Olof Sisask: Strong upper bounds on the sizes of caps over finite fields via linear algebra over polynomial rings

Abstract: Earlier this year, work of Croot-Lev-Pach and Ellenberg-Gijswijt showed how a question that's been vexing researchers in additive combinatorics for decades could be solved (in some sense) using linear algebraic arguments over polynomial rings. This stunned the community, who until then had been mostly pushing forward using methods from Fourier analysis. The basic question is as follows: how large can a subset of an n-dimensional vector space over \(F_3\) be if it is not allowed to contain a solution to the equation \(x+y+z=0\) in distinct variables? The aim of the talk is to give some background to the problem, to indicate how the new argument goes, and to point out some future directions that I hope some audience members would be interested in.