This is the first volume of the proceedings of the third European Congress of Mathematics. Volume I presents the speeches delivered at the Congress, the list of lectures, and short summaries of the achievements of the prize winners as well as papers by plenary and parallel speakers. The second volume collects articles by prize winners and speakers of the mini-symposia. This two-volume set thus gives an overview of the state of the art in many fields of mathematics and is therefore of interest to every professional mathematician. Contributors: R. Ahlswede, V. Bach, V. Baladi, J. Bruna, N. Burq, X. Cabre, P.J. Cameron, Z. Chatzidakis, C. Ciliberto, G. Dal Maso, J. Denef, R. Dijkgraaf, B. Fantechi, H. Follmer, A.B. Goncharov, A. Grigor'yan, M. Harris, R. Iturriaga, K. Johansson, K. Khanin, P. Koskela, H.W. Lenstra, Jr., F. Loeser, Y.I. Manin, N.S. Manton, Y. Meyer, I. Moerdijk, E.M. Opdam, T. Peternell, B.M.A.G. Piette, A. Reznikov, H. Schlichtkrull, B. Schmidt, K. Schmidt, C. Simo, B. Toth, E. van den Ban, M.-F. Vigneras, O. Viro
Table of Contents of Volume I.- Articles by Plenary Speakers.- The Mathematics of M-Theory.- Probabilistic Aspects of Financial Risk.- Flags and Lattice Basis Reduction.- Moduli, Motives, Mirrors.- The Role of Oscillations in Some Nonlinear Problems.- New Families of Solutions in N-Body Problems.- Irreducible Modular Representations of a Reductive p-Adic Group and Simple Modules for Hecke Algebras.- Dequantization of Real Algebraic Geometry on Logarithmic Paper.- Articles by Parallel Speakers.- Advances on Extremal Problems in Number Theory and Combinatorics.- Spectral Analysis of Nonrelativistic Quantum Electrodynamics.- Spectrum and Statistical Properties of Chaotic Dynamics.- Sampling in Complex and Harmonic Analysis.- Lower Bounds for Shape Resonance Widths of Schrodinger Operators.- A Conjecture of De Giorgi on Symmetry for Elliptic Equations in ?n.- The Random Graph Revisited.- Difference Fields: Model Theory and Applications to Number Theory.- Geometric Aspects of Polynomial Interpolation in More Variables and of Waring's Problem.- The Calibration Method for Free Discontinuity Problems.- Geometry on Arc Spaces of Algebraic Varieties.- Stacks for Everybody.- Multiple ?-Values, Galois Groups, and Geometry of Modular Varieties.- Heat Kernels on Manifolds, Graphs and Fractals.- Local Langlands Correspondences and Vanishing Cycles on Shimura Varieties.- Burgers Turbulence and Dynamical Systems.- Random Growth and Random Matrices.- Sobolev Spaces and Quasiconformal Mappings on Metric Spaces.- Understanding Skyrmions Using Rational Maps.- Models for the Leaf Space of a Foliation.- Multivariable Hypergeometric Functions.- Contact Structures, Rational Curves and Mori Theory.- Analytic Topology.- Towards Ryser's Conjecture.- The Dynamics of Algebraic ? d-Actions.- Self-Interacting Random Motions.- Harmonic Analysis on Reductive Symmetric Spaces.