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 2013; 824 pp; hardcover ISBN-13: 978-3-03719-120-0 List Price: US$128 Member Price: US$102.40 Order Code: EMSEMC/2012 The European Congress of Mathematics, held every four years, has become a well-established major international mathematical event. Following those in Paris (1992), Budapest (1996), Barcelona (2000), Stockholm (2004), and Amsterdam (2008), the Sixth European Congress of Mathematics (6ECM) took place in Kraków, Poland, July 2-7, 2012, with about 1000 participants from all over the world. Ten plenary, thirty-three invited lectures, and three special lectures formed the core of the program. As at all the previous EMS congresses, ten outstanding young mathematicians received the EMS prizes in recognition of their research achievements. In addition, two more prizes were awarded: the Felix Klein Prize for a remarkable solution of an industrial problem, and--for the first time--the Otto Neugebauer Prize for a highly original and influential piece of work in the history of mathematics. The program was complemented by twenty-four minisymposia with nearly 100 talks that covered all areas of mathematics. Six panel discussions were organized, covering a variety of issues ranging from the financing of mathematical research to gender imbalance in mathematics. These proceedings, which present extended versions of most of the invited talks delivered during the congress, provide a permanent record of the best of what mathematics offers today. Readership Graduate students and research mathematicians. Table of Contents Plenary Lectures A. Constantin -- Some mathematical aspects of water waves C. De Lellis and L. Székelyhidi -- Continuous dissipative Euler flows and a conjecture of Onsager H. Edelsbrunner and D. Morozov -- Persistent homology: theory and practice M. Gromov -- In a search for a structure, Part 1: On entropy C. Hacon -- Classification of algebraic varieties A. Braverman and D. Kazhdan -- Representations of affine Kac-Moody groups over local and global fields: A survey of some recent results S. Serfaty -- Emergence of the Abrikosov lattice in several models with two dimensional Coulomb interaction S. Shelah -- Dependent classes, E72 M. Talagrand -- Chaining and the geometry of stochastic processes Invited Lectures A. Alekseev -- Duflo isomorphism, the Kashiwara-Vergne conjecture and Drinfeld associators J. Bertoin -- Coagulation with limited aggregations S. Cantat -- The Cremona group in two variables V. Caselles -- Variational models for image inpainting A. Celletti -- KAM theory and its applications: From conservative to dissipative systems P. Colmez -- Le programme de Langlands p-adique T. Coates, A. Corti, S. Galkin, V. Golyshev, and A. Kasprzyk -- Mirror symmetry and Fano manifolds H. Esnault -- On flat bundles in characteristic $$0$$ and $$p \gt 0$$ A. A. Gaifullin -- Combinatorial realisation of cycles and small covers I. Gallagher -- Remarks on the global regularity for solutions to the incompressible Navier-Stokes equations O. Häggström -- Why the empirical sciences need statistics so desperately A. Iserles -- Computing the Schrödinger equation with no fear of commutators A. S. Kechris -- Dynamics of non-archimedean Polish groups B. Keller -- Cluster algebras and cluster monomials S. Kołodziej -- Weak solutions to the complex Monge-Ampère equation G. Kozma -- Reinforced random walk F. Merle -- On blow-up curves for semilinear wave equations A. E. Mironov -- Commuting higher rank ordinary differential operators D. Nualart -- Stochastic calculus with respect to the fractional Brownian motion A. Olevskii -- Sampling, interpolation, translates L. Parnovski -- Multidimensional periodic and almost-periodic spectral problems B. Schlein -- Effective equations for quantum dynamics P. Śniady -- Combinatorics of asymptotic representation theory H. Jia and V. Sveřák -- On scale-invariant solutions of the Navier-Stokes equations S. Todorčević -- Ramsey-theoretic analysis of the conditional structure of weakly-null sequences Prize Winners' Lectures S. Brendle -- Uniqueness results for minimal surfaces and constant mean curvature surfaces in Riemannian manifolds A. Figalli -- Stability in geometric and functional inequalities A. Ioana -- Classification and rigidity for von Neumann algebras M. Lewin -- A nonlinear variational problem in relativistic quantum mechanics C. Manolescu -- Grid diagrams in Heegaard Floer theory G. Miermont -- Random maps and continuum random $$2$$-dimensional geometries T. Sanders -- Approximate (Abelian) groups C. Ulcigrai -- Shearing and mixing in parabolic flows E. Trelat -- Optimal control theory and some applications to aerospace problems J. P. Hogendijk -- Mathematics and geometric ornamentation in the medieval Islamic world Special Lectures J. F. Rodrigues -- Some mathematical aspects of the Planet Earth P. Welch -- Turing's mathematical work A. Siemaszko and M. P. Wojtkowski -- Counting Berg partitions via Sturmian words and substitution tilings