Olof Sisask: The recent breakthrough on capsets using polynomial algebra

Set-free configurations in the card game SET; subsets of $\mathbb{F}_3^n$ without lines (so-called affine caps); sets of integers without three-term arithmetic progressions --- what do they have in common? One can view them all as instances of the problem of finding the largest subset of an abelian group that does not contain a solution to x+z=2y with distinct x, y, z. Klaus Roth made ingenious inroads in this questions in the 1950s using Fourier analysis, and his method has set the standard ever since, both for this problem and numerous others in related topics. Last month, however, the bounds produced for this problem by Fourier-analytic methods were exponentially smashed through work of Croot--Lev--Pach and Ellenberg--Gijswijt using a completely different method involving polynomial algebra. This talk will be a self-contained exposition of this work.