The proceedings of the Séminaire Bourbaki appear regularly in Astérisque. This book contains fourteen written versions of seminars held in the academic year 2003/2004. They are taken from diverse branches of mathematics. There are contributions from Y. André (finite-dimensional motives), A. Beauville (the Green conjecture), E. Peyre (obstructions for the Hasse principle), S. Alinhac (geometric methods for the Einstein equation), I. Gallagher (the Euler equations and the incompressible limit), P. Gérard (mean field dynamics for the quantum N-body problem), N. Tzvetkov (long-time behaviour of KdV-type equations), K. Belabas (parametrization of algebraic structures), J. Bertoin (Schramm-Loewner evolution and conformal invariance), R. Krikorian (deviations of ergodic averages for area preserving flows), B. Maurey (geometric inequalities), H. Pajot (analytic capacity and the Painlevé problem), J.-P. Serre (complete reducibility in group theory) and S. Vaes (free quasi-free states and type III factors).

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