European Congress of Mathematics. Proceedings of the 6th ECM congress, Kraków, Poland, July 2–7 July, 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 Krakow, 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. 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 corti_proc_ems.pdf Introduction Local systems Local systems from Laurent polynomials Local systems from quantum cohomology Extremal local systems and extremal Laurent polynomials Examples in low dimensions Minkowski polynomials and Fano 3-folds Fano 4-folds? esnault_proc_ems.pdf Introduction Characteristic 0 Characteristic p>0 Results Question 3.2, ii) Question 3.3 Rank 1 The affine space Some comments and questions Moduli From characteristic p to characteristic 0 and vice-versa Question 3.1 ii) Finiteness kozma_proc_ems.pdf Introduction Phase transitions Pólya's urn Partial exchangeability The magic formula The hyperbolic sigma model The solution Where now?