Session: SDPs and their applications across mathematics
Chair: Christoph Spiegel
Cluster: Conic and Semidefinite Optimization
Talk 1: Semidefinite programming bounds for packing in geometrical graphs and hypergraphs
Speaker: Fernando Mario de Oliveira Filho
Abstract: Packing problems in geometry, like the sphere-packing problem, can be modeled as independent-set problems in infinite graphs. The Lovász theta number, a semidefinite programming upper bound for the independence number of a graph, can often be extended to such infinite graphs, providing some of the best upper bounds for several geometrical parameters. Perhaps the most famous such application is Viazovska's solution of the sphere-packing problem in dimension 8. In this talk I will discuss the problem of packing regular polygons on the surface of the 2-dimensional sphere and how the Lovász theta number can be used to give upper bounds. I will also discuss the problem of finding large almost-equiangular sets, which is a hypergraph extension of packing.
Talk 2: TBD
Speaker: Pablo A. Parrilo
Abstract: TBD
Talk 3: Shattering triples with permutations
Speaker: Jan Volec
Abstract: Given a 6-tuple of permutations of [n], we say that a triple of points T of [n] is totally shattered if each of the six possible relative orderings of T appears in exactly one of the permutations. We set F(n) to be the maximum fraction of triples that can be totally shattered by a 6-tuple of permutations of [n]. In 2023, Johnson and Wickes showed that the limit of F(n) is at least 17/42 and at most 11/14, and they asked to determine the limit. In this talk, we use the flag algebra method to prove that the limit of F(n) = 10/21. This is a joint work with Daniela Opocenska.
