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 2005; 900 pp; hardcover ISBN-10: 3-03719-009-4 ISBN-13: 978-3-03719-009-8 List Price: US$138 Member Price: US$110.40 Order Code: EMSEMC/2004 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 prizewinners were independently chosen by the 4ECM Scientific Committee as plenary or invited speakers. The other five prizewinners 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. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Graduate students and research mathematicians. Table of Contents Invited Speakers G. Alberti, M. Csörnyei, and D. Preiss -- Structure of null sets in the plane and applications D. Auroux -- Some open questions about symplectic 4-manifolds, singular plane curves and braid group factorizations D. Beliaev and S. Smirnov -- Harmonic measure on fractal sets S. Bianchini -- Singular approximations to hyperbolic systems of conservation laws in one space dimension A. Borodin and G. Olshanski -- Representation theory and random point processes F. Bouchut -- Stability of relaxation models for conservation laws B. H. Bowditch -- Hyperbolic $$3$$-manifolds and the geometry of the curve complex E. Friedgut -- Proof of an intersection theorem via Fourier analysis P. Gérard -- Nonlinear Schrödinger equations on compact manifolds A. Guionnet -- A probabilistic approach to some problems in von Neumann algebras S. Helmke and P. Slodowy -- Singular elements of affine Kac-Moody groups H. Holden -- On the Camassa-Holm and Hunter-Saxton equations R. Klein, E. Mikusky, and A. Owinoh -- Multiple scales asymptotics for atmospheric flows J. Krajicek -- Proof complexity D. Krammer -- Horizontal configurations of points in link complements E. Lindenstrauss -- Invariant measures for multiparameter diagonalizable algebraic actions - A short survey T. Luczak -- Phase transition phenomena in random discrete structures T. Lyons -- Systems controlled by rough paths I. Madsen and M. Weiss -- The stable mapping class group and stable homotopy theory P. Massart -- A non-asymptotic theory for model selection P. Mihailescu -- Reflection, Bernoulli numbers and the proof of Catalan's conjecture M. Mustata, S. Takagi, and K. Watanabe -- $$F$$-thresholds and Bernstein-Sato polynomials K. G. O'Grady -- Hyperkähler manifolds and algebraic geometry I. Z. Ruzsa -- Sumsets Y. Shalom -- Measurable group theory M. Shcherbina -- Some mathematical problems of Neural networks theory M. Sodin -- Zeroes of Gaussian analytic functions X. Tolsa -- Painlevé's problem, analytic capacity and curvature of measures A.-K. Tornberg -- Regularization techniques for singular source terms in differential equations V. Totik -- Equilibrium measures and polynomials W. Werner -- SLE, conformal restriction, loops U. Zannier -- On the integral points on certain algebraic varieties Network Lectures A. Bonami -- Some problems related with holomorphic functions on tube domains over light cones Y. Brenier -- Hyperbolic PDEs, kinetic formulation and geometric measure theory F. den Hollander -- Random dynamics in spatially extended systems J. Esterle -- Analysis and operators 2000-2004. Four years of network activity B. Helffer -- Analysis of the bottom of the spectrum of Schrödinger operators with magnetic potentials and applications J. P. Keating -- Mathematical aspects of quantum chaos C. Krattenthaler -- The research training network "Algebraic Combinatorics in Europe" M. Monsurro -- Algebras with involution and adjoint groups M. Reid -- Constructing algebraic varieties via commutative algebra J. P. Solovej -- Mathematical problems of large quantum systems J. Stix -- The Grothendieck-Teichmüller group and Galois theory of the rational numbers - European network GTEM Plenary Speakers F. Golse -- Hydrodynamic limits F. Guerra -- Mathematical aspects of mean field spin glass theory J. Hastad -- Complexity theory, proofs and approximation A. Okounkov -- Random surfaces enumerating algebraic curves P. Ozsvath and Z. Szabo -- On Heegaard diagrams and holomorphic disks O. Schramm -- Emergence of symmetry: Conformal invariance in scaling limits of random systems C. Voisin -- Recent progresses in Kähler and complex algebraic geometry Prize Lectures F. Barthe -- Isoperimetric inequalities, probability measures and convex geometry P. Biran -- Symplectic topology and algebraic families S. Serfaty -- Vortices in the Ginzburg-Landau model of superconductivity W. Tucker -- Validated numerics for pedestrians O. Venjakob -- From classical to non-commutative Iwasawa theory: An introduction to the $$GL_2$$ main conjecture Index of authors