The Third European Congress of Mathematics (3ecm) took place from July 10th to July 14th, 2000 in Barcelona. It was organised by the Societat Catalana de Matematiques (Catalan Mathematical Society), under the auspices of the Euro pean Mathematical Society (EMS). As foreseen by the EMS and the International Mathematical Union, this Congress was a major event in World Mathematical Year 2000. In addition to reviewing outstanding research achievements, important aspects of the life of European mathematics were discussed. Mathematics is undergoing a period of rapid changes. Effective computation and applications in science and technology go ever more hand in hand with con ceptual developments. It was one of the aims of 3ecm to reflect this mutual enrich ment, while steering present and future trends of mathematical sciences. In fact, the motto of the Congress, Shaping the 21st Century, was meant to capture these views. Nearly 1400 people from 87 countries gathered together in the Palau de Con gressos of Barcelona in order to take part in the activities of the 3ecm scientific programme: Nine plenary lectures, thirty invited lectures in parallel sessions, lec tures given by EMS prize winners, ten mini-symposia on special topics, seven round tables, poster sessions, presentations of mathematical software and video exhibitions. Twenty events were satellites of 3ecm in various countries.

Inhalt
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 Schrödinger 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.