Benny Sudakov — Equiangular lines and spherical codes in Euclidean spaces

A set of lines in ℝ^d is called equiangular if the angles between any two of them are the same. The problem of estimating the size of the maximum family of equiangular lines has had a long history. The problem was formally posed by van Lint and Seidel in 1966, although some earlier results in this area were obtained already in late 40s. A closely related notion is that of a spherical code, which is a collection C of unit vectors in ℝ^d such that x·y ∈ L for any distinct x, y in C and some set of real numbers L. Spherical codes have been extensively studied since their introduction in the 1970s by Delsarte, Goethals and Seidel. Despite a lot of attention in the last forty years, there are still many open interesting questions about equiangular lines and spherical codes. In this talk we report recent progress on some of them.

Joint work with I. Balla, F. Drexler and P. Keevash.

Tibor Szabó — Michael's favorite games

Positional games are classic objects of combinatorics, which served as a catalyst to a number of important developments in the field, as well as in theoretical computer science. Throughout the times they also developed a strong back and forth connection with random graph theory. This is somewhat of a surprise as positional games represent a fully deterministic concept.

Michael has been playing the game for more than a decade and obtained some of the most spectacular recent advances of the field. As I usually had the pleasure of either playing along or watching from a front row seat, I plan to give a somewhat subjective historical overview of the main results of the subject and formulate some of the vague and less vague main challenges of the future.

Ron Aharoni — Matchings and covers in duals of d-interval hypergraphs

A classical theorem of Gallai is that in any finite family of closed intervals on the real line, the maximal number of disjoint intervals from the family is equal to the minimal number of points piercing all intervals. Tardos and Kaiser extended this result (appropriately modified) to families of d-intervals, namely hypergraphs in which each edge is the union of d intervals, each on a separate copy of the real line. We prove an analogous result for dual d-interval hypergraphs, in which the roles of the points and the edges are reversed. The proof is topological.

Joint with Ron Holzman and Shira Zerbib.

Tali Kaufman — Bounded degree high dimensional expanders

Expander graphs are widely studied; Various methods are known to obtain bounded degree expander graphs. Recently, there is a growing interest in understanding combinational expansion in higher dimensions (higher dimensional simplicial complexes). However, bounded degree combinatorial expanders (random or explicit) were not known till our work.

We present a local to global criterion on a complex that implies combinatorial expansion. We use our criterion to present explicit bounded degree high dimensional expanders. This solves in the affirmative an open question raised by Gromov, who asked whether bounded degree high dimensional expanders could at all exist.

Based on joint works with David Kazhdan and Alex Lubotzky, and with Shai Evra.