Mathematical Conversations

In 2016, Ellenberg and Gijswijt made a breakthrough on the famous cap-set problem, which asks about the maximum size of a subset of \mathbb{F}_3^n not containing a three-term arithmetic progression. Ellenberg and Gijswijt proved that any such set has size at most 2.756^n. Their proof used a new polynomial method introduced by Croot, Lev and Pach just a few weeks earlier. In this talk, we will discuss the proof of the Ellenberg-Gijswijt bound, following a reformulation of their original proof by Tao. In this reformulation, Tao introduced what is now called the slice rank polynomial method. At the end of the talk, we will also discuss further applications as well as limitations of the slice rank polynomial method.