Proceedings of the International Congress of Mathematicians, Madrid

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.

Table of Contents:
Determinacy and large cardinals; The art of ordinal analysis; Analytic difference rings; Borel superrigidity and the classification problem for the torsion-free abelian groups of finite rank; Zeta functions of groups and rings; On differential graded categories; Derived equivalences and finite dimensional algebras; Algorithmic and asymptotic properties of groups; A. unified approach to computations with permutation and matrix groups; Some results in noncommutative ring theory; Hecke orbits as Shimura varieties in positive characteristic; Heegner points, Stark-Heegner points and values of $L$-series; Galois deformations and arithmetic geometry of Shimura varieties; Generalising the Hardy-Littlewood method for primes; Aspects geometriques du Lemme Fondamental de Langlands-Shelstad; Equidistribution, $L$-functions and ergodic theory: on some problems of Yu. Linnik; $p$-adic motivic cohomology in arithmetic; Vanishing of $L$-functions and ranks of Selmer groups; Special values of $L$-functions modulo $p$; Evaluation maps, slopes, and algebraicity criteria; Derived categories of coherent sheaves; Invariants of singularities of pairs; Rational curves and rational points; Rigidity of rational homogeneous spaces; Geometry of multiple zeta values; Geometry over nonclosed fields; Algebraic Morse theory and the weak factorization theorem; Elliptic and parabolic problems in conformal geometry; The topology and geometry of contact structures in dimension three; Generalized triangle inequalities and their applications; The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity; Lagrangian submanifolds: from the local model to the cluster complex; Gromov-Witten invariants and moduli spaces of curves; Extremal metrics and stabilities on polarized manifolds; Tropical geometry and its applications; Embedded minimal surfaces; Floer homology in symplectic geometry and in mirror symmetry; Properly embedded minimal surfaces with finite topology; Applications of loop group factorization to geometric soliton equations; Non-positive curvature and complexity for finitely presented groups; Link homology and categorification; Curve complexes, surfaces and 3-manifolds; $mathbb{A}1$-algebraic topology; Development in symplectic Floer theory; Heegaard diagrams and Floer homology; The cohomology of automorphism groups of free groups; Spaces of quasi-maps into the flag varieties and their applications; On the local Langlands and Jacquet-Langlands correspondences; An invitation to bounded cohomology; Fibration de Hitchin et structure endoscopique de la formule des traces; Hecke algebras and harmonic analysis; Continuous representation theory of $p$-adic Lie groups; The algebraization of Kazhdan's property (T); Rankin-Selberg integrals, the descent method, and Langlands functorality; Representation theory and the cohomology of arithmetic groups; Some results on compactifications of semisimple groups; Local $Tb$ theorems and applications in PDE; Almost everywhere convergence and divergence of Fourier series; Iterated Segre mappings of real submanifolds in complex space and applications; Towards conformal invariance of 2D lattice models; Aspects of the $L2$-Sobolev theory of the $overlinepartial$-Neumann problem; Greedy approximations with regard to bases; Analytic capacity, rectifiability, and the Cauchy integral; Isomorphic and almost-isometric problems in high-dimensional convex geometry; Amenable actions and applications; Structure and classification of $Cast$-algebras; Convexity, complexity, and high dimensions; Higher index theory of elliptic operators and geometry of groups; Ergodic Ramsey theory: a dynamical approach to static theorems; Hyperbolic billiards and statistical physics; Some recent progress in geometric methods in the instability problem in Hamiltonian mechanics; Diagonalizable flows on locally homogeneous spaces and number theory; Author index; Newton interpolation polynomials, discretization method, and certain prevalent properties in dynamical systems; From combinatorics to ergodic theory and back again; From Brouwer theory to the study of homeomorphisms of surfaces; All, most, some differentiable dynamical systems; Geodesics on flat surfaces; Nonlinear Schrodinger equations in inhomogeneous media: wellposedness and illposedness of the Cauchy problem; The periodic Lorentz gas in the Boltzmann-Grad limit; Conformal invariants and nonlinear elliptic equations; Asymptotic solutions for large time of Hamilton-Jacobi equations; The weak-coupling limit of large classical and quantum systems; Symmetry of entire solutions for a class of semilinear elliptic equations; Vortices in the Ginzburg-Landau model of superconductivity; Recent developments in elliptic partial differential equations of Monge-Ampere type; The initial value problem for nonlinear Schrodinger equations; Singular solutions of partial differential equations modelling chemotactic aggregation; Matrix ansatz and large deviations of the density in exclusion processes; Correlation functions of the $XXZ$ Heisenberg spin chain: Bethe anastz approach; Gromov-Witten invariants and topological strings: a progress report; The Cauchy problem in general relativity; Categorification and correlation functions in conformal field theory; Soliton dynamics and scattering; Hypocoercive diffusion operators; On Ising droplets; Simple random covering, disconnection, late and favorite points; Modelling genes: mathematical and statistical challenges in genomics; Geometric stochastic analysis on path spaces; Statistical challenges with high dimensionality: feature selection in knowledge discovery; Random matrices and enumeration of maps; The weak/strong survival transition on trees and nonamenable graphs; New developments in stochastic dynamics; Stochastic classification models; Random partitions and instanton counting; Estimation in inverse problems and second-generation wavelets; Conformal restriction properties; Rational and algebraic series in combinatorial enumeration; Towards a structure theory for matrices and matroids; Cherednik algebras, Macdonald polynomials and combinatorics; Poisson cloning model for random graphs; Randomness and regularity; Additive combinatorics and geometry of numbers; Geometric bistellar flips: the setting, the context and a construction; A survey of Pfaffian orientations of graphs; The additivity problem in quantum information theory; Complex networks and decentralized search algorithms; On expander graphs and connectivity in small space; Potential functions and the inefficiency of equilibria; Sublinear time algorithms; Pseudorandomness and combinatorial constructions; A posteriori error analysis and adaptive methods for partial differential equations; Error estimates for anisotropic finite elements and applications; Linear subdivision schemes for the refinement of geometric objects; Wave propagation software, computational science, and reproducible research; Reduced basis method for the rapid and reliable solution of partial differential equations; Finite element algorithms for transport-diffusion problems: stability, adaptivity, tractability; Convex optimization of graph Laplacian eigenvalues; Controllability of evolution equations of fluid dynamics; Port-Hamiltonian systems: an introductory survey; Passive linear discrete time-invariant systems; Control and numerical approximation of the wave and heat equations; Compressive sampling; Total variation based image denoising and restoration; A wavelet based sparse grid method for the electronic Schrodinger equation; Mathematical and numerical analysis for molecular simulation: accomplishments and challenges; Evolutionary dynamics of cooperation; Fractional Brownian motion: stochastic calculus and applications; Atomistic and continuum models for phase change dynamics; Understanding and misunderstanding the Third International Mathematics and Science Study: what is at stake and why K-12 education studies matter; Mathematics, the media, and the public; Panel A: Controversial issues in K-12 mathematical education; Panel B: What are PISA and TIMSS? What do they tell us?; Panel C: The role of mathematicians in K-12 mathematics education; Method versus calculus in Newton's criticisms of Descartes and Leibniz; Author index