Some mathematical aspects of water waves / Adrian Constantin -- Continuous dissipative Euler flows and a conjecture of Onsager / Camillo De Lellis, László Székelyhidi Jr. -- Persistent Homology: Theory and Practice / Herbert Edelsbrunner, Dmitriy Morozov -- In a Search for a Structure, Part 1: On Entropy / Misha Gromov -- Classification of Algebraic Varieties / Christopher D. Hacon -- Representations of affine Kac–Moody groups over local and global fields: a survey of some recent results / Alexander Braverman, David Kazhdan -- Emergence of the Abrikosov lattice in several models with two dimensional Coulomb interaction / Sylvia Serfaty -- Dependent Classes E72 / Saharon Shelah -- Chaining and the Geometry of Stochastic Processes / Michel Talagrand -- Duflo isomorphism, the Kashiwara–Vergne conjecture and Drinfeld associators / Anton Alekseev -- Coagulation with limited aggregations / Jean Bertoin -- The Cremona group in two variables / Serge Cantat -- Variational models for image inpainting / Vicent Caselles -- KAM theory and its applications: from conservative to dissipative systems / Alessandra Celletti -- Le programme de Langlands $p$-adique / Pierre Colmez -- Mirror Symmetry and Fano Manifolds / Tom Coates, Alessio Corti, Sergey Galkin, Vasily Golyshev, Alexander Kasprzyk -- On flat bundles in characteristic 0 and $p > 0$ / Hélène Esnault -- Combinatorial realisation of cycles and small covers / Alexander A. Gaifullin -- Remarks on the global regularity for solutions to the incompressible Navier–Stokes equations / Isabelle Gallagher -- Why the empirical sciences need statistics so desperately / Olle Häggström -- Computing the Schrodinger equation with no fear of commutators / Arieh Iserles -- Dynamics of non-archimedean Polish groups / Alexander S. Kechris -- Cluster algebras and cluster monomials / Bernhard Keller -- Weak solutions to the complex Monge–Ampère equation / Sławomir Kołodziej -- Reinforced random walk / Gady Kozma -- On blow-up curves for semilinear wave equations / Frank Merle -- Commuting higher rank ordinary differential operators / Andrey Mironov -- Stochastic calculus with respect to the fractional Brownian motion / David Nualart -- Sampling, Interpolation, Translates / Alexander Olevskii -- Multidimensional periodic and almost-periodic spectral problems / Leonid Parnovski -- Effective equations for quantum dynamics / Benjamin Schlein -- Combinatorics of asymptotic representation theory / Piotr Śniady -- On scale-invariant solutions of the Navier–Stokes equations / Hao Jia, Vladimír Šverák -- Ramsey-theoretic analysis of the conditional structure of weakly-null sequences / Stevo Todorčević -- Uniqueness results for minimal surfaces and constant mean curvature surfaces in Riemannian manifolds / Simon Brendle -- Stability in geometric and functional inequalities / Alessio Figalli -- Classification and rigidity for von Neumann algebras / Adrian Ioana -- A nonlinear variational problem in relativistic quantum mechanics / Mathieu Lewin -- Grid Diagrams in Heegaard Floer Theory / Ciprian Manolescu -- Random maps and continuum random 2-dimensional geometries / Grégory Miermont -- Approximate (Abelian) groups / Tom Sanders -- Shearing and mixing in parabolic flows / Corinna Ulcigrai -- Optimal control theory and some applications to aerospace problems / Emmanuel Trélat -- Mathematics and geometric ornamentation in the medieval Islamic world / Jan P. Hogendijk -- Some mathematical aspects of the planet Earth / José Francisco Rodrigues -- Turing's Mathematical Work / P.D. Welch -- Counting Berg partitions via Sturmian words and substitution tilings / Artur Siemaszko, Maciej P. Wojtkowski.

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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, spread over 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 present extended versions of most of the invited talks which were delivered during the congress, providing a permanent record of the best what mathematics offers today.