2007; 4500 pp; hardcover ISBN10: 3037190221 ISBN13: 9783037190227 List Price: US$428 Member Price: US$342.40 Order Code: PICM/2006
 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 threevolume 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. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Undergraduates, graduate students, and research mathematicians. Table of Contents Volume I The work of the Fields Medalists, the Rolf Nevanlinna Prize Winner and the Gauss Prize Winner  G. Felder  The work of Andrei Okounkov
 J. Lott  The work of Grigory Perelman
 C. Fefferman  The work of Terence Tao
 C. M. Newman  The work of Wendelin Werner
 J. Hopcroft  The work of Jon Kleinberg
 H. Föllmer  On Kiyosi Itô's work and its impact
Plenary lectures  P. Deift  Universality for mathematical and physical systems
 J.P. Demailly  Kähler manifolds and transcendental techniques in algebraic geometry
 R. A. Devore  Optimal computation
 Y. Eliashberg  Symplectic field theory and its applications
 E. Ghys  Knots and dynamics
 H. Iwaniec  Prime numbers and \(L\)functions
 I. M. Johnstone  High dimensional statistical inference and random matrices
 K. Kato  Iwasawa theory and generalizations
 R. V. Kohn  Energydriven pattern formation
 I. Madsen  Moduli spaces from a topological viewpoint
 A. Nemirovski  Advances in convex optimization: conic programming
 S. Popa  Deformation and rigidity for group actions and von Neumann algebras
 A. Quarteroni  Cardiovascular mathematics
 O. Schramm  Conformally invariant scaling limits: an overview and a collection of problems
 R. P. Stanley  Increasing and decreasing subsequences and their variants
 T. Tao  The dichotomy between structure and randomness, arithmetic progressions, and the primes
 J. L. Vázquez  Perspectives in nonlinear diffusion: between analysis, physics and geometry
 M. Vergne  Applications of equivariant cohomology
 A. Wigderson  \(\mathcal{P, NP}\) and mathematicsa computational complexity perspective
Special activities  J. W. Morgan  The Poincaré Conjecture
 Panel discussion organised by the European Mathematical Society
 ICM 2006 closing round table
Cultural activities  J. M. SánchezRon  From the private to the public: The road from Zurich (1897) to Madrid (2006)
 List of participants
 Participants by country
 Author index
Volume II Logic and foundations  R. Downey  Algorithmic randomness and computability
 I. Neeman  Determinacy and large cardinals
 M. Rathjen  The art of ordinal analysis
 T. Scanlon  Analytic difference rings
 S. Thomas  Borel superrigidity and the classification problem for the torsionfree abelian groups of finite rank
Algebra  W. CrawleyBoevey  Quiver algebras, weighted projective lines, and the DeligneSimpson problem
 M. du Sautoy and F. Grunewald  Zeta functions of groups and rings
 B. Keller  On differential graded categories
 R. Rouquier  Derived equivalences and finite dimensional algebras
 M. Sapir  Algorithmic and asymptotic properties of groups
 Seress  A. unified approach to computations with permutation and matrix groups
 A. Smoktunowicz  Some results in noncommutative ring theory
Number theory  M. Bhargava  Higher composition laws and applications
 C.L. Chai  Hecke orbits as Shimura varieties in positive characteristic
 H. Darmon  Heegner points, StarkHeegner points and values of \(L\)series
 K. Fujiwara  Galois deformations and arithmetic geometry of Shimura varieties
 B. Green  Generalising the HardyLittlewood method for primes
 G. Laumon  Aspects géométriques du Lemme Fondamental de LanglandsShelstad
 P. Michel and A. Venkatesh  Equidistribution, \(L\)functions and ergodic theory: on some problems of Yu. Linnik
 W. Nizioł  \(p\)adic motivic cohomology in arithmetic
 C. Skinner and E. Urban  Vanishing of \(L\)functions and ranks of Selmer groups
 V. Vatsal  Special values of \(L\)functions modulo \(p\)
Algebraic and complex geometry  V. Alexeev  Higherdimensional analogues of stable curves
 J.B. Bost  Evaluation maps, slopes, and algebraicity criteria
 T. Bridgeland  Derived categories of coherent sheaves
 L. Ein and M. Mustaţǎ  Invariants of singularities of pairs
 T. Graber  Rational curves and rational points
 J.M. Hwang  Rigidity of rational homogeneous spaces
 T. Terasoma  Geometry of multiple zeta values
 Y. Tschinkel  Geometry over nonclosed fields
 J. Włodarczyk  Algebraic Morse theory and the weak factorization theorem
Geometry  C. Böhm and B. Wilking  Manifolds with positive curvature operators are space forms
 S. Brendle  Elliptic and parabolic problems in conformal geometry
 K. Honda  The topology and geometry of contact structures in dimension three
 M. Kapovich  Generalized triangle inequalities and their applications
 B. Kleiner  The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity
 F. Lalonde  Lagrangian submanifolds: from the local model to the cluster complex
 X. Liu  GromovWitten invariants and moduli spaces of curves
 T. Mabuchi  Extremal metrics and stabilities on polarized manifolds
 G. Mikhalkin  Tropical geometry and its applications
 W. P. Minicozzi II  Embedded minimal surfaces
 Y.G. Oh and K. Fukaya  Floer homology in symplectic geometry and in mirror symmetry
 A. Ros  Properly embedded minimal surfaces with finite topology
 C.L. Terng  Applications of loop group factorization to geometric soliton equations
Topology  I. Agol  Finiteness of arithmetic Kleinian reflection groups
 M. R. Bridson  Nonpositive curvature and complexity for finitely presented groups
 M. Khovanov  Link homology and categorification
 Y. N. Minsky  Curve complexes, surfaces and 3manifolds
 F. Morel  \(\mathbb{A}^1\)algebraic topology
 K. Ono  Development in symplectic Floer theory
 P. Ozváth and Z. Szabó  Heegaard diagrams and Floer homology
 K. Vogtmann  The cohomology of automorphism groups of free groups
Lie groups and Lie algebras  R. Bezrukavnikov  Noncommutative counterparts of the Springer resolution
 A. Braverman  Spaces of quasimaps into the flag varieties and their applications
 G. Henniart  On the local Langlands and JacquetLanglands correspondences
 N. Monod  An invitation to bounded cohomology
 B.C. Ngô  Fibration de Hitchin et structure endoscopique de la formule des traces
 E. M. Opdam  Hecke algebras and harmonic analysis
 P. Schneider  Continuous representation theory of \(p\)adic Lie groups
 Y. Shalom  The algebraization of Kazhdan's property (T)
 D. Soudry  RankinSelberg integrals, the descent method, and Langlands functorality
 B. Speh  Representation theory and the cohomology of arithmetic groups
 T. A. Springer  Some results on compactifications of semisimple groups
Analysis  M. Bonk  Quasiconformal geometry of fractals
 S. Hofmann  Local \(Tb\) theorems and applications in PDE
 S. V. Konyagin  Almost everywhere convergence and divergence of Fourier series
 L. P. Rothschild  Iterated Segre mappings of real submanifolds in complex space and applications
 S. Smirnov  Towards conformal invariance of 2D lattice models
 E. J. Straube  Aspects of the \(L^2\)Sobolev theory of the \(\overline\partial\)Neumann problem
 V. N. Temlyakov  Greedy approximations with regard to bases
 X. Tolsa  Analytic capacity, rectifiability, and the Cauchy integral
Operator algebras and functional analysis  F. Barthe  The BrunnMinkowski theorem and related geometric and functional inequalities
 B. Klartag  Isomorphic and almostisometric problems in highdimensional convex geometry
 N. Ozawa  Amenable actions and applications
 M. Rørdam  Structure and classification of \(C^\ast\)algebras
 S. J. Szarek  Convexity, complexity, and high dimensions
 G. Yu  Higher index theory of elliptic operators and geometry of groups
Ordinary differential equations and dynamical systems  O. N. Ageev  On spectral invariants in modern ergodic theory
 V. Bergelson  Ergodic Ramsey theory: a dynamical approach to static theorems
 N. Chernov and D. Dolgopyat  Hyperbolic billiards and statistical physics
 R. de la Llave  Some recent progress in geometric methods in the instability problem in Hamiltonian mechanics
 M. Einsiedler and E. Lindenstrauss  Diagonalizable flows on locally homogeneous spaces and number theory
 Author index
Volume III Ordinary differential equations and dynamical systems (continued)  R. Ghrist  Braids and differential equations
 A. Gorodetski, B. Hunt, and V. Kaloshin  Newton interpolation polynomials, discretization method, and certain prevalent properties in dynamical systems
 B. Kra  From combinatorics to ergodic theory and back again
 P. Le Calvez  From Brouwer theory to the study of homeomorphisms of surfaces
 M. Shub  All, most, some differentiable dynamical systems
 A. Zorich  Geodesics on flat surfaces
Partial differential equations  S. Bianchini  Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems and relaxation approximation
 P. Gérard  Nonlinear Schrödinger equations in inhomogeneous media: wellposedness and illposedness of the Cauchy problem
 F. Golse  The periodic Lorentz gas in the BoltzmannGrad limit
 M. J. Gursky  Conformal invariants and nonlinear elliptic equations
 H. Ishii  Asymptotic solutions for large time of HamiltonJacobi equations
 M. Pulvirenti  The weakcoupling limit of large classical and quantum systems
 O. Savin  Symmetry of entire solutions for a class of semilinear elliptic equations
 S. Serfaty  Vortices in the GinzburgLandau model of superconductivity
 N. S. Trudinger  Recent developments in elliptic partial differential equations of MongeAmpère type
 L. Vega  The initial value problem for nonlinear Schrödinger equations
 J. J. L. Velázquez  Singular solutions of partial differential equations modelling chemotactic aggregation
Mathematical physics  A. S. Cattaneo  From topological field theory to deformation quantization and reduction
 B. Derrida  Matrix ansatz and large deviations of the density in exclusion processes
 J.M. Maillet  Correlation functions of the \(XXZ\) Heisenberg spin chain: Bethe anastz approach
 M. Mariño  GromovWitten invariants and topological strings: a progress report
 I. Rodnianski  The Cauchy problem in general relativity
 C. Schweigert, J. Fuchs, and I. Runkel  Categorification and correlation functions in conformal field theory
 A. Soffer  Soliton dynamics and scattering
 C. Villani  Hypocoercive diffusion operators
Probability and statistics  A. Bovier  Metastability: a potential theoretic approach
 R. Cerf  On Ising droplets
 A. Dembo  Simple random covering, disconnection, late and favorite points
 P. Donnelly  Modelling genes: mathematical and statistical challenges in genomics
 K. D. Elworthy and X.M. Li  Geometric stochastic analysis on path spaces
 J. Fan and R. Li  Statistical challenges with high dimensionality: feature selection in knowledge discovery
 A. Guionnet  Random matrices and enumeration of maps
 S. P. Lalley  The weak/strong survival transition on trees and nonamenable graphs
 Y. Le Jan  New developments in stochastic dynamics
 P. McCullagh and J. Yang  Stochastic classification models
 A. Okounkov  Random partitions and instanton counting
 D. Picard and G. Kerkyacharian  Estimation in inverse problems and secondgeneration wavelets
 W. Werner  Conformal restriction properties
Combinatorics  A. Barvinok  The complexity of generating functions for integer points in polyhedra and beyond
 M. BousquetMélou  Rational and algebraic series in combinatorial enumeration
 J. Geelen, B. Gerards, and G. Whittle  Towards a structure theory for matrices and matroids
 M. Haiman  Cherednik algebras, Macdonald polynomials and combinatorics
 J. H. Kim  Poisson cloning model for random graphs
 T. Łuczak  Randomness and regularity
 I. Z. Ruzsa  Additive combinatorics and geometry of numbers
 F. Santos  Geometric bistellar flips: the setting, the context and a construction
 R. Thomas  A survey of Pfaffian orientations of graphs
Mathematical aspects of computer science  M. Agrawal  Determinant versus permanent
 A. S. Holevo  The additivity problem in quantum information theory
 J. Kleinberg  Complex networks and decentralized search algorithms
 O. Reingold  On expander graphs and connectivity in small space
 T. Roughgarden  Potential functions and the inefficiency of equilibria
 R. Rubinfeld  Sublinear time algorithms
 L. Trevisan  Pseudorandomness and combinatorial constructions
Numerical analysis and scientific computing  P. Bochev and M. Gunzburger  Leastsquares finite element methods
 Z. Chen  A posteriori error analysis and adaptive methods for partial differential equations
 R. G. Durán  Error estimates for anisotropic finite elements and applications
 N. Dyn  Linear subdivision schemes for the refinement of geometric objects
 R. J. LeVeque  Wave propagation software, computational science, and reproducible research
 Y. Maday  Reduced basis method for the rapid and reliable solution of partial differential equations
 E. Süli  Finite element algorithms for transportdiffusion problems: stability, adaptivity, tractability
Control theory and optimization  V. S. Borkar  Ergodic control of diffusion processes
 S. Boyd  Convex optimization of graph Laplacian eigenvalues
 O. Y. Emanouilov (Imanuvilov)  Controllability of evolution equations of fluid dynamics
 A. van der Schaft  PortHamiltonian systems: an introductory survey
 O. J. Staffans  Passive linear discrete timeinvariant systems
 E. Zuazua  Control and numerical approximation of the wave and heat equations
Application of mathematics in the sciences  R. E. Caflisch  Multiscale modeling for epitaxial growth
 E. J. Candès  Compressive sampling
 V. Caselles  Total variation based image denoising and restoration
 M. Griebel and J. Hamaekers  A wavelet based sparse grid method for the electronic Schrödinger equation
 C. Le Bris  Mathematical and numerical analysis for molecular simulation: accomplishments and challenges
 M. A. Nowak  Evolutionary dynamics of cooperation
 D. Nualart  Fractional Brownian motion: stochastic calculus and applications
 A. Szepessy  Atomistic and continuum models for phase change dynamics
Mathematics education and popularization of mathematics  P. S. Kenderov  Competitions and mathematics education
 A. Siegel  Understanding and misunderstanding the Third International Mathematics and Science Study: what is at stake and why K12 education studies matter
 I. Stewart  Mathematics, the media, and the public
 M. Artigue, E. de Shalit, and A. Ralston  Panel A: Controversial issues in K12 mathematical education
 L. P. Yee, J. de Lange, and W. Schmidt  Panel B: What are PISA and TIMSS? What do they tell us?
 Fr. Nebres, S.Y. Cheng, K. Osterwalder, and H.H. Wu  Panel C: The role of mathematicians in K12 mathematics education
History of mathematics  L. Corry  On the origins of Hilbert's sixth problem: physics and the empiricist approach to axiomatization
 N. Guicciardini  Method versus calculus in Newton's criticisms of Descartes and Leibniz
Special activity  S. X. Descamps, H. Bass, G. B. Evia, R. Seiler, and M. Seppälä  elearning mathematics
 Author index
