Discrete Geometry 1137 Workshop : Discrete Geometry Table of Contents

A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics or algebraic geometry. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry. Mathematics Subject Classification (2010): 52Bxx, 52Cxx. Introduction by the Organisers Discrete Geometry deals with the structure and complexity of discrete geometric objects, from finite point sets in the plane to intersection patterns of convex sets in high dimensional spaces. It goes back to classical problems such as Kepler’s conjecture on the density of packings of balls in space, and Hilbert’s third problem on decomposing polyhedra, as well as works by Minkowski, Steinitz, Hadwiger, and Erdős form the heritage of this area. Over the past years, several outstanding problems were solved, for example: (1) Andrew Suk proved a near-tight bound for the Happy-ending problem open from 1935 about the largest point set in the plane without k points in convex position, (2) Florian Frick gave, completing an approach of Mabillard and Wagner, a counter-example to the topological Tverberg conjecture from the early 1980’s, (3) Maryna Viazovska solved the sphere packing problem in 8 dimension (and later, with others, in 24 dimensions), (4) János Pach, Natan Rubin and Gábor Tardos proved the Richter-Thomassen conjecture from 1995, a breakthrough which was reported during the workshop, and (5) Jǐŕı Matoušek, Aleksandar Nikolov, and Kunai Talwar proved a nearly tight lower bound 1136 Oberwolfach Report 19/2017 for Tusnády’s problem about the discrepancy of finite point sets with respect to axis-parallel boxes. This list illustrates the interdisciplinary nature of discrete geometry and its many relations to other fields of mathematics like algebra, topology, combinatorics, computational geometry, probability and discrepancy theory. It is also in the front line of applications like geographic information systems, mathematical programming, coding theory, solid modeling, computational structural biology, and crystallography. The workshop gathered 51 participants. The outstanding contributions by young scholars include the lecture by Rubin on his proof (with Pach and Tardos) of the Richter-Thomassen conjecture from 1995: In the plane, n pairwise intersecting Jordan curves (no three intersecting in a point, but being allowed to touch) have not much less than n intersection points. Another one was the lecture by Karim Adiprasito, who outlined his proof of a long-standing conjecture of Grünbaum, that the number of of triangles in a 2-dimensional simplicial complex embedded in R is at most 4 times the number of edges. There were 28 other, mostly short, lectures presenting new connections to classical topics in combinatorics (Holmsen, Padrol) as well as developments in classical topics such as the analysis of point configurations (Füredi, Swanepoel, Vogtenhuber), incidence geometry (Aronov, Lund, Raz, Zahl), polytope theory (Nevo, Pournin, Santos), topological methods (Blagojević, Paták, Akopyan), or algorithmic issues (Mustafa, Mulzer). József Solymosi gave a survey about the hypergraph container method of Balogh, Morris and Samotij and of Saxton and Thomason, which has recently lead to spectacular breakthroughs in extremal combinatorics, and outlined some of its geometric applications. Edgardo Roldán-Pensado (with Luis Montejano) presented their systematic method for constructing families of three-dimensional constant-width bodies (Meissner polyhedra) in a talk which was beside the main stream of topics but was nevertheless very much appreciated for the intuitive appeal of the result. In addition, there was an open problem session chaired by János Pach on Tuesday evening; the collection of open problems resulting from this session can be found in this report. The program left ample time for research and discussions in the stimulating atmosphere of the Oberwolfach Institute. In particular, there were several special informal sessions, attended by smaller groups of the participants, on specific topics of common interest. On Wednesday afternoon, most participants joined the traditional outing to St. Roman with the black forest cherry cake, enjoying the beautiful spring air despite adverse weather predictions. Acknowledgment: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1049268, “US Junior Oberwolfach Fellows”. Discrete Geometry 1137 Workshop: Discrete Geometry