Thematic Seminar: Applications and limitations of the slice rank polynomial method

Seminar | January 24 | 4:10-5 p.m. | 740 Evans Hall

 Lisa Sauermann, Stanford University

 Department of Mathematics

In 2016, Tao introduced the slice rank polynomial method as a reformulation of a technique that first appeared in work of Croot, Lev and Pach, and was the basis of a breakthrough of Ellenberg and Gijswijt on the upper bound for the famous cap-set problem. The cap-set problem asks about the maximum size of a subset of F_p^n not containing a three-term arithmetic progression. The slice rank polynomial method and its consequences have also had many other spectacular applications in combinatorics and additive number theory.

In this talk, we will discuss two applications. The first application is to prove polynomial bounds for arithmetic k-cycle removal lemmas in F_p^n (which is joint work with Jacob Fox and Laszlo Miklos Lovasz). The second application is to prove upper bounds on the so-called Erdös-Ginzburg-Ziv constant of F_p^n. We will also discuss current limitations of the slice rank polynomial method, and how studying these applications can help to overcome these limitations.