and generalizations Several complex variables and analytic spaces Partial differential equations Primary: 14N15, 20F55, 32G34, 32S22, 33E17, 34Mxx, 35C10, 35Q15, 70H05; Secondary: 12Hxx, 14H10, 14Pxx, 16W30, 17B37, 17B40, 17B67, 32S40, 35C20, 53C15, 53D30, 70G65, 82B23 This special volume is dedicated to the memory of Andrey A. Bolibrukh. It contains two expository articles devoted to some aspects of Bolibrukh's work, followed by ten refereed research articles. Topics cover complex linear and nonlinear differential equations as well as quantum groups: monodromy, Fuchsian linear systems, Riemann–Hilbert problem, differential Galois theory, differential algebraic groups, multisummability, isomonodromy, Painlevé equations, Schlesinger equations, integrable systems, KZ equations, complex reflection groups, root systems. 12 1 2006 978-3-03719-020-3 978-3-03719-520-8 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/020 http://www.ems-ph.org/doi/10.4171/020 9

Yulij Ilyashenko Cornell University, ITHACA, UNITED STATES Algebraic geometry Group theory and generalizations Several complex variables and analytic spaces Partial differential equations Two classical results of Bolibrukh are exposed. We try to present clearly the main ideas that are parallel in both proofs, and reduce to a minimum the technical part. 1 8 1 10.4171/020-1/1 http://www.ems-ph.org/doi/10.4171/020-1/1

Claude Sabbah Ecole Polytechnique, PALAISEAU CEDEX, FRANCE Algebraic geometry Group theory and generalizations Several complex variables and analytic spaces Partial differential equations We give a description of the work of Andrey Bolibrukh on isomonodromic deformations and relate it to existing results in this domain. 9 25 1 10.4171/020-1/2 http://www.ems-ph.org/doi/10.4171/020-1/2

Michèle Audin Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE Algebraic geometry Group theory and generalizations Several complex variables and analytic spaces Partial differential equations We investigate the relation between two notions of integrability, to have enough first integrals on the one hand, and to have meromorphic solutions on the other, that are present in Kowalevskaya’s famous mémoire on the rigid body problem. We concentrate on the examples of the rigid body and of the system of Hénon–Heiles. 27 47 1 10.4171/020-1/3 http://www.ems-ph.org/doi/10.4171/020-1/3

Werner Balser Universität Ulm, ULM, GERMANY Algebraic geometry Group theory and generalizations Several complex variables and analytic spaces Partial differential equations In this article we investigate formal power series solutions of the heat equation in one spatial variable. In previous work of Lutz, Miyake, and Schäfke, resp. of W. Balser, solutions of a Cauchy problem have been shown to be k-summable in a direction d if, and only if, the initial condition satisfies a certain condition. Here, we investigate the initial value problem for the spatial variable, finding new results especially for the case when the initial values are Gevrey functions of order larger than one, so that the corresponding power series solution diverges. 49 58 1 10.4171/020-1/4 http://www.ems-ph.org/doi/10.4171/020-1/4

Prakash Belkale University of North Carolina at Chapel Hill, CHAPEL HILL, UNITED STATES Evgeny Mukhin Indiana University Purdue University Indianapolis, INDIANAPOLIS, UNITED STATES Alexander Varchenko University of North Carolina at Chapel Hill, CHAPEL HILL, UNITED STATES Algebraic geometry Group theory and generalizations Several complex variables and analytic spaces Partial differential equations In [MV], some correspondences were defined between critical points of master functions associated to slN + 1 and subspaces of ℂ[x] with given ramification properties. In this paper we show that these correspondences are in fact scheme theoretic isomorphisms of appropriate schemes. This gives relations between multiplicities of critical point loci of the relevant master functions and multiplicities in Schubert calculus. 59 84 1 10.4171/020-1/5 http://www.ems-ph.org/doi/10.4171/020-1/5

Philip Boalch Ecole Normale Superieure, PARIS CEDEX 05, FRANCE Algebraic geometry Group theory and generalizations Several complex variables and analytic spaces Partial differential equations Explicit solutions to the Riemann–Hilbert problem will be found realising some irreducible non-rigid local systems. The relation to isomonodromy and the sixth Painlevé equation will be described. 85 112 1 10.4171/020-1/6 http://www.ems-ph.org/doi/10.4171/020-1/6

Phyllis Cassidy The City College of CUNY, NEW YORK, UNITED STATES Michael Singer North Carolina State University, RALEIGH, UNITED STATES Algebraic geometry Group theory and generalizations Several complex variables and analytic spaces Partial differential equations We present a Galois theory of parameterized linear differential equations where the Galois groups are linear differential algebraic groups, that is, groups of matrices whose entries are functions of the parameters and satisfy a set of differential equations with respect to these parameters. We present the basic constructions and results, give examples, discuss how isomonodromic families fit into this theory and show how results from the theory of linear differential algebraic groups may be used to classify systems of second order linear differential equations. 113 155 1 10.4171/020-1/7 http://www.ems-ph.org/doi/10.4171/020-1/7

Boris Dubrovin SISSA, TRIESTE, ITALY Marta Mazzocco University of Manchester, MANCHESTER, UNITED KINGDOM Algebraic geometry Group theory and generalizations Several complex variables and analytic spaces Partial differential equations The Schlesinger equations S(n,m) describe monodromy preserving deformations of order m Fuchsian systems with n + 1 poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of n copies of m × m matrix algebras equipped with the standard linear Poisson bracket. In this paper we address the problem of reduction of particular solutions of “more complicated” Schlesinger equations S(n,m) to “simpler” S(n',m') having n' ≤ n, m' ≤ m. 157 187 1 10.4171/020-1/8 http://www.ems-ph.org/doi/10.4171/020-1/8

Valentina Golubeva VINITI, MOSCOW, RUSSIAN FEDERATION Algebraic geometry Group theory and generalizations Several complex variables and analytic spaces Partial differential equations The paper is devoted to the restricted Riemann–Hilbert problem in a class of generalized Knizhnik–Zamolodchikov equations. The Knizhnik–Zamolodchikov equation has singularities on the set of reflection hyperplanes of the root system An−1. I. Cherednik proposed to consider also similar systems of equations, associated to the other root systems and gave examples of corresponding physical models. The Cherednik systems now are called the generalized Knizhnik–Zamolodchikov equations. In this class of equations the Riemann–Hilbert problem consists in investigating the correspondence between the equations and the representations of the fundamental group of the complement of the singular locus of the equation. For the case of the root system An this correspondence was investigated by V. Drinfeld and T. Kohno. In the paper an exposition of the results on the generalization of the Drinfeld–Kohno theory for the root system Bn obtained in collaboration with V. P. Lexin is given. Some principal elements of the proof of the main theorems are discussed. Besides, the equations of the Knizhnik–Zamolodchikov type for the other root systems and some equations close to them are touched. 189 207 1 10.4171/020-1/9 http://www.ems-ph.org/doi/10.4171/020-1/9

Vladimir Petrov Kostov Université de Nice, NICE CEDEX 2, FRANCE Algebraic geometry Group theory and generalizations Several complex variables and analytic spaces Partial differential equations We consider the Riemann–Hilbert problem (or Hilbert’s 21st problem) for Fuchsian linear systems of ordinary differential equations on the Riemann sphere: Prove that for any prescribed monodromy group and poles there exists a Fuchsian system with this monodromy group and with these poles. In this setting the problem has been given a negative answer due to A. A. Bolibrukh. We give sufficient conditions for the realizability of a monodromy group by a Fuchsian system, e.g. if one of the monodromy operators in its Jordan normal form has at most one block of size 2 the rest being of size 1, or if the monodromy group is irreducible (the proof of this result has been obtained simultaneously and independently of the one of Bolibrukh). We discuss invariants of matrix groups considered as monodromy groups. We also give the codimension in the space of couples (monodromy group, poles) of the set for which the answer to the Riemann–Hilbert problem is negative, and we describe the couples on which this codimension is attained. 209 254 1 10.4171/020-1/10 http://www.ems-ph.org/doi/10.4171/020-1/10

Vladimir Leksin Kolomna State Pedagogical Institute, KOLOMNA, RUSSIAN FEDERATION Algebraic geometry Group theory and generalizations Several complex variables and analytic spaces Partial differential equations A special class of flat logarithmic connections ∇R on ℂn associated with finite complex vector configurations R ⊂ ℂn generating ℂn is considered. The connection ∇R acts on the trivial holomorphic bundle with fiber ℂn and it has logarithmic poles on hyperplanes that are orthogonal to vectors of R with respect to the standard Hermitian form of ℂn. We prove that Veselov’s ∨-conditions for a complex vector configuration R are equivalent to the Frobenius integrability of the connection ∇R. If R is a root system with finite complex reflection group W(R), then R satisfies Veselov’s conditions and ∇R is an integrable connection. In the case of some root systems R, we describe the monodromy representation of the generalized braid group Bn(R) defined by the associated logarithmic connection ∇R on the quotient space ℂn/W(R). These representations are deformations of the standard representations of the corresponding complex reflection groups. They are generalizations of the Burau representations for some complex root systems which were earlier defined by Squier and Givental only for real root systems. 255 268 1 10.4171/020-1/11 http://www.ems-ph.org/doi/10.4171/020-1/11

Hiroshi Umemura Nagoya University, NAGOYA, CHIKUSA-KU, JAPAN Algebraic geometry Group theory and generalizations Several complex variables and analytic spaces Partial differential equations This note is introductory and prepared for non-specialists. We explain by using examples the basic ideas of a general differential Galois theory of ordinary differential field extensions. 269 289 1 10.4171/020-1/12 http://www.ems-ph.org/doi/10.4171/020-1/12

Hans Triebel University of Jena, Germany Functional analysis Measure and integration Fourier analysis 46-02; 46E35, 42C40; 42B35, 28A80 Function spaces, wavelets, bases and frames, extension problem, thick domains, cellular domains Wavelets have emerged as an important tool in analyzing functions containing discontinuities and sharp spikes. They were developed independently in the fields of mathematics, quantum physics, electrical engineering, and seismic geology. Interchanges between these fields during the last ten years have led to many new wavelet applications such as image compression, turbulence, human vision, radar, earthquake prediction, and pure mathematics applications such as solving partial differential equations. This book develops a theory of wavelet bases and wavelet frames for function spaces on various types of domains. Starting with the usual spaces on Euclidean spaces and their periodic counterparts, the exposition moves on to so-called thick domains (including Lipschitz domains and snowflake domains). Especially, wavelet expansions and extensions to corresponding spaces on Euclidean n-spaces are developed. Finally, spaces on smooth and cellular domains and related manifolds are treated. Although the presentation relies on the recent theory of function spaces, basic notation and classical results are repeated in order to make the text self-contained. The book is addressed to two types of readers: researchers in the theory of function spaces who are interested in wavelets as new effective building blocks for functions, and scientists who wish to use wavelet bases in classical function spaces for various applications. Adapted to the second type of readers, the preface contains a guide to where one finds basic definitions and key assertions. 9 2 2008 978-3-03719-019-7 978-3-03719-519-2 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/019 http://www.ems-ph.org/doi/10.4171/019 7 Martin Mohlenkamp Ohio University, Athens, USA María Cristina Pereyra University of New Mexico, Albuquerque, USA Fourier analysis Numerical analysis 42-01; 42C40, 65T60 Wavelet, time-frequency analysis, multiresolution analysis, fast wavelet transform, filter banks So what is all the fuss about wavelets? You can find out by reading these notes. They will introduce you to the central concepts surrounding wavelets and their applications. By focusing on the essential ideas and arguments, they enable you to get to the heart of the matter as quickly as possible. They then point you to the appropriate places in the literature for detailed proofs and real applications, so you can continue your study. They begin with the notion of time-frequency analysis, present the multiresolution analysis and basic wavelet construction, introduce you to the many friends, relatives and mutations of wavelets, and finally give a selection of applications. They are suitable for beginning graduate students and above. A preliminary chapter containing some of the prerequisite concepts and definitions is included for reference. 6 26 2008 978-3-03719-018-0 978-3-03719-518-5 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/018 http://www.ems-ph.org/doi/10.4171/018 Analysis and Numerical Solution Peter Kunkel University of Leipzig, Germany Volker Mehrmann Technical University of Berlin, Germany Ordinary differential equations Numerical analysis 34A09, 65L80 Differential-algebraic equations are a widely accepted tool for the modeling and simulation of constrained dynamical systems in numerous applications, such as mechanical multibody systems, electrical circuit simulation, chemical engineering, control theory, fluid dynamics and many others. This is the first comprehensive textbook that provides a systematic and detailed analysis of initial and boundary value problems for differential-algebraic equations. The analysis is developed from the theory of linear constant coefficient systems via linear variable coefficient systems to general nonlinear systems. Further sections on control problems, generalized inverses of differential-algebraic operators, generalized solutions, and differential equations on manifolds complement the theoretical treatment of initial value problems. Two major classes of numerical methods for differential-algebraic equations (Runge--Kutta and BDF methods) are discussed and analyzed with respect to convergence and order. A chapter is devoted to index reduction methods that allow the numerical treatment of general differential-algebraic equations. The analysis and numerical solution of boundary value problems for differential-algebraic equations is presented, including multiple shooting and collocation methods. A survey of current software packages for differential-algebraic equations completes the text. The book is addressed to graduate students and researchers in mathematics, engineering and sciences, as well as practitioners in industry. A prerequisite is a standard course on the numerical solution of ordinary differential equations. Numerous examples and exercises make the book suitable as a course textbook or for self-study. 2 28 2006 978-3-03719-017-3 978-3-03719-517-8 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/017 http://www.ems-ph.org/doi/10.4171/017 Markus Stroppel University of Stuttgart, Germany Topological groups, Lie groups Field theory and polynomials Group theory and generalizations Abstract harmonic analysis 22D05, 22-01; 20E18, 22A25, 22B05, 22C05, 22D10, 22D45, 22F05, 12J10, 43A05, 54H15, 22A15 Locally compact groups play an important role in many areas of mathematics as well as in physics. The class of locally compact groups admits a strong structure theory, which allows to reduce many problems to groups constructed in various ways from the additive group of real numbers, the classical linear groups and from finite groups. The book gives a systematic and detailed introduction to the highlights of that theory. In the beginning, a review of fundamental tools from topology and the elementary theory of topological groups and transformation groups is presented. Completions, Haar integral, applications to linear representations culminating in the Peter–Weyl Theorem are treated. Pontryagin duality for locally compact Abelian groups forms a central topic of the book. Applications are given, including results about the structure of locally compact Abelian groups, and a structure theory for locally compact rings leading to the classification of locally compact fields. Topological semigroups are discussed in a separate chapter, with special attention to their relations to groups. The last chapter reviews results related to Hilbert's Fifth Problem, with the focus on structural results for non-Abelian connected locally compact groups that can be derived using approximation by Lie groups. The book is self-contained and is addressed to advanced undergraduate or graduate students in mathematics or physics. It can be used for one-semester courses on topological groups, on locally compact Abelian groups, or on topological algebra. Suggestions on course design are given in the preface. Each chapter is accompanied by a set of exercises that have been tested in classes. 2 28 2006 978-3-03719-016-6 978-3-03719-516-1 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/016 http://www.ems-ph.org/doi/10.4171/016 Torsten Ekedahl University of Stockholm, Sweden Convex and discrete geometry Algebraic geometry 14H52; 14K25 These lecture notes grew out of a one semester introductory course on elliptic curves given to an audience of computer science and mathematics students, and assume only minimal background knowledge. After having covered basic analytic and algebraic aspects, putting special emphasis on explaining the interplay between algebraic and analytic formulas, they go on to some more specialized topics. These include the j-function from an algebraic and analytic perspective, a discussion of elliptic curves over finite fields, derivation of recursion formulas for the division polynomials, the algebraic structure of the torsion points of an elliptic curve, complex multiplication, and modular forms. In an effort to motivate basic problems the book starts very slowly, but considers some aspects such as modular forms of higher level which are not usually treated. It presents more than 100 exercises and a Mathematica™ notebook that treats a number of calculations involving elliptic curves. The book is aimed at students of mathematics with a general interest in elliptic curves but also at students of computer science interested in their cryptographic aspects. 2 14 2006 978-3-03719-015-9 978-3-03719-515-4 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/015 http://www.ems-ph.org/doi/10.4171/015 a practical guide Jerzy Trzeciak This booklet is intended to provide practical help for authors of mathematical papers. It is written mainly for non-English speaking writers but should prove useful even to native speakers of English who are beginning their mathematical writing and may not yet have developed a command of the structure of mathematical discourse. The first part provides a collection of ready-made sentences and expressions occurring in mathematical papers. The examples are divided into sections according to their use (in introductions, definitions, theorems, proofs, comments, references to the literature, acknowledgements, editorial correspondence and referee's reports). Typical errors are also pointed out. The second part concerns selected problems of English grammar and usage, most often encountered by mathematical writers. Just as in the first part, an abundance of examples are presented, all of them taken from actual mathematical texts. "The author has packed an awful lot in a few pages and has obviously been collecting his best (or worst) examples for a long time." Edwin F. Beschler About the author: Jerzy Trzeciak, formerly of Polish Scientific Publishers, is now the senior copy editor at the Institute of Mathematics, Polish Academy of Sciences. He is responsible for journals including Studia Mathematica, Fundamenta Mathematicae, Acta Arithmetica and others. He is also the author of "Mathematical English Usage - a Dictionary", available at www.impan.gov.pl/Dictionary. 8 30 2005 978-3-03719-014-2 978-3-03719-514-7 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/014 http://www.ems-ph.org/doi/10.4171/014 Olivier Biquard IRMA, Strasbourg, France Differential geometry Partial differential equations Quantum theory Relativity and gravitational theory 53Cxx, 81Txx, 53A30, 58J60, 83D05 Since its discovery in 1997 by Maldacena, AdS/CFT correspondence has become one of the prime subjects of interest in string theory, as well as one of the main meeting points between theoretical physics and mathematics. On the physical side it provides a duality between a theory of quantum gravity and a field theory. The mathematical counterpart is the relation between Einstein metrics and their conformal boundaries. The correspondence has been intensively studied, and a lot of progress emerged from the confrontation of viewpoints between mathematics and physics. Written by leading experts and directed at research mathematicians and theoretical physicists as well as graduate students, this volume gives an overview of this important area both in theoretical physics and in mathematics. It contains survey articles giving a broad overview of the subject and of the main questions, as well as more specialized articles providing new insight both on the Riemannian side and on the Lorentzian side of the theory. 5 31 2005 978-3-03719-013-5 978-3-03719-513-0 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/013 http://www.ems-ph.org/doi/10.4171/013 8

Michael Anderson Stony Brook University, STONY BROOK, UNITED STATES Differential geometry Partial differential equations Quantum theory Relativity and gravitational theory We discuss classical gravitational aspects of the AdS/CFT correspondence, with the aim of obtaining a rigorous (mathematical) understanding of the semi-classical limit of the gravitational partition function. The paper surveys recent progress in the area, together with a selection of new results and open problems. 1 31 1 10.4171/013-1/1 http://www.ems-ph.org/doi/10.4171/013-1/1

Jan de Boer University of Amsterdam, AMSTERDAM, NETHERLANDS Liat Maoz University of Amsterdam, AMSTERDAM, NETHERLANDS Asad Naqvi University of Amsterdam, AMSTERDAM, NETHERLANDS Differential geometry Partial differential equations Quantum theory Relativity and gravitational theory This is a very brief review of some aspects of the AdS/CFT correspondence with an emphasis on the role of the topology of the boundary and the meaning of the sum over bulk geometries. 33 57 1 10.4171/013-1/2 http://www.ems-ph.org/doi/10.4171/013-1/2

C. Robin Graham Seattle University, SEATTLE, UNITED STATES Kengo Hirachi University of Tokyo, TOKYO, JAPAN Differential geometry Partial differential equations Quantum theory Relativity and gravitational theory 59 71 1 10.4171/013-1/3 http://www.ems-ph.org/doi/10.4171/013-1/3

Ioannis Papadimitriou University of Amsterdam, AMSTERDAM, NETHERLANDS Kostas Skenderis University of Amsterdam, AMSTERDAM, NETHERLANDS Differential geometry Partial differential equations Quantum theory Relativity and gravitational theory In the first part of this paper we provide a short introduction to the AdS/ renormalization. We discuss how QFT correlation encoded in the bulk geometry. In the second part we method of holographic renormalization, with the radial this approach regularized correlation functions are boundary expansions of the standard approach are replaced various terms are organized according to their dilatation expressions for counterterms and one-point functions (in the dimensions. The new approach combines optimally supersedes them in efficiency. 73 101 1 10.4171/013-1/4 http://www.ems-ph.org/doi/10.4171/013-1/4

Marc Herzlich Université de Montpellier II, MONTPELLIER CEDEX 5, FRANCE Differential geometry Partial differential equations Quantum theory Relativity and gravitational theory 103 121 1 10.4171/013-1/5 http://www.ems-ph.org/doi/10.4171/013-1/5

Sergey Solodukhin International University Bremen, BREMEN, GERMANY Differential geometry Partial differential equations Quantum theory Relativity and gravitational theory Minkowski space is a physically important space-time for which it is an urgent problem to find an adequate holographic description. In this paper we develop further the proposal made in [10] for the description as a duality between Minkowski space-time and a Conformal Field Theory defined on the boundary of the light-cone. We focus on the gravitational aspects of the duality. Specifically, we identify the gravitational holographic data and provide the way Minkowski space-time (understood in a more general context as a Ricci-flat space) is reconstructed from the data. In order to avoid the complexity of non-linear Einstein equations we consider linear perturbations and do the analysis for the perturbations. The analysis proceeds in two steps. We first reduce the problem in Minkowski space to an infinite set of field equations on de Sitter space one dimension lower. These equations are quite remarkable: they describe massless and massive gravitons in de Sitter space. In particular, the partially massless graviton appears naturally in this reduction. In the second step we solve the graviton field equations and identify the holographic boundary data. Finally, we consider the asymptotic form of the black hole space-time and identify the way the information about the mass of the static gravitational configuration is encoded in the holographic data. 123 163 1 10.4171/013-1/6 http://www.ems-ph.org/doi/10.4171/013-1/6

Michael Anderson Stony Brook University, STONY BROOK, UNITED STATES Piotr Chruściel Université François Rabelais, TOURS, FRANCE Erwann Delay Université d'Avignon, AVIGNON, FRANCE Differential geometry Partial differential equations Quantum theory Relativity and gravitational theory We show that the recent work of Lee [24] implies existence of a large class of new singularity-free strictly static Lorentzian vacuum solutions of the Einstein equations with a negative cosmological constant. This holds in all space-time dimensions greater than or equal to four, and leads both to strictly static solutions and to black hole solutions. The construction allows in principle for metrics (whether black hole or not) with Yang–Mills-dilaton fields interacting with gravity through a Kaluza–Klein coupling. 165 204 1 10.4171/013-1/7 http://www.ems-ph.org/doi/10.4171/013-1/7

Charles Frances Université Paris Sud, ORSAY CEDEX, FRANCE Differential geometry Partial differential equations Quantum theory Relativity and gravitational theory 205 216 1 10.4171/013-1/8 http://www.ems-ph.org/doi/10.4171/013-1/8

Jerome Gauntlett Imperial College, LONDON, UNITED KINGDOM Dario Martelli CERN Theory Division, GENÈVE, SWITZERLAND James Sparks Harvard University, CAMBRIDGE, UNITED STATES Daniel Waldram Imperial College, LONDON, UNITED KINGDOM Differential geometry Partial differential equations Quantum theory Relativity and gravitational theory We first present a short review of general supersymmetric compactifications in string and M-theory using the language of G-structures and intrinsic torsion. We then summarize recent work on the generic conditions for supersymmetric AdS5 backgrounds in M-theory and the construction of classes of new solutions. Turning to AdS5 compactifications in type IIB, we summarize the construction of an infinite class of new Sasaki–Einstein manifolds in dimension 2k + 3 given a positive curvature Kähler–Einstein base manifold in dimension 2k. For k = 1 these describe new supergravity duals for N = 1 superconformal field theories with both rational and irrational R-charges and central charge. We also present a generalization of this construction, that has not appeared elsewhere in the literature, to the case where the base is a product of Kähler–Einstein manifolds. 217 252 1 10.4171/013-1/9 http://www.ems-ph.org/doi/10.4171/013-1/9

Stéphane Cordier Orléans, France Thierry Goudon Lille, France Michael Gutnic Strasbourg, France Eric Sonnendrücker Strasbourg, France Numerical analysis 65M, 76-xx, 82-xx Hyperbolic and kinetic equations arise in a large variety of industrial problems. For this reason, the CEMRACS summer research center held at CIRM in Luminy in 2003 was devoted to this topic. During a six-week period, junior and senior researchers worked full time on several projects proposed by industry and academia. Most of this work was completed later on, and the results are now reported in the present book. The articles address modelling issues as well as the development and comparisons of numerical methods in different situations. The applications include multi-phase flows, plasma physics, quantum particle dynamics, radiative transfer, sprays and aeroacoustics. The text is aimed at researchers and engineers interested in modelling and numerical simulation of hyperbolic and kinetic problems arising from applications. 5 31 2005 978-3-03719-012-8 978-3-03719-512-3 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/012 http://www.ems-ph.org/doi/10.4171/012 7

R. Barthelmé Université de Strasbourg, STRASBOURG CEDEX, FRANCE C. Parzani Université Paul Sabatier, TOULOUSE CEDEX, FRANCE Numerical analysis General A difficulty in electromagnetic PIC simulations of plasmas modeled by the Vlasov–Maxwell system is that the continuity equation may not be satisfied and that then the Maxwell system isn’t any more well-posed. Indeed, in PIC codes, fields and densities are defined at spatial grid points while charged particles can take arbitrary positions. Therefore, the current mapped from particles to the mesh may not satisfy the continuity equation at the discrete level. In this paper, we are interested in charge conserving current deposition algorithms. We study the efficiency of two methods ensuring that the continuity equation is satisfied at the discrete level: an extension of the Villasenor–Buneman method and the Esirkepov method. 7 28 1 10.4171/012-1/1 http://www.ems-ph.org/doi/10.4171/012-1/1

J.-P. Chehab CNRS-Université Lille I, VILLENEUVE D'ASCQ CEDEX, FRANCE Albert Cohen Université Pierre et Marie Curie, PARIS, FRANCE D. Jennequin Centre Universitaire de la Mi-Voix, CALAIS, CEDEX, FRANCE J. J. Nieto Universidad de Granada, GRANADA, SPAIN J. R. Roche Université Henri Poincaré, VANDOEUVRE-LES-NANCY, FRANCE Ch. Roland CNRS-Université Lille I, VILLENEUVE D'ASCQ CEDEX, FRANCE Numerical analysis General In this paper we introduce a new PIC method based on an adaptive multi-resolution scheme for solving the one dimensional Vlasov–Poisson equation. Our approach is based on a description of the solution by particles of unit weight and on a reconstruction of the density at each time step of the numerical scheme by an adaptive wavelet technique: the density is firstly estimated in a proper wavelet basis as a distribution function from the current empirical data and then “de-noised” by a thresholding procedure. The so-called Landau damping problem is considered for validating our method. The numerical results agree with those obtained by the classical PIC scheme, suggesting that this multi-resolution procedure could be extended with success to plasma dynamics in higher dimensions. 29 42 1 10.4171/012-1/2 http://www.ems-ph.org/doi/10.4171/012-1/2

M. Campos Pinto Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE M. Mehrenberger Université Louis Pasteur, STRASBOURG CEDEX, FRANCE Numerical analysis General A fully adaptive scheme (based on hierarchical continuous finite element decomposition) is derived from a semi-Lagrangian method for solving a periodic Vlasov–Poisson system. The first numerical results establish the validity of such a scheme. 43 58 1 10.4171/012-1/3 http://www.ems-ph.org/doi/10.4171/012-1/3

N. Crouseilles Université Paul Sabatier – Toulouse 3, TOULOUSE CEDEX, FRANCE F. Filbet Université d'Orléans, ORLÉANS CEDEX 2, FRANCE Numerical analysis General In this paper we compute numerical solutions to the Vlasov equation coupled with the Fokker–Planck–Landau collision operator using a phase space grid. On the one hand, the transport is treated by a flux conservative method and the distribution function is reconstructed to preserve energy and to control numerical oscillations. On the other hand, the collision operators are treated to preserve the conservation properties. Some results are presented in one dimension in space and three dimensions in velocity. 59 70 1 10.4171/012-1/4 http://www.ems-ph.org/doi/10.4171/012-1/4

Christophe Besse Université Paul Sabatier, TOULOUSE CEDEX, FRANCE Norbert Mauser Universität Wien, WIEN, AUSTRIA Hans Peter Stimming Universität Wien, WIEN, AUSTRIA Numerical analysis General We study the numerical approximation and analysis of two different time dependent nonlinear Schrödinger equations (NLS) : the “Schrödinger–Poisson-Xα” equation, which plays a role in the modeling of quantum particle dynamics, and the Davey–Stewartson system, which is a 2-d equation modeling unidirectional water surface waves. For both equations we use a Time Splitting Spectral scheme, which had previously shown to be a good tool for numerical simulation of cubic NLS. This scheme is particularly useful for calculations in the “semi-classical regime”, where the scaled Planck constant is taken to be small. Extensive numerical results of position density and Wigner measures in 1d, 2d and 3d for the S-P-Xα model with/without an external potential are presented. These results give an insight to understand the interplay between the nonlocal (“weak”) and the local (“strong”) nonlinearity. For the Davey–Stewartson system, we give a convergence analysis for the semi-discrete version of the scheme. Numerical results are presented for various blowup phenomena of the equation, including blowup of defocusing, elliptic-elliptic Davey–Stewartson systems and simultaneous blowup at multiple locations in the focusing elliptic-elliptic system. Also the modeling of exact soliton type solutions for the hyperbolic-elliptic (DS2) system is studied. 71 99 1 10.4171/012-1/5 http://www.ems-ph.org/doi/10.4171/012-1/5

Christophe Besse Université Paul Sabatier, TOULOUSE CEDEX, FRANCE Jean Claudel 2CEA–Centre d’Etudes Scientifiques et Techniques d, LE BARP, FRANCE P. Degond Université Paul Sabatier, TOULOUSE CEDEX, FRANCE F. Deluzet Université Paul Sabatier, TOULOUSE CEDEX, FRANCE Gerard Gallice 2CEA–Centre d’Etudes Scientifiques et Techniques d, LE BARP, FRANCE Christian Tessieras 2CEA–Centre d’Etudes Scientifiques et Techniques d, LE BARP, FRANCE Numerical analysis General This paper is concerned with a review of models of ionospheric plasma. Two hierarchies of models are presented, both of them leading to the so-called striation model. A linear stability analysis is performed and shows that the striation model can developed plasma instabilities. A dissipation process has disappeared during the limiting process. In order to add a dissipation mechanism to the striation model, we propose to use a statistical approach to turbulence and derive a so-called ’turbulent striation model’. Numerical simulations are proposed to show the gain of stability. 101 129 1 10.4171/012-1/6 http://www.ems-ph.org/doi/10.4171/012-1/6

L. Gosse Sezione di Bari, BARI, ITALY Numerical analysis General We present a short overview of high-frequency asymptotics for the 1D Schrödinger equation while emphasizing the computational aspects. We show that presumably different techniques like stationary phase methods, WKB or Wigner analysis give essentially the same macroscopic behaviour. Moment systems and K-branch entropy solutions are introduced in order to derive well-suited numerical methods. Finally, we display some computations supporting these ideas on a classical example. 131 141 1 10.4171/012-1/7 http://www.ems-ph.org/doi/10.4171/012-1/7

Céline Baranger , CACHAN CEDEX, FRANCE G. Baudin , CACHAN CEDEX, FRANCE L. Boudin Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE Bruno Després Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE Frédéric Lagoutière Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE E. Lapébie , CACHAN CEDEX, FRANCE Takéo Takahashi Université Henri Poincaré, VANDOEUVRE-LES-NANCY, FRANCE Numerical analysis General This work is motivated by the numerical simulation of the generation and break-up of droplets after the impact of a rigid body on a tank filled with a compressible fluid. This paper splits into two very different parts. The first part deals with the modeling and the numerical resolution of a spray of liquid droplets in a compressible medium like air. Phenomena taken into account are the breakup effects due to the velocity and pressure waves in the compressible ambient fluid. The second part is concerned with the transport of a rigid body in a compressible liquid, involving reciprocal effects between the two components. A new one-dimensional algorithm working on a fixed Eulerian mesh is proposed. 149 176 1 10.4171/012-1/8 http://www.ems-ph.org/doi/10.4171/012-1/8

B. Després DSSI/SNEC/LPNP, BRUYERES LE CHATEL, FRANCE S. Jaouen DSSI/SNEC/LPNP, BRUYERES LE CHATEL, FRANCE C. Mazeran DSSI/SNEC/LPNP, BRUYERES LE CHATEL, FRANCE Takéo Takahashi Université Henri Poincaré, VANDOEUVRE-LES-NANCY, FRANCE Numerical analysis General The present work is devoted to the study of a system of conservation laws which arises when modelling compressible bifluid flows, in view of preliminary inertial confinement fusion simulations. By opposition to usual multiphase models, the present system of PDE’s turns out to be in fully conservative form, what is of great practical interest in the development of usual numerical approaches and for the capture of shocks. The study is strongly conditioned by the so-called isobar-isothermal closure. We propose a numerical scheme for the approximate resolution of the model which respects positivity of mass fractions. Numerical results illustrate the interest of the model and the method on classical bifluid test-cases. 177 207 1 10.4171/012-1/9 http://www.ems-ph.org/doi/10.4171/012-1/9

F. Caro ONERA Châtillon, CHATILLON CEDEX, FRANCE F. Coquel Université Pierre et Marie Curie, PARIS, FRANCE D. Jamet CEA-Grenoble, GRENOBLE CEDEX 9, FRANCE S. Kokh CEA Saclay, GIF-SUR-YVETTE CEDEX, FRANCE Numerical analysis General We present an isothermal model for liquid-vapor phase transition. We use a Hamilton principle to derive the conservative part of the system and we propose additional stiff phase change source terms that are compatible with the second thermodynamic principle. These terms are considered as relaxation terms and define an equilibrium manifold for the system. We formally study the equilibrium return process. While the original system is hyperbolic, the equilibrium limit system is only weakly hyperbolic. Finally, we propose a numerical scheme based on the relaxed system. 209 237 1 10.4171/012-1/10 http://www.ems-ph.org/doi/10.4171/012-1/10

F. Coquel Université Pierre et Marie Curie, PARIS, FRANCE D. Diehl Universität Freiburg, FREIBURG I BR, GERMANY C. Merkle Universität Freiburg, FREIBURG I BR, GERMANY Christian Rohde Universität Stuttgart, STUTTGART, GERMANY Numerical analysis General We consider the dynamics of a compressible fluid exhibiting phase transitions between a liquid and a vapour phase. As the basic mathematical model we use the Euler equations for a sharp interface approach and local and global versions of the Navier–Stokes–Korteweg equations for the diffuse interface approach. The mathematical models are discussed and we introduce discretization methods for both approaches. Finally numerical simulations in one and two space dimensions are presented. 239 270 1 10.4171/012-1/11 http://www.ems-ph.org/doi/10.4171/012-1/11

J. Cartier Direction des Applications Militaires, BRUYERES LE CHATEL, FRANCE A. Munnier Université de Paris Dauphine, PARIS CEDEX 16, FRANCE Numerical analysis General In this paper a geometric closure method for radiative transfer equations has been developed and investigated using particular geometric configurations. Then we propose a new formulation of the Eddington factor (and the related flux limiter) adapted to radiative transfer calculations, whereas classical Eddington’s approximation cannot be applied. Moreover, a numerical scheme and numerical results for the new flux limiter are presented in two dimensions configurations. 271 293 1 10.4171/012-1/12 http://www.ems-ph.org/doi/10.4171/012-1/12

Michael Dumbser Università degli Studi di Trento, TRENTO, ITALY C.-D. Munz Universität Stuttgart, STUTTGART, GERMANY Numerical analysis General In this paper we apply the ADER one step time discretization to the discontinuous Galerkin framework for hyperbolic conservation laws. In the case of linear hyperbolic systems we obtain a quadrature-free explicit single-step scheme of arbitrary order of accuracy in space and time on Cartesian and triangular meshes. The ADER-DG scheme does not need more memory than a first order explicit Euler time-stepping scheme. This becomes possible because of an extensive use of the governing equations inside the numerical scheme. In the nonlinear case, quadrature of the ADER-DG scheme in space and time is performed with Gaussian quadrature formulae of suitable order of accuracy. We show numerical convergence results for the linearized Euler equations up to 10th order of accuracy in space and time on Cartesian and triangular meshes. Numerical results for the nonlinear Euler equations up to 6th order of accuracy in space and time are provided as well. In this paper we also show the possibility of applying a linear reconstruction operator of the order 3N + 2 to the degrees of freedom of the DG method resulting in a numerical scheme of the order 3N + 3 on Cartesian grids where N is the order of the original basis functions before reconstruction. 295 333 1 10.4171/012-1/13 http://www.ems-ph.org/doi/10.4171/012-1/13

C.-D. Munz Universität Stuttgart, STUTTGART, GERMANY Michael Dumbser Università degli Studi di Trento, TRENTO, ITALY M. Zucchini Corporate Research and Development, GERLINGEN, GERMANY Numerical analysis General The incompressible limit of a compressible fluid, when the Mach number tends to zero, is a singular limit due to the fact that the governing equations change their type. The propagation rate of the pressure waves become infinite. Based on asymptotic results the multiple pressure variables (MPV-) method is reviewed as it may be used for the construction of numerical methods in this low Mach number regime. The MPV-schemes formally converge to incompressible methods, if the Mach number tends to zero. The insight into the behavior in the incompressible limit is used to introduce a perturbation method to calculate the noise generation and propagation in the flow region. In this approach the fluid flow is calculated by an incompressible method. The introduction of a perturbation about this incompressible solution leads to a system describing the generation and propagation of the compressible corrections to the incompressible flow field. The wave propagation model are the linearized Euler equations and the noise generation is given by source terms for these equations. This relates very well to the EIF-approach (Expansion about Incompressible Flow) as proposed by Hardin and Pope. The new part is a scaling motivated by the MPV-method and M = 0 limit. By that a linearization is introduced and the most important terms in the sources can easily be identified. For a simple one- dimensional problem with compression from the boundary the perturbation method is validated. Numerical results are also presented for the noise generation by a rotating vortex pair and for the applied problem of the noise generation of a plane jet. 335 359 1 10.4171/012-1/14 http://www.ems-ph.org/doi/10.4171/012-1/14

A Bodil Branner Festschrift Poul Hjorth University of Southern Denmark, Odense, Denmark Carsten Lunde Petersen University of Roskilde, Denmark Dynamical systems and ergodic theory 37F45 Dynamics on the Riemann Sphere presents a collection of original research articles by leading experts in the area of holomorphic dynamics. These papers arose from the symposium Dynamics in the Complex Plane, held on the occasion of the 60th birthday of Bodil Branner. Topics covered range from Lattès maps to cubic polynomials over rational maps with Sierpinsky Carpets and Gaskets as Julia sets, as well as rational and entire transcendental maps with Herman rings. Contributors include Artur Avila (Paris VI, France), Arnault Chéritat (Toulouse, France), Robert L. Devaney (Boston, USA), Adrien Douady (Orsay, France), Nuria Fagella (Barcelona, Spain), Christian Henriksen (Lyngby, Denmark), Wolf Jung (Aachen, Germany), Tomoki Kawahira (Kyoto, Japan), Tan Lei (Cergy Pontoise, France), Mikhail Lyubich (Stony Brook, USA), Carsten Lunde Petersen (Roskilde, Denmark), John Milnor (Stony Brook, USA), Pascale Roesch (Lille, France). 1 10 2006 978-3-03719-011-1 978-3-03719-511-6 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/011 http://www.ems-ph.org/doi/10.4171/011

John Milnor State Univ of New York at Stony Brook, STONY BROOK, UNITED STATES Dynamical systems and ergodic theory General An exposition of the 1918 paper of Lattès, together with its historical antecedents, and its modern formulations and applications. 9 43 1 10.4171/011-1/1 http://www.ems-ph.org/doi/10.4171/011-1/1

Tan Lei Université de Cergy-Pontoise, CERGY-PONTOISE CEDEX, FRANCE Carsten Lunde Petersen Roskilde Universitetscenter, ROSKILDE, DENMARK Dynamical systems and ergodic theory General Branner–Hubbard motion is a systematic way of deforming an attracting holomorphic dynamical system f into a family (fs)s ∈ L, via a holomorphic motion which is also a group action. We establish the analytic dependence of fs on s (a result first stated by Lyubich) and the injectivity of fs on f. We prove that the stabilizer of f (in terms of s) is either the full group L (rigidity), or a discrete subgroup (injectivity). The first case means that fs is Möbius conjugate to f for all s ∈ L, and it happens for instance at the center of a hyperbolic component. In the second case the map s ↦ fs is locally injective. We show that BH-motion induces a periodic holomorphic motion on the parameter space of cubic polynomials, and that the corresponding quotient motion has a natural extension to its isolated singularity. We give another application in the setting of Lavaurs enriched dynamical systems within a parabolic basin. 45 70 1 10.4171/011-1/2 http://www.ems-ph.org/doi/10.4171/011-1/2

Artur Avila Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE Mikhail Lyubich Stony Brook University, STONY BROOK, UNITED STATES Dynamical systems and ergodic theory General We give examples of infinitely renormalizable quadratic polynomials Fc : z ↦ z2 + c with stationary combinatorics whose Julia sets have Hausdorff dimension arbitrary close to 1. The combinatorics of the renormalization involved is close to the Chebyshev one. The argument is based upon a new tool, a "Recursive Quadratic Estimate" for the Poincaré series of an infinitely renormalizable map. 71 87 1 10.4171/011-1/3 http://www.ems-ph.org/doi/10.4171/011-1/3

Arnaud Cheritat Université Paul Sabatier, TOULOUSE CEDEX 4, FRANCE Dynamical systems and ergodic theory General For an irrational number θ, let r(θ) be the conformal radius at f=0 of the polynomial z ↦ ei2πθz + θ2. Let Y be Yoccoz’s arithmetic Brjuno function. The linearization theorem of Yoccoz implies that there exists a real constant C1 such that for all θ irrational with Y(θ) ≤ +∞, C1 ≤ Y(θ) + log(r(θ)). Yoccoz also proved a similar but weaker inequality on the other side, which in particular implies non linearizability of the polynomial when Y(θ) = +∞. The author found a new and independent proof of non linearizability. One of the tools of this new proof is the control on parabolic explosion. In a joint work with X. Buff, we pushed this further and were able to prove the existence of a real constant C2 such that for all θ with r(θ) > 0, Y(θ) + log(r(θ)) ≤ C2, enhancing Yoccoz’s second inequality. 89 96 1 10.4171/011-1/4 http://www.ems-ph.org/doi/10.4171/011-1/4

Paul Blanchard Boston University, BOSTON, UNITED STATES Robert Devaney Boston University, BOSTON, UNITED STATES Daniel Look Boston University, BOSTON, UNITED STATES Monica Moreno Rocha Tufts University, MEDFORD, UNITED STATES Pradipta Seal Boston University, BOSTON, UNITED STATES Stefan Siegmund J. W. Goethe-Universität, FRANKFURT A.M., GERMANY David Uminsky Boston University, BOSTON, UNITED STATES Dynamical systems and ergodic theory General 97 119 1 10.4171/011-1/5 http://www.ems-ph.org/doi/10.4171/011-1/5

P. Roesch CNRS-Université Lille I, VILLENEUVE D'ASCQ CEDEX, FRANCE Dynamical systems and ergodic theory General 121 129 1 10.4171/011-1/6 http://www.ems-ph.org/doi/10.4171/011-1/6

Tomoki Kawahira Nagoya University, NAGOYA, CHIKUSA-KU, JAPAN Dynamical systems and ergodic theory General For a geometrically finite rational map f, there exists a perturbation into another geometrically finite rational map which preserves the dynamics on the Julia set nicely. Indeed, we can perturb parabolic cycles of f into attracting cycles and repelling cycles in any combination. 131 138 1 10.4171/011-1/7 http://www.ems-ph.org/doi/10.4171/011-1/7

Wolf Jung Gesamtschule Aachen-Brand, AACHEN, GERMANY Dynamical systems and ergodic theory General On subsets of the Mandelbrot set, EM ⊂ M, homeomorphisms are constructed by quasi-conformal surgery. When the dynamics of quadratic polynomials is changed piecewise by a combinatorial construction, a general theorem yields the corresponding homeomorphism h : EM → EM in the parameter plane. Each h has two fixed points in EM, and a countable family of mutually homeomorphic fundamental domains. Possible generalizations to other families of polynomials or rational mappings are discussed. The homeomorphisms on subsets EM ⊂ M constructed by surgery are extended to homeomorphisms of M, and employed to study groups of non-trivial homeomorphisms h : M → M. It is shown that these groups have the cardinality of ℝ, and they are not compact. 139 159 1 10.4171/011-1/8 http://www.ems-ph.org/doi/10.4171/011-1/8

Núria Fagella Universitat de Barcelona, BARCELONA, SPAIN Christian Henriksen Technical University of Denmark, LYNGBY, DENMARK Dynamical systems and ergodic theory General We consider the Arnold family of analytic diffeomorphisms of the circle x ↦ x + t + a /2π sin(2πx) mod (1), where a, t ∈ [0, 1) and its complexification fλ,a(z) = λzea/2(z−1/z), with λ = e2πit a holomorphic self map of ℂ*. The parameter space contains the well known Arnold tongues Tα for α ∈ [0, 1) being the rotation number. We are interested in the parameters that belong to the irrational tongues and in particular in those for which the map has a Herman ring. Our goal in this paper is twofold. First we are interested in studying how the modulus of this Herman ring varies in terms of the parameter a, when a tends to 0 along the curve Tα. We survey the different results that describe this variation including the complexification of part of the Arnold tongues (called Arnold disks) which leads to the best estimate. To work with this complex parameter values we use the concept of the twist coordinate, a measure of how far from symmetric the Herman rings are. Our second goal is to investigate the slice of parameter space that contains all maps in the family with twist coordinate equal to one half, proving for example that this is a plane in ℂ2. We show a computer picture of this slice of parameter space and we also present some numerical algorithms that allow us to compute new drawings of non-symmetric Herman rings of various moduli. 161 182 1 10.4171/011-1/9 http://www.ems-ph.org/doi/10.4171/011-1/9

Tan Lei Université de Cergy-Pontoise, CERGY-PONTOISE CEDEX, FRANCE Dynamical systems and ergodic theory General Pia Willumsen did her graduate work with Bodil Branner. Willumsen has written a very beautiful Ph.D. thesis, [W], containing many interesting results. They have, however, remained unpublished up to today. We present here a short account of some of Willumsen’s results, sketch some of the proofs, as well as some immediate extensions. The main topic is stretching rays, which is the analogue in higher dimensional parameter space of external rays of the Mandelbrot set. In the space of cubic polynomials, the interaction of the two critical points create new and interesting phenomena. A typical case deals with maps with a parabolic basin containing both critical points. The results we present here provide necessary conditions for accumulation and landing of stretching rays to these maps. Also, discontinuity of the wring operator on the cubic (in contrast to quadratic) polynomials, is proven. 183 208 1 10.4171/011-1/10 http://www.ems-ph.org/doi/10.4171/011-1/10

Adrien Douady , PARIS, FRANCE Dynamical systems and ergodic theory General 209 222 1 10.4171/011-1/11 http://www.ems-ph.org/doi/10.4171/011-1/11

Ari Laptev Royal Institute of Technology, Stockholm, Sweden The European Congress of Mathematics, held every four years, has established itself as a major international mathematical event. Following those in Paris, 1992, Budapest, 1996 and Barcelona, 2000, the Fourth European Congress of Mathematics took place in Stockholm, Sweden, June 27 to July 2, 2004 with 913 participants from 65 countries. Apart from seven plenary and thirty three invited lectures, there were six “Science Lectures” covering the most relevant aspects of mathematics in science and technology. Moreover, twelve projects of the EU Research Training Networks in Mathematics and Information Sciences, as well as Programmes from the European Science Foundation in Physical and Engineering Sciences were presented. Ten EMS Prizes were awarded to young European mathematicians who have made a particular contribution to the progress of mathematics. Five of the prize winners were independently chosen by the 4ECM Scientific Committee as plenary or invited speakers. The other five prize winners gave their lectures in parallel sessions. Most of these contributions are now collected in this volume, providing a permanent record of so much that is best in mathematics today. Plenary lectures François Golse (Paris, France) Francesco Guerra (Roma, Italy) Johan Håstad (Stockholm, Sweden) Andrei Okounkov (Princeton, USA) Oded Schramm (Microsoft Research, USA) Zoltán Szabó (Princeton, USA) Claire Voisin (Paris, France) Invited Lectures Giovanni Alberti (Pisa, Italy) Denis Auroux (MIT, USA and Palaiseau, France) Stefano Bianchini (Rome, Italy) François Bouchut (Paris, France) Brian Bowditch (Southampton, UK) Ehud Friedgut (Jerusalem, Israel) Patrick Gérard (Orsay, France) Alice Guionnet (Lyon, France) Stefan Helmke (Kyoto, Japan) Helge Holden (Trondheim, Norway) Rupert Klein (Berlin, Germany) Jan Krajícek (Prague, Czech Republic) Daan Krammer (Warwick, UK) Elon Lindenstrauss (Clay Mathematics Institute, USA) Tomasz Luczak (Poznan, Poland) Terry Lyons (Oxford, UK) Pascal Massart (Orsay, France) Preda Mihailescu (Paderborn, Germany) Mircea Mustata (Clay Mathematics Institute, USA) Kieran O'Grady (Roma , Italy) Grigori Olshanski (Moscow, Russia) Imre Ruzsa (Budapest, Hungary) Yehuda Shalom (Tel-Aviv, Israel) Maria Shcherbina (Kharkov, Ukraine) Stanislav Smirnov (Stockholm, Sweden) Mikhail Sodin (Tel-Aviv, Israel) Xavier Tolsa (Barcelona, Spain) Anna-Karin Tornberg (New York, USA and Stockholm, Sweden) Vilmos Totik (Tampa, USA and Szeged, Hungary) Michael Weiss (Aberdeen, UK) Wendelin Werner (Orsay, France) Umberto Zannier (Venice, Italy) Network lectures Aline Bonami (Orléans, France) Yann Brenier (Nice, France) Jean Esterle (Bordeaux, France) Bernard Helffer (Orsay, France) Frank den Hollander (Eindhoven, Netherlands) Jonathan Keating (Bristol, UK) Christian Krattenthaler (Lyon, France) Marina Monsurrò (Lausanne, Switzerland) Jan Philip Solovej (Copenhagen, Denmark) Miles Reid (Warwick, UK) Jakob Stix (Bonn, Germany) Prize Lectures Franck Barthe (Toulouse, France) Paul Biran (Tel-Aviv, Israel) Sylvia Serfaty (New York, USA) Warwick Tucker (Uppsala, Sweden) Otmar Venjakob (Heidelberg, Germany) 6 30 2005 978-3-03719-009-8 978-3-03719-509-3 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/009 http://www.ems-ph.org/doi/10.4171/009

Giovanni Alberti Università di Pisa, PISA, ITALY Marianna Csörnyei University College London, LONDON, UNITED KINGDOM David Preiss University of Warwick, COVENTRY, UNITED KINGDOM Real functions Order, lattices, ordered algebraic structures Calculus of variations and optimal control; optimization Convex and discrete geometry We describe a decomposition result for Lebesgue negligible sets in the plane, and outline some applications to real analysis and geometric measure theory. These results are contained in [2]. 3 22 1 10.4171/009-1/1 http://www.ems-ph.org/doi/10.4171/009-1/1

Denis Auroux Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES Manifolds and cell complexes Group theory and generalizations Differential geometry General The topology of symplectic 4-manifolds is related to that of singular plane curves via the concept of branched covers. Thus, various classification problems concerning symplectic 4-manifolds can be reformulated as questions about singular plane curves. Moreover, using braid monodromy, these can in turn be reformulated in the language of braid group factorizations. While the results mentioned in this paper are not new, we hope that they will stimulate interest in these questions, which remain essentially wide open. 23 40 1 10.4171/009-1/2 http://www.ems-ph.org/doi/10.4171/009-1/2

D. Beliaev Royal Institute of Technology, STOCKHOLM, SWEDEN Stanislav Smirnov Université de Genève, GENÈVE 4, SWITZERLAND Functions of a complex variable General Many problems in complex analysis can be reduced to the evaluation of the universal spectrum: the supremum of multifractal spectra of harmonic measures for all planar domains. Its exact value is still unknown, with very few estimates available. We start with a brief survey of related problems and available estimates from above. Then we discuss in more detail estimates from below, describing the search for a fractal domain which attains the maximal possible spectrum. 41 59 1 10.4171/009-1/3 http://www.ems-ph.org/doi/10.4171/009-1/3

Stefano Bianchini SISSA-ISAS, TRIESTE, ITALY Functions of a complex variable General Consider an n × n hyperbolic system of conservation laws of the form ut + f(u)x = 0,  (t,x) ∈ ℝ+ × ℝ, u ∈ ℝn. Here u = (u1,…,un) is the vector of conserved quantities, while the components of f = (f1,…,fn) are the luxes. The system is said strictly hyperbolic if at each point u the Jacobian matrix Df(u) has n real, distinct eigenvalues λ1(u)< ··· < λn(u). A fundamental ingredient to prove existence and stability in BV is the introduction of a functional, the Glimm–Liu interaction functional, which controls the interactions among non linear waves.   Aim of this note is to present a simple interpretation of (the scalar part of) the interaction functional, and show how it can be extended to the following equation: a parabolic equation of the form ut + f(u)x = uxx; scalar semidiscrete schemes, for example the upwind scheme ut(t,x) + (f(u(t,x)) − f(u(t,x −1)) = 0, or the backward scheme (u(t,x) − u(t −1,x)) + f(u(t,x)) = 0; 2 × 2 relaxation approximation, in particular ut + vx = 0, vt + ux = f(u) − v. All these approximations are interesting from the physical and numerical point of view. Finding an interaction is one of the key steps toward the proof of BV bounds. 61 71 1 10.4171/009-1/4 http://www.ems-ph.org/doi/10.4171/009-1/4

Alexei Borodin California Institute of Technology, PASADENA, UNITED STATES Grigori Olshanski Institute for Information Transmission Problems, Moscow GSP-4, RUSSIAN FEDERATION Group theory and generalizations General On a particular example we describe how to state and to solve the problem of harmonic analysis for groups with infinite-dimensional dual space. The representation theory for such groups differs in many respects from the conventional theory. We emphasize a remarkable connection with random point processes that arise in random matrix theory. The paper is an extended version of the second author’s talk at the Congress. 73 94 1 10.4171/009-1/5 http://www.ems-ph.org/doi/10.4171/009-1/5

François Bouchut Ecole Normale Superieure, PARIS CEDEX 05, FRANCE Partial differential equations General These notes intend to give an introduction to the recent development of relaxation models, the associated stability conditions, and discrete approximations. 95 101 1 10.4171/009-1/6 http://www.ems-ph.org/doi/10.4171/009-1/6

Brian Bowditch University of Warwick, COVENTRY, UNITED KINGDOM Manifolds and cell complexes Group theory and generalizations General We give a brief survey of some recent work on 3-manifolds, notably towards proving Thurston’s ending lamination conjecture. We describe some applications to the theory of surfaces and mapping class groups. 103 115 1 10.4171/009-1/7 http://www.ems-ph.org/doi/10.4171/009-1/7

Ehud Friedgut Hebrew University, JERUSALEM, ISRAEL Functional analysis Group theory and generalizations General Let p ≤ ½ and let μp be the product measure on {0,1}n, where μp(x) = p∑xi (1 − p)n − ∑xi. Let A ⊂ {0,1}n be an intersecting family, i.e., for every x, y ∈ A there exists 1 ≤ i ≤ n such that xi = yi = 1. Then μp(A) ≤ p. The proof uses discrete harmonic analysis. 117 120 1 10.4171/009-1/8 http://www.ems-ph.org/doi/10.4171/009-1/8

Patrick Gérard Université Paris-Sud, ORSAY CX, FRANCE Partial differential equations Dynamical systems and ergodic theory Quantum theory General Nonlinear Schrödinger equations have been studied by mathematicians for about thirty years. However, until recently, most of the contributions concerned the equation on the whole Euclidean space, with the notable exception of J. Bourgain’s contributions on tori. In the case of general Riemannian manifolds, the interaction of geometry with nonlinear operations leads to new phenomena, particularly if the manifold is compact. Here we review the state of the art concerning the Cauchy problem on such manifolds, and we describe optimal results on spheres, where new estimates on spherical harmonics play a crucial role. The matter of this paper is based on a series of results in collaboration with N. Burq and N. Tzvetkov. 121 139 1 10.4171/009-1/9 http://www.ems-ph.org/doi/10.4171/009-1/9

Alice Guionnet Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES Commutative rings and algebras General 141 154 1 10.4171/009-1/10 http://www.ems-ph.org/doi/10.4171/009-1/10

Stefan Helmke Kyoto University, KYOTO, JAPAN Peter Slodowy Universität Hamburg, HAMBURG, GERMANY Group theory and generalizations Algebraic geometry General 155 172 1 10.4171/009-1/11 http://www.ems-ph.org/doi/10.4171/009-1/11

Helge Holden University of Trondheim, TRONDHEIM, NORWAY Partial differential equations Algebraic geometry General We survey recent results for the Camassa–Holm equation ut − uxxt + 2κux + 3uux − 2uxuxx − uuxxx = 0, in particular convergence of a carefully selected finite difference scheme in the case of periodic initial data, and a detailed description of algebro-geometric solutions of the Camassa–Holm hierarchy. Furthermore, we present results for the generalized hyperelastic-rod wave equation ut − uxxt + ½ g(u)x = γ(2uxuxx + uuxxx). Finally, we discuss convergence of finite difference schemes for the Hunter–Saxton equation (ut + uux)x = ½ (ux)2 and describe semi-discrete, implicit as well as explicit upwind schemes that converge to diffusive solutions of the Hunter–Saxton equation. 173 200 1 10.4171/009-1/12 http://www.ems-ph.org/doi/10.4171/009-1/12

Rupert Klein Freie Universität Berlin, BERLIN, GERMANY Eileen Mikusky Potsdam Institute for Climate Impact Research (PIK, POTSDAM, GERMANY Antony Owinoh Potsdam Institute for Climate Impact Research (PIK, POTSDAM, GERMANY Fluid mechanics Biology and other natural sciences General One important activity of theoretical meteorology involves the development of simplified model equations that describe selected scale-dependent phenomena observed in atmospheric flows. This paper summarizes a unified mathematical approach to the derivation of such models, based on multiple scales asymptotic techniques.   First we motivate the approach by an example from fluid mechanics, the interaction of small-scale quasi-incompressible flow with long-wave acoustics. In this case, the analysis proceeds via multiple scales asymptotics in terms of the Mach number, M, as the small expansion parameter. Then we discuss the particular setting of meteorology, where there is a host of singular small parameters to be taken into account. Examples are the Rossby, Froude, Mach, and Strouhal numbers. A particular distinguished limit among these parameters is introduced in combination with systematic multiple scales asymptotics. A wide range of simplified meteorological models can then be recovered by specializing this general ansatz to a single horizontal, a single vertical, and a single time coordinate.   As a concrete example we report on the multiple scales derivation of boundary layer theories. In particular, we recover the classical Ekman boundary layer equations for flows on synoptic scales (∼ 500 km, 12 hours), and find an extension of the nonlinear Prandtl boundary layer equations to atmospheric mesoscales (∼ 70 km, 2 hours). 201 220 1 10.4171/009-1/13 http://www.ems-ph.org/doi/10.4171/009-1/13

Jan Krajícek Czech Academy of Sciences, PRAGUE 1, CZECH REPUBLIC Computer science General This note, based on my 4ECM lecture, exposes few basic points of proof complexity in a way accessible to any mathematician. 221 232 1 10.4171/009-1/14 http://www.ems-ph.org/doi/10.4171/009-1/14

Daan Krammer University of Warwick, COVENTRY, UNITED KINGDOM Group theory and generalizations General For any tangle T (up to isotopy) and integer k ≥ 1 we construct a group F(T) (up to isomorphism). It is the fundamental group of the configuration space of k points in a horizontal plane avoiding the tangle, provided the tangle is in what we call Heegaard position. This is analogous to the first half of Lawrence’s homology construction of braid group representations. We briefly discuss the second half: homology groups of F(T) . 233 246 1 10.4171/009-1/15 http://www.ems-ph.org/doi/10.4171/009-1/15

Elon Lindenstrauss The Hebrew University of Jerusalem, JERUSALEM, ISRAEL Dynamical systems and ergodic theory Number theory General 247 256 1 10.4171/009-1/16 http://www.ems-ph.org/doi/10.4171/009-1/16

Tomasz Łuczak Adam Mickiewicz University, POZNAN, POLAND Mathematical logic and foundations Game theory, economics, social and behavioral sciences General We present a few results concerning the phase transition phenomenon in the theory of random graphs, mathematical logic, and game theory. 257 268 1 10.4171/009-1/17 http://www.ems-ph.org/doi/10.4171/009-1/17

Terry Lyons Oxford University, OXFORD, UNITED KINGDOM Probability theory and stochastic processes General It is a matter of observation that many complex and important systems evolve, and that this evolution depends, to an extent, on external stimuli (which we call controls). These stimuli are frequently varying with time, and on normal scales are often highly oscillatory and potentially non-differentiable. To model them mathematically one has to go beyond the classical theory of differential equations and find a meaning for dy 269 282 1 10.4171/009-1/18 http://www.ems-ph.org/doi/10.4171/009-1/18

Jørgen Ellegaard Andersen University of Aarhus, AARHUS C, DENMARK Michael Weiss University of Aberdeen, ABERDEEN, UNITED KINGDOM Algebraic geometry General This overview is intended as a lightweight companion to the long article [20]. One of the main results there is the determination of the rational cohomology of the stable mapping class group, in agreement with the Mumford conjecture [26]. This is part of a recent development in surface theory which was set in motion by Ulrike Tillmann’s discovery [34] that Quillen’s plus construction turns the classifying space of the stable mapping class group into an infinite loop space. Tillmann’s discovery depends heavily on Harer’s homological stability theorem [15] for mapping class groups, which can be regarded as one of the high points of geometric surface theory. 283 308 1 10.4171/009-1/19 http://www.ems-ph.org/doi/10.4171/009-1/19

Pascal Massart Université Paris Sud, ORSAY CEDEX, FRANCE Probability theory and stochastic processes Information and communication, circuits General Model selection is a classical topic in statistics. The idea of selecting a model via penalizing a log-likelihood type criterion goes back to the early seventies with the pioneering works of Mallows and Akaike. One can find many consistency results in the literature for such criteria. These results are asymptotic in the sense that one deals with a given number of models and the number of observations tends to infinity. We shall give an overview of a non asymtotic theory for model selection which has emerged during these last ten years. In various contexts of function estimation it is possible to design penalized log-likelihood type criteria with penalty terms depending not only on the number of parameters defining each model (as for the classical criteria) but also on the “complexity” of the whole collection of models to be considered. For practical relevance of these methods, it is desirable to get a precise expression of the penalty terms involved in the penalized criteria on which they are based. Our approach heavily relies on concentration inequalities, the prototype being Talagrand's inequality for empirical processes which leads to explicit penalties. Simultaneously, we derive non asymptotic risk bounds for the corresponding penalized estimators showing that they perform almost as well as if the “best model“ (i.e., with minimal risk) were known. Our purpose will be to give an account of the theory and discuss some selected applications such as variable selection or change point detection. 309 324 1 10.4171/009-1/20 http://www.ems-ph.org/doi/10.4171/009-1/20

Preda Mihailescu Georg-August-Universität Göttingen, GÖTTINGEN, GERMANY Number theory General Catalan's conjecture states that the equation xp − yq = 1 has no other integer solutions but 32 − 23 = 1. We prove a theorem which simplifies the proof of this conjecture. 325 340 1 10.4171/009-1/21 http://www.ems-ph.org/doi/10.4171/009-1/21

Mircea Mustaţă University of Michigan, ANN ARBOR, UNITED STATES Shunsuke Takagi Kyushu University, FUKUOKA, JAPAN Kei-ichi Watanabe College of Humanities and Sciences, TOKYO, JAPAN Algebraic geometry General 341 364 1 10.4171/009-1/22 http://www.ems-ph.org/doi/10.4171/009-1/22

Kieran O'Grady Università di Roma La Sapienza, ROMA, ITALY Algebraic geometry Manifolds and cell complexes General We survey some recent and less recent results on hyperkähler manifolds, i.e., irreducible (holomorphic) symplectic manifolds. The point of view will be that of an algebraic (or complex) geometer. 365 380 1 10.4171/009-1/23 http://www.ems-ph.org/doi/10.4171/009-1/23

Imre Ruzsa Hungarian Academy of Sciences, BUDAPEST, HUNGARY Group theory and generalizations General Highlights in the theory of sets with small sumset and related problems. 381 390 1 10.4171/009-1/24 http://www.ems-ph.org/doi/10.4171/009-1/24

Yehuda Shalom University of California, Los Angeles, LOS ANGELES, UNITED STATES Group theory and generalizations General Measurable group theory aims at understanding how much of the algebraic structure of a countable group can be recovered solely from the equivalence relation of “being in the same orbit”, induced by (specific, or all) finite measure preserving actions of the group. It turns out that some groups (e.g., abelian) lose “most” of their structure, while for others the opposite happens, to the extent that occasionally both the group and the action can be entirely reconstructed from the equivalence relation. This theme turns out to be a common playground for diverse areas of research including ergodic theory, operator algebra, and descriptive set theory. It can also be viewed as the “measurable younger brother” of geometric group theory; a perspective (due to Gromov), which is found to be fruitful in both disciplines. Nowadays, following five to six years of rapid progress involving the introduction of diverse and deep tools, it may be considered as an independent – even if interdisciplinary – area of research in its own right.   The purpose of this survey is to describe the foundations of the discipline on one hand, and its most recent exciting developments on the other, in as friendly and non-technical a manner as possible. The vast majority of results presented here appeared over the past six years, and of those many had not been published at the time of writing this paper. Proofs are not given, and occasionally even the results themselves are not stated in their full generality. Rather, the emphasis is on the main ideas, approaches, and concepts underlying the results, along with the connections between them. Each of the sections of the paper actually deserves a considerably more detailed exposition, and we hope that the inevitable omission of some results will be received with understanding. We do try, however, to offer a fairly complete list of relevant bibliography to which, at appropriate places, readers are referred for further details. We hope that while helping to bridge the “cultural differences” present in this interdisciplinary area, the exposition will also serve as an attractive and welcoming invitation for mathematicians working in neighboring fields, as well as graduate students taking their first steps. 391 424 1 10.4171/009-1/25 http://www.ems-ph.org/doi/10.4171/009-1/25

M. Shcherbina Ukrainian Academy of Sciences, KHARKOV, UKRAINE Computer science General We discuss some problems of the dynamics of neural networks, in particular, the rigorous results on the critical capacity of the Hopfield model, the cavity method for spin glass and the rigorous solution of the Gardner problem. 425 444 1 10.4171/009-1/26 http://www.ems-ph.org/doi/10.4171/009-1/26

Mikhail Sodin Tel-Aviv University, TEL-AVIV, ISRAEL Number theory General Geometrically, zeroes of a Gaussian analytic function are intersection points of an analytic curve in a Hilbert space with a randomly chosen hyperplane. Mathematical physics provides another interpretation as a gas of interacting particles. In the last decade, these interpretations influenced progress in understanding statistical patterns in the zeroes of Gaussian analytic functions, and led to the discovery of canonical models with invariant zero distribution. We shall discuss some of recent results in this area and mention several open questions. 445 458 1 10.4171/009-1/27 http://www.ems-ph.org/doi/10.4171/009-1/27

Xavier Tolsa Universitat Autónoma de Barcelona, BARCELONA, SPAIN Functional analysis General In this paper we survey some recent results in connection with the so-called Painlevé’s problem, the semiadditivity of analytic capacity and other related results. 459 476 1 10.4171/009-1/28 http://www.ems-ph.org/doi/10.4171/009-1/28

Anna-Karin Tornberg New York University, NEW YORK, UNITED STATES Partial differential equations General Regularization of singular source terms in partial differential equations is a widely used approach to deal with the challenge of numerically approximating such equations. In this paper, we analyze regularization techniques in one and multi-dimensions. We consider general numerical grids as well as the special case of a uniform grid, in which case discrete properties allow for reduced errors.   All regularizations are based on a regularization of the Dirac delta function in one dimension, that are extended to more dimensions. It is critical how this extension is done. While it can produce a multi-dimensional approximation with the same accuracy as given by the one-dimensional analysis, we also show that a technique commonly used in connection to level-set methods may lead to O(1) errors. Modifications to this inconsistent technique are introduced. 477 500 1 10.4171/009-1/29 http://www.ems-ph.org/doi/10.4171/009-1/29

Vilmos Totik University of Szeged, SZEGED, HUNGARY Approximations and expansions Potential theory General Two areas of approximation theory are reviewed which utilize potential theoretical methods. These are approximation by weighted polynomials with varying weights and the so-called polynomial inverse image method. We also illustrate the latter method by finding the form of some equilibrium measures. 501 514 1 10.4171/009-1/30 http://www.ems-ph.org/doi/10.4171/009-1/30

Wendelin Werner ETH Zürich, ZÜRICH, SWITZERLAND Mechanics of particles and systems General There have been quite a few surveys or proceedings papers on the present subject of Schramm-Loewner Evolutions and 2D critical phenomena in recent years, such for instance as [38], [17], [18], [8], [42], as well as lecture notes [39], [16], [41], and a book [19] is in preparation. We will here try to review the subject including some recent and ongoing work, in an informal style close to that of an actual seminar talk. 515 528 1 10.4171/009-1/31 http://www.ems-ph.org/doi/10.4171/009-1/31

Umberto Zannier Scuola Normale Superiore, PISA, ITALY Algebraic geometry General We shall briefly present some recent results concerning S-integral points on certain varieties of dimension > 1. Some of them concern non-singular surfaces, some other ones arise from diophantine equations with linear recurrences. These results, obtained jointly with P. Corvaja, use the Schmidt Subspace Theorem applied to suitable auxiliary linear forms, constructed as values of certain regular functions on the variety. For the methods to apply, it is often necessary that the divisor at infinity is highly reducible; we give examples showing that on dropping this restriction very difficult problem arise even in simple contexts. We then show some instances when the reducibility condition may be achieved by taking an unramified cover, as in recent work by Faltings.   Generally speaking, we shall not assume prior knowledge of the subject and we shall often illustrate the theorems by quite concrete examples. 529 546 1 10.4171/009-1/32 http://www.ems-ph.org/doi/10.4171/009-1/32

Aline Bonami Université d'Orléans, ORLÉANS CEDEX 2, FRANCE Abstract harmonic analysis General In this survey, we consider two kinds of problems on tube domains over light cones. The first one is related with Poisson-Szegő integrals F. When F is a bounded real function and satisfies an appropriate smoothness up to the boundary, then F is necessarily the real part of a holomorphic function. The second one is the Lp boundedness of the Bergman projection, for which known positive and negative results leave a gap between them. This gives an illustration of activities within HARP. 549 554 1 10.4171/009-1/33 http://www.ems-ph.org/doi/10.4171/009-1/33

Yann Brenier Université de Nice, NICE CEDEX 2, FRANCE Partial differential equations General One of the most fruitful interaction between kinetic theory and non-linear hyperbolic PDEs has been the so-called kinetic formulation of conservation laws. Very recently, within the HYKE network, a beautiful structure theorem for non-smooth solutions of some non-linear hyperbolic PDEs has been obtained by de Lellis, Otto and Westdickenberg [DOW]. The proof is an unusual blend of kinetic ideas, blow-up techniques and tools coming from geometric measure theory. 555 560 1 10.4171/009-1/34 http://www.ems-ph.org/doi/10.4171/009-1/34

Frank den Hollander Universiteit Leiden, LEIDEN, NETHERLANDS Probability theory and stochastic processes General This short contribution describes the scientific programme “Random Dynamics in Spatially Extended Systems” that is supported by the European Science Foundation. In this programme, which runs over the period 2002–2006, 13 European countries participate. The main activities of the programme are listed, and a brief sketch is given of some of the main developments and future challenges in each of the eight research themes the programme is targeting. 561 572 1 10.4171/009-1/35 http://www.ems-ph.org/doi/10.4171/009-1/35

Jean Esterle Université de Bordeaux I et C.N.R.S., TALENCE CEDEX, FRANCE Operator theory General 573 596 1 10.4171/009-1/36 http://www.ems-ph.org/doi/10.4171/009-1/36

Bernard Helffer Université de Nantes, NANTES, FRANCE Partial differential equations Quantum theory General The aim of this article is to present a review on the analysis of the discrete spectrum of the Schrödinger operator with magnetic field, with particular emphasis on the bottom of the spectrum. 597 618 1 10.4171/009-1/37 http://www.ems-ph.org/doi/10.4171/009-1/37

J.P. Keating University of Bristol, BRISTOL, UNITED KINGDOM Quantum theory General This is a brief report on some of the highlights of the research supported by the EC Training Research Network Mathematical Aspects of Quantum Chaos, as described in my lecture at the 2004 European Mathematical Congress. The Network consisted of teams from Bristol (UK), Bologna (Italy), Paris (France), Tel Aviv (Israel), Ulm (Germany) and Uppsala (Sweden). It covered a broad spectrum of interests, from mathematics to theoretical physics. My focus here will be on a personal selection of some of the main mathematical achievements. 619 624 1 10.4171/009-1/38 http://www.ems-ph.org/doi/10.4171/009-1/38

Christian Krattenthaler Universität Wien, WIEN, AUSTRIA Combinatorics General I present four highlights from the scientific activities of the eight teams of the Research Training Network “Algebraic Combinatorics in Europe” in Bordeaux, Jerusalem, Linköping, Lyon, Marburg, Marne-la-Vallée, Roma, and Wien: the proof of the combinatorial invariance conjecture for Kazhdan–Lusztig polynomials in an important special case; the proof of the conjectures on the random assignment problem; important progress on the Neggers–Stanley Conjecture on posets and the Charney–Davis conjecture on (certain) simplicial complexes; new bijections between maps and trees with interesting applications to the Ising and the hard particle model on maps. 625 642 1 10.4171/009-1/39 http://www.ems-ph.org/doi/10.4171/009-1/39

Marina Monsurrò Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND Group theory and generalizations General 643 654 1 10.4171/009-1/40 http://www.ems-ph.org/doi/10.4171/009-1/40

Miles Reid University of Warwick, COVENTRY, UNITED KINGDOM Algebraic geometry General Problems on the existence and moduli of abstract varieties in the classification of varieties can often be studied by embedding the variety X into projective space, preferably in terms of an intrinsically determined ample line bundle L such as the (anti-) canonical class or its submultiples. A comparatively modern twist on this old story is to study the graded coordinate ring R(X,L) = ⊕n ≥ 0 H0(X, L⊗n), which in interesting cases is a Gorenstein ring; this makes available theoretical and computations tools from commutative algebra and computer algebra. The varieties of interest are curves, surfaces, 3-folds, and historical results of Enriques, Fano and others are sometimes available to serve as a guide. This has been a prominent area of work within European algebraic geometry in recent decades, and the lecture will present the current state of knowledge, together with some recent examples. 655 668 1 10.4171/009-1/41 http://www.ems-ph.org/doi/10.4171/009-1/41

Ryszard Nest University of Copenhagen, COPENHAGEN, DENMARK Quantum theory General 669 680 1 10.4171/009-1/42 http://www.ems-ph.org/doi/10.4171/009-1/42

Jakob Stix Universität Bonn, BONN, GERMANY Group theory and generalizations Number theory General GTEM is an acronym for Galois Theory and Explicit Methods. A selection from the activities within the network is presented. We focus on nonabelian Galois action, Grothendieck–Teichmüller theory and anabelian geometry. The style is expository and proofs are omitted. 681 696 1 10.4171/009-1/43 http://www.ems-ph.org/doi/10.4171/009-1/43

François Golse École Polytechnique, PARIS, FRANCE Fluid mechanics General This article reviews recent progress on the derivation of the fundamental PDE models in fluid mechanics from the Boltzmann equation. 699 718 1 10.4171/009-1/44 http://www.ems-ph.org/doi/10.4171/009-1/44

Francesco Guerra Università di Roma La Sapienza, ROMA, ITALY Statistical mechanics, structure of matter General A comprehensive review will be given about the rich mathematical structure of mean field spin glass theory, mostly developed, until now, in the frame of the methods of theoretical physics, based on deep physical intuition and hints coming from numerical simulation. Central to our treatment is a very simple and yet powerful interpolation method, allowing to compare different probabilistic schemes, by using convexity and positivity arguments. In this way we can prove the existence of the thermodynamic limit for the free energy density of the system, a long standing open problem. Moreover, in the frame of a generalized variational principle, we can show the emergency of the Derrida-Ruelle random probability cascades, leading to the form of free energy given by the celebrated Parisi Ansatz. All these results seem to be in full agreement with the mechanism of spontaneous replica symmetry breaking as developed by Giorgio Parisi. 719 732 1 10.4171/009-1/45 http://www.ems-ph.org/doi/10.4171/009-1/45

Johan Håstad KTH, STOCKHOLM, SWEDEN Computer science General We give a short introduction to some questions in complexity theory and proceed to describe some recent developments. In particular, we discuss probabilistically checkable proofs and their applications in establishing inapproximability results. In a traditional proof the proof-checker reads the entire proof and decides deterministically whether the proof is correct. In a probabilistically checkable proof the proof-checker randomly verifies only a very small portion of the proof but still cannot be fooled into accepting a false claim except with small probability. 733 750 1 10.4171/009-1/46 http://www.ems-ph.org/doi/10.4171/009-1/46

Elliott Lieb Princeton University, PRINCETON, UNITED STATES Algebraic geometry General 751 768 1 10.4171/009-1/47 http://www.ems-ph.org/doi/10.4171/009-1/47

Peter Ozsváth Princeton University, PRINCETON, UNITED STATES Zoltán Szabó Princeton University, PRINCETON, UNITED STATES Manifolds and cell complexes General The aim of this paper is to give a quick introduction to Heegaard Floer homology [31] for three-dimensional manifolds and a related Floer homology invariant for knots [37], [41] and discuss some recent results. 769 782 1 10.4171/009-1/48 http://www.ems-ph.org/doi/10.4171/009-1/48

Michael Freedman University of California, SANTA BARBARA, UNITED STATES Probability theory and stochastic processes General 783 786 1 10.4171/009-1/49 http://www.ems-ph.org/doi/10.4171/009-1/49

Claire Voisin Institut de Mathématiques de Jussieu, PARIS, FRANCE Algebraic geometry Differential geometry General 787 807 1 10.4171/009-1/50 http://www.ems-ph.org/doi/10.4171/009-1/50

Franck Barthe Université Toulouse III, TOULOUSE CEDEX, FRANCE Probability theory and stochastic processes General We survey recent developments of the isoperimetric problem for product probability measures. These questions were considered with motivations coming from different areas (differential geometry, Banach spaces, probability theory) and substantial progress was enabled by combining geometric, probabilistic and analytic approaches. 811 826 1 10.4171/009-1/51 http://www.ems-ph.org/doi/10.4171/009-1/51

Paul Biran ETH Zürich, ZÜRICH, SWITZERLAND Manifolds and cell complexes General In this paper we outline a recent direction of research interrelating symplectic geometry and algebraic geometry. We show how methods and ideas from symplectic geometry can be used to study classical algebro-geometric problems on hyperplane sections and degenerations of algebraic varieties. 827 836 1 10.4171/009-1/52 http://www.ems-ph.org/doi/10.4171/009-1/52

Sylvia Serfaty Université Pierre et Marie Curie (Paris VI), PARIS CEDEX 05, FRANCE Partial differential equations General 837 850 1 10.4171/009-1/53 http://www.ems-ph.org/doi/10.4171/009-1/53

Warwick Tucker Uppsala Universitet, UPPSALA, SWEDEN Computer science General The aim of this paper is to give a very brief introduction to the emerging area of validated numerics. This is a rapidly growing field of research faced with the challenge of interfacing computer science and pure mathematics. Most validated numerics is based on interval analysis, which allows its users to account for both rounding and discretization errors in computer-aided proofs. We will illustrate the strengths of these techniques by converting the well-known bisection method into a efficient, validated root finder. 851 860 1 10.4171/009-1/54 http://www.ems-ph.org/doi/10.4171/009-1/54

Otmar Venjakob Universität Heidelberg, HEIDELBERG, GERMANY Number theory General This paper, which is an extended version of my talk ‘The GL2 main conjecture for elliptic curves without complex multiplication’ given on the 4ECM, aims to give a survey on recent developments in non-commutative Iwasawa theory. It is written mainly for non-experts and contains neither proofs nor any new results, but hopefully serves as introduction to the original articles [8], [41]. Also, technical details are sometimes placed into footnotes in order to keep the main text as easily accessible as possible. 861 879 1 10.4171/009-1/55 http://www.ems-ph.org/doi/10.4171/009-1/55

Alan Carey Australian National University, Canberra Global analysis, analysis on manifolds Number theory Functional analysis Quantum theory 58B34, 11M55, 11G09 11M06, 11M32, 47G30, 81Q30, 81T15, 17A30; 16T30, 18D50, 46L80, 46L87, 19K33, 19K56, 58J42, 58J20, 81Q30 Arithmetic geometry, motives, noncommutative geometry, index theory, K-theory, pre-Lie algebras, Feynman integrals, zeta functions, Birkhoff–Hopf factorisation, renormalisation This collection of expository articles grew out of the workshop “Number Theory and Physics” held in March 2009 at the The Erwin Schrödinger International Institute for Mathematical Physics, Vienna. The common theme of the articles is the influence of ideas from noncommutative geometry (NCG) on subjects ranging from number theory to Lie algebras, index theory, and mathematical physics. Matilde Marcolli’s article gives a survey of relevant aspects of NCG in number theory, building on an introduction to motives for beginners by Jorge Plazas and Sujatha Ramdorai. A mildly unconventional view of index theory from the viewpoint of NCG is described in the article by Alan Carey, John Phillips and Adam Rennie. As developed by Alain Connes and Dirk Kreimer, NCG also provides insight into novel algebraic structures underlying many analytic aspects of quantum field theory. Dominique Manchon's article on pre-Lie algebras fits into this developing research area. This interplay of algebraic and analytic techniques also appears in the articles by Christoph Bergbauer, who introduces renormalisation theory and Feynman diagram methods, and Sylvie Paycha, who focuses on relations between renormalisation and zeta function techniques. 7 6 2011 978-3-03719-008-1 978-3-03719-508-6 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/008 http://www.ems-ph.org/doi/10.4171/008

Christoph Bergbauer Johannes Gutenberg-Universität, MAINZ, GERMANY Quantum theory Algebraic geometry Several complex variables and analytic spaces General Various aspects of Feynman integrals, regularization and renormalization are reviewed. Following Bloch, the focus is upon a linear algebraic approach to the Feynman rules. Using resolution of singularities, several renormalization methods are brought together. In the second part, the work of Belkale and Brosnan resp. Bloch, Esnault and Kreimer on the motivic nature of Feynman integrals is briefly sketched. 1 40 1 10.4171/008-1/1 http://www.ems-ph.org/doi/10.4171/008-1/1

Sujatha Ramdorai Tata Institute of Fundamental Research, MUMBAI, INDIA Jorge Plazas University of Utrecht, UTRECHT, NETHERLANDS Matilde Marcolli California Institute of Technology, PASADENA, UNITED STATES Global analysis, analysis on manifolds Number theory Algebraic geometry Category theory; homological algebra This article is based on the lectures of the same tittle given by the first author during the instructional workshop of the program “number theory and physics” at ESI Vienna during March 2009. An account of the topics treated during the lectures can be found in [50] where the categorical aspects of the theory are stressed. Although naturally overlapping, these two independent articles serve as complements to each other. In the present article we focus on the construction of the category of pure motives starting from the category of smooth projective varieties. The necessary preliminary material is discussed. Early accounts of the theory were given in Manin [42] and Kleiman [36], the material presented here reflects to some extent their treatment of the main aspects of the theory. We also survey the theory of endomotives developed in [11], this provides a link between the theory of motives and tools from quantum statistical mechanics which play an important role in results connecting number theory and noncommutative geometry. An extended appendix (by Matilde Marcolli) further elaborates these ideas and reviews the role of motives in noncommutative geometry. 41 87 1 10.4171/008-1/2 http://www.ems-ph.org/doi/10.4171/008-1/2

Dominique Manchon Université Blaise Pascal, AUBIÈRE CEDEX, FRANCE Nonassociative rings and algebras Associative rings and algebras Category theory; homological algebra General We give an account of fundamental properties of pre-Lie algebras, and provide several examples borrowed from various domains of mathematics and physics: algebra, combinatorics, quantum field theory and numerical analysis. 89 102 1 10.4171/008-1/3 http://www.ems-ph.org/doi/10.4171/008-1/3

Sylvie Paycha Université Blaise Pascal, AUBIÈRE CEDEX, FRANCE Number theory Operator theory Quantum theory General We investigate the meromorphic behaviour of multiple discrete sums of tensor products of symbols with conical constraints, multiple integrals of tensor products of symbols with linear constraints, and compare their pole structures. We discuss how to extract reasonable finite parts at poles leading to “renormalised” multiple discrete sums and integrals which factorise on tensor products. The first generalise “renormalised”' multiple zeta values at non positive integers, whereas the second relate to “renormalised” Feynman integrals in physics. These lectures, which are essentially self-contained, start with a review of regularised integrals and sums in one variable. 103 174 1 10.4171/008-1/4 http://www.ems-ph.org/doi/10.4171/008-1/4

Alan Carey The Australian National University, CANBERRA, AUSTRALIA John Phillips University of Victoria, VICTORIA, B.C., CANADA Adam Rennie University of Wollongong, WOLLONGONG, AUSTRALIA Functional analysis $K$-theory Global analysis, analysis on manifolds General The main objective of these notes is to give some intuition about spectral triples and the role they play in index theory. The notes are basically a road map, with much detail omitted. 175 265 1 10.4171/008-1/5 http://www.ems-ph.org/doi/10.4171/008-1/5

Carel Faber Royal Institute of Technology, Stockholm, Sweden Gerard van der Geer University of Amsterdam, The Netherlands Eduard Looijenga University of Utrecht, The Netherlands Algebraic geometry 14E, 14E30, 14D, 14J10 Classification of algebraic varieties, minimal model program, birational geometry Fascinating and surprising developments are taking place in the classification of algebraic varieties. Work of Hacon and McKernan and many others is causing a wave of breakthroughs in the Minimal Model Program: we now know that for a smooth projective variety the canonical ring is finitely generated. These new results and methods are reshaping the field. Inspired by this exciting progress, the editors organized a meeting at Schiermonnikoog and invited leading experts to write papers about the recent developments. The result is the present volume, a lively testimony of the sudden advances that originate from these new ideas. This volume will be of interest to a wide range of pure mathematicians, but will appeal especially to algebraic and analytic geometers. 12 30 2010 978-3-03719-007-4 978-3-03719-507-9 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/007 http://www.ems-ph.org/doi/10.4171/007

Dan Abramovich Brown University, PROVIDENCE, UNITED STATES Brendan Hassett Brown University, PROVIDENCE, UNITED STATES Algebraic geometry General We describe a new approach to the definition of the moduli functor of stable varieties. While there is wide agreement as to what classes of varieties should appear, the notion of a family of stable surfaces is quite subtle, as key numerical invariants may fail to be constant in flat families. Our approach is to add natural stack structure to stable varieties. For example, given a canonical model we take the global-quotient stack structure arising from its realization as Proj of the canonical ring. Deformations of the stack structure preserve key numerical invariants of the stable variety, including the top self-intersection of the canonical divisor. This approach yields a transparent construction of the moduli stack of stable varieties as a global quotient of a suitable Hilbert scheme of weighted projective stacks. 1 38 1 10.4171/007-1/1 http://www.ems-ph.org/doi/10.4171/007-1/1

Florin Ambro Romanian Academy, BUCHAREST, ROMANIA Algebraic geometry General We present the elementary properties of log canonical centers of log varieties. 39 48 1 10.4171/007-1/2 http://www.ems-ph.org/doi/10.4171/007-1/2

Ingrid Bauer Universität Bayreuth, BAYREUTH, GERMANY Fabrizio Catanese Universität Bayreuth, BAYREUTH, GERMANY Algebraic geometry Group theory and generalizations Several complex variables and analytic spaces General This is the first in a series of articles on the moduli spaces of Burniat surfaces. We prove that Burniat surfaces are Inoue surfaces and we calculate their fundamental groups. As a main result of this paper we prove that every smooth projective surface which is homotopically equivalent to a primary Burniat surface (i.e., a Burniat surface with K2 = 6) is indeed a primary Burniat surface. 49 76 1 10.4171/007-1/3 http://www.ems-ph.org/doi/10.4171/007-1/3

Caucher Birkar University of Cambridge, CAMBRIDGE, UNITED KINGDOM Mihai Paun Université Henri Poincaré, VANDOEUVRE LES NANCY, FRANCE Algebraic geometry General In this paper, we give a proof of the existence of log minimal models for klt pairs (X/Z,B) with B big/Z. This then implies existence of klt log flips, finite generation of klt log canonical rings, and most of the other results of the Birkar–Cascini–Hacon–McKernan paper [3]. 77 113 1 10.4171/007-1/4 http://www.ems-ph.org/doi/10.4171/007-1/4

Frédéric Campana Université Henri Poincaré, VANDOEUVRE LES NANCY, FRANCE Algebraic geometry General In [U75], Example 16.15, p. 207, Kenji Ueno constructs the threefold X described below, and says that it is unknown, whether or not it is unirational. After recalling Ueno’s construction, we show that X is rationally connected. The (much deeper) question of whether X is unirational (or even rational) remains open. We found it worthwhile drawing attention to this seemingly unnoticed example, on which several questions (such as unirationality vs rational connectedness), might possibly be tested more easily than on the test cases usually considered (such as quartics or double ℙ3’s ramified along sextics). We formulate some of these questions in §4. 115 121 1 10.4171/007-1/5 http://www.ems-ph.org/doi/10.4171/007-1/5

Frédéric Campana Université Henri Poincaré, VANDOEUVRE LES NANCY, FRANCE Algebraic geometry General Among other things, we show the existence, for any $n$-dimensional connected complex projective manifold $X$, of a unique canonical and functorial fibration $c\colon X \to C(X)$, called ‘the core of $X$’, such that its general fibre is ‘special’, and its ‘orbifold base’ is of general type. Assuming an orbifold version of Iitaka’s $C_{n,m}$-conjecture, we next show that $c=(jr)^n$, where $j$ (resp. $r$) is an orbifold version of the Moishezon-Iitaka fibration (resp. of the ‘rational quotient’ – sometimes also called the MRC-fibration). In particular, the ‘special’ manifolds are also simply the orbifold combination of orbifolds with either $\kappa=0$ or $\kappa_+=-\infty$ (a weak version of rational connectedness). The class of ‘special’ manifolds may be seen of the class of manifolds which are ‘opposite’ to those of general type. It is for example conjectured that they are exactly the ones having an identically vanishing Kobayashi pseudometric, and also exactly the ones which are ‘potentially dense’, when defined over a number field. Because it is necessary to work with orbifolds pairs in order to get meaningful fibrations, we start by developing the geometry of such orbifold pairs, and obtain the above decomposition results in this enlarged orbifold context.This text is intended to be an introduction to the papers [Ca04] and [Ca07]. 123 170 1 10.4171/007-1/6 http://www.ems-ph.org/doi/10.4171/007-1/6

Jungkai Alfred Chen National Taiwan University, TAIPEI, TAIWAN Algebraic geometry General In this article, we survey various recent results on explicit birational geometry in dimension three. We explain the notion of packings of baskets introduced by Meng Chen and the author. We also demonstrate applications including pluricanonical maps of threefolds of general type, Fano threefolds and weighted complete intersections. Second part of the article is devoted to survey the joint work of Christopher Hacon and the author on factoring birational maps in dimension three. 171 187 1 10.4171/007-1/7 http://www.ems-ph.org/doi/10.4171/007-1/7

Ciro Ciliberto Università di Roma, ROMA, ITALY Olivia Dumitrescu Colorado State Univeristy, FORT COLLINS, UNITED STATES Rick Miranda Colorado State University, FORT COLLINS, UNITED STATES Joaquim Roé Universitat Autònoma de Barcelona, BELLATERRA, SPAIN Algebraic geometry General In this paper we give a new proof of the fact that for all pairs of positive integers (d,m) with d/m < 117/37, the linear system of plane curves of degree d with ten general base points of multiplicity m is empty. 189 195 1 10.4171/007-1/8 http://www.ems-ph.org/doi/10.4171/007-1/8

Alessio Corti Imperial College London, LONDON, UNITED KINGDOM Algebraic geometry General This note is an introduction to all the key ideas of Lazic's recent proof of the theorem on the finite generation of adjoint rings [Laz09]. (The theorem was first proved in [BCHM09].) I try to convince you that, despite technical issues that are not yet adequately optimised, nor perhaps fully understood, Lazic's argument is a self-contained and transparent induction on dimension based on lifting lemmas and relying on none of the detailed general results of Mori theory. On the other hand, it is shown in [CL10] that all the fundamental theorems of Mori theory follow easily from the finite generation statement discussed here: together, these results give a new and more efficient organisation of higher dimensional algebraic geometry. The approach presented here is ultimately inspired by a close reading of the work of Shokurov [Sho03], I mean specifically his proof of the existence of 3-fold flips. Siu was the first to believe in the possibility of a direct proof of finite generation, and believing that something is possible is, of course, a big part of making it happen. All mathematical detail is taken from [Laz09]; my contribution is merely exegetic. I begin with a few key definitions leading to the statement of the main result. 197 220 1 10.4171/007-1/9 http://www.ems-ph.org/doi/10.4171/007-1/9

Tommaso de Fernex University of Utah, SALT LAKE CITY, UNITED STATES Lawrence Ein University of Illinois at Chicago, CHICAGO, UNITED STATES Mircea Mustaţă University of Michigan, ANN ARBOR, UNITED STATES Algebraic geometry General We consider pairs (X,A), where X is a variety with klt singularities and A is a formal product of ideals on X with exponents in a fixed set that satisfies the Descending Chain Condition. We also assume that X has (formally) bounded singularities, in the sense that it is, formally locally, a subvariety in a fixed affine space defined by equations of bounded degree. We prove in this context a conjecture of Shokurov, predicting that the set of log canonical thresholds for such pairs satisfies the Ascending Chain Condition. 221 257 1 10.4171/007-1/10 http://www.ems-ph.org/doi/10.4171/007-1/10

Gavril Farkas Humboldt-Universität zu Berlin, BERLIN, GERMANY Algebraic geometry General We develop a theory of Brill-Noether divisors on the moduli space of stable spin curves of genus g, and compute the classes of these loci. These spin Brill–Noether cycles are defined in terms of the relative position of theta-characteristics with respect to certain Raynaud theta-divisors in the Jacobian of the curve. 259 276 1 10.4171/007-1/11 http://www.ems-ph.org/doi/10.4171/007-1/11

Kristina Frantzen Universität Bayreuth, BAYREUTH, GERMANY Thomas Peternell Universität Bayreuth, BAYREUTH, GERMANY Algebraic geometry General We prove that a compact contact threefold which is bimeromorphically equivalent to a Kähler manifold and not rationally connected is the projectivised tangent bundle of a Kähler surface. 277 288 1 10.4171/007-1/12 http://www.ems-ph.org/doi/10.4171/007-1/12

Osamu Fujino Kyoto University, KYOTO, JAPAN Algebraic geometry General This paper is a gentle introduction to the theory of quasi-log varieties by Ambro. We explain the fundamental theorems for the log minimal model program for log canonical pairs. More precisely, we give a proof of the base point free theorem for log canonical pairs in the framework of the theory of quasi-log varieties. 289 303 1 10.4171/007-1/13 http://www.ems-ph.org/doi/10.4171/007-1/13

Osamu Fujino Kyoto University, KYOTO, JAPAN Algebraic geometry General We give an alternate proof of the main theorem of Kawamata’s paper: Pluricanonical systems on minimal algebraic varieties. Our proof also works for varieties in class C. We note that our proof is completely different from Kawamata’s. 305 315 1 10.4171/007-1/14 http://www.ems-ph.org/doi/10.4171/007-1/14

Yujiro Kawamata University of Tokyo, TOKYO, JAPAN Algebraic geometry General We prove two theorems on the locally finite decompositions of the cones of divisors by the cones which correspond to canonical and minimal models. We introduce the concept of numerical linear systems in order to simplify the argument on the Zariski decompositions. 317 325 1 10.4171/007-1/15 http://www.ems-ph.org/doi/10.4171/007-1/15

Yuri Prokhorov Faculty of Mathematics, MOSCOW, RUSSIAN FEDERATION Algebraic geometry General For the subgroups of the Cremona group $\mathrm{Cr}_3(\mathbb C)$ having the form $(\boldsymbol{\mu}_p)^s$, where $p$ is prime, we obtain an upper bound for $s$. Our bound is sharp if $p\ge 17$. 327 338 1 10.4171/007-1/16 http://www.ems-ph.org/doi/10.4171/007-1/16

Sun-Yung Alice Chang Princeton University, USA Differential geometry Partial differential equations Manifolds and cell complexes Global analysis, analysis on manifolds 53Axx; 58Jxx Non-linear elliptic partial differential equations are an important tool in the study of Riemannian metrics in differential geometry, in particular for problems concerning the conformal change of metrics in Riemannian geometry. In recent years the role played by the second order semi-linear elliptic equations in the study of Gaussian curvature and scalar curvature has been extended to a family of fully non-linear elliptic equations associated with other symmetric functions of the Ricci tensor. A case of particular interest is the second symmetric function of the Ricci tensor in dimension four closely related to the Pfaffian. In these lectures, starting from the background material, the author reviews the problem of prescribing Gaussian curvature on compact surfaces. She then develops the analytic tools (e.g. higher order conformal invariant operators, Sobolev inequalities, blow-up analysis) in order to solve a fully nonlinear equation in prescribing the Chern-Gauss-Bonnet integrand on compact manifolds of dimension four. 10 25 2004 978-3-03719-006-7 978-3-03719-506-2 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/006 http://www.ems-ph.org/doi/10.4171/006 Demetrios Christodoulou ETH Zurich, Switzerland Relativity and gravitational theory 83C05; 35L70, 35Q75, 53C50, 58J45, 83C40, 83C45 General Relativity is a theory proposed by Einstein in 1915 as a unified theory of space, time and gravitation. It is based on and extends Newton’s theory of gravitation as well as Newton’s equations of motion. It is thus fundamentally rooted in classical mechanics. The theory can be seen as a development of Riemannian geometry, itself an extension of Gauss’ intrinsic theory of curved surfaces in Euclidean space. The domain of application of the theory is astronomical systems. One of the mathematical methods analyzed and exploited in the present volume is an extension of Noether’s fundamental principle connecting symmetries to conserved quantities. This is involved at a most elementary level in the very definition of the notion of hyperbolicity for an Euler–Lagrange system of partial differential equations. Another method is the study and systematic use of foliations by characteristic (null) hypersurfaces, and is in the spirit of the approach of Roger Penrose in his incompleteness theorem. The methods have applications beyond general relativity to problems in fluid mechanics and, more generally, to the mechanics and electrodynamics of continuous media. The book is intended for advanced students and researchers seeking an introduction into the methods and applications of general relativity. 2 5 2008 978-3-03719-005-0 978-3-03719-505-5 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/005 http://www.ems-ph.org/doi/10.4171/005 Katrin Wehrheim MIT, Cambrigde, USA Global analysis, analysis on manifolds Partial differential equations Functional analysis Quantum theory 53C05, 53C07; 58J32 This book gives a detailed account of the analytic foundations of gauge theory – Uhlenbeck’s compactness theorems for general connections and for Yang–Mills connections. It intends to guide graduate students into the analysis of Yang–Mills theory as well as to serve as a reference for researchers in the field. The book is largely self-contained. It contains a number of appendices (e.g. on Sobolev spaces of maps between manifolds) and an introductory part covering the Lp-regularity theory for the inhomogenous Neumann problem. The two main parts contain the full proofs of Uhlenbeck’s weak and strong compactness theorems on closed manifolds as well as their generalizations to manifolds with boundary and noncompact manifolds. These parts include a number of useful analytic tools such as general patching constructions and local slice theorems. 1 31 2004 978-3-03719-004-3 978-3-03719-504-8 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/004 http://www.ems-ph.org/doi/10.4171/004 Yakov Pesin Pennsylvania State University, USA Dynamical systems and ergodic theory 37-xx; 58-xx This book is an introduction to the modern theory of partial hyperbolicity with applications to stable ergodicity theory of smooth dynamical systems. It provides a systematic treatment of the theory and describes all the basic concepts and major results that have been obtained in the area since its creation around the early 1970s. It can be used as a textbook for a graduate student course and is also of interest to professional mathematicians working in the field of dynamical systems and their applications. 5 31 2004 978-3-03719-003-6 978-3-03719-503-1 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/003 http://www.ems-ph.org/doi/10.4171/003 Arkady Onishchik Yaroslavl State University, Russia Nonassociative rings and algebras Topological groups, Lie groups 17-01, 17B10, 17B20; 22E46 In 1914, E. Cartan posed the problem to find all irreducible real linear Lie algebras. An updated exposition of his work was given by Iwahori (1959). This theory reduces the classification of irreducible real representations of a real Lie algebra to a description of the so-called self-conjugate irreducible complex representations of this algebra and to the calculation of an invariant of such a representation (with values +1 or -1) which is called the index. Moreover, these two problems were reduced to the case when the Lie algebra is simple and the highest weight of its irreducible complex representation is fundamental. A complete case-by-case classification for all simple real Lie algebras was given (without proof) in the tables of Tits (1967). But actually a general solution of these problems is contained in a paper of Karpelevich (1955) (written in Russian and not widely known), where inclusions between real forms induced by a complex representation were studied. We begin with a simplified (and somewhat extended and corrected) exposition of the main part of this paper and relate it to the theory of Cartan-Iwahori. We conclude with some tables, where an involution of the Dynkin diagram which allows us to find self-conjugate representations is described and explicit formulas for the index are given. In a short addendum, written by J. v. Silhan, this involution is interpreted in terms of the Satake diagram. The book is aimed at students in Lie groups, Lie algebras and their representations, as well as researchers in any field where these theories are used. The reader is supposed to know the classical theory of complex semisimple Lie algebras and their finite dimensional representation; the main facts are presented without proofs in Section 1. In the remaining sections the exposition is made with detailed proofs, including the correspondence between real forms and involutive automorphisms, the Cartan decompositions and the conjugacy of maximal compact subgroups of the automorphism group. 2 28 2004 978-3-03719-002-9 978-3-03719-502-4 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/002 http://www.ems-ph.org/doi/10.4171/002 Richard Arratia University of Southern California, USA A. Barbour University of Zürich, Switzerland Simon Tavaré University of Southern California, USA Probability theory and stochastic processes Number theory Field theory and polynomials 05-02, 60-02; 05A16, 60C05 The elements of many classical combinatorial structures can be naturally decomposed into components. Permutations can be decomposed into cycles, polynomials over a finite field into irreducible factors, mappings into connected components. In all of these examples, and in many more, there are strong similarities between the numbers of components of different sizes that are found in the decompositions of `typical' elements of large size. For instance, the total number of components grows logarithmically with the size of the element, and the size of the largest component is an appreciable fraction of the whole. This book explains the similarities in asymptotic behaviour as the result of two basic properties shared by the structures: the conditioning relation and the logarithmic condition. The discussion is conducted in the language of probability, enabling the theory to be developed under rather general and explicit conditions; for the finer conclusions, Stein's method emerges as the key ingredient. The book is thus of particular interest to graduate students and researchers in both combinatorics and probability theory. 12 15 2003 978-3-03719-000-5 978-3-03719-500-0 European Mathematical Society Publishing House Zuerich, Switzerland 10.4171/000 http://www.ems-ph.org/doi/10.4171/000