Final Report Summary - DISCONV (DISCRETE AND CONVEX GEOMETRY: CHALLENGES, METHODS, APPLICATIONS) During the six years of the project DISCONV several important results were achieved, most of them are in the direction set out in the original proposal. One great loss is the untimely passing away of Jiri Matousek in March 2015, one of the main researcher in this project. His contribution to the success of the project is enormous and cannot be overestimated. The Micro- and Mini-Workshops have proved to be extremely useful, and very productive. This is four-five researchers focusing on two or three preselected open problems without formal lectures or program. Such a workshop provides an excellent research atmosphere equally suitable for established researchers and students and postdocs. During the project DISCONV full or partial solutions have been reached in almost all of the 23 original problems. The conference "Mathematics of Jiri Matousek" was celebrating and commemorating Matousek's extraordinary achievements and his legacy. It was held at Charles university, Prague (July 23-28, 2016) and was a great success with more than 250 participants, many of them students and postdocs. The thematic semester (July 2016 to March 2017) attracted many mathematicians and was very productive and successful. Some of the most significant achievements directly connected to the project DISCONV: (1) new results on geometric Ramsey numbers, (2) establishing the computational complexity of classical problems in algebraic topology, including equivariant ones, (3) several breakthrough theorems in combinatorial convexity, where the use of algebraic topology was crucial, (4) almost optimal bounds on the size of separators in string graphs, (5) universality of vector sequences and Tverberg partitions, (6) novel methods were developed in geometric discrepancy theory, (7) several new results on random polytopes, (8) critical points on Alexandrov surfaces, (9) establishing the limiting shape of integral zonotopes, (10) progress in incidence problems with the use of algebraic geometry, (11) new results around the Erdos-Szekeres theorem, (12) solving several instances of Turan's extremal hypergraph problem, and (13) significant new results on list colourings, and (14) in geometric binary codes. In summary, the ERC Advanced Research grant DISCONV was very productive and successful, and its achievements will have significant impact on future research in discrete and convex geometry and in combinatorics.
