CanaDAM 2017
Ryerson University, June 12 - 16, 2017

Public Interest Lecture

Around the cap set conjecture  [PDF]

A very popular question in extremal combinatorics, inspired by Roth's theorem on sets of integers with no three in arithmetic progression, is the following:

Let S be a subset of $(\mathbf{Z}/3\mathbf{Z})^n$ such that no three distinct elements of S sum to 0. Such an S is called a cap set. How large can S be?

This simple problem has generated a rich stream of mathematics; it seems to sit at the intersection of many fields, from combinatorics to geometry to harmonic analysis. It was notable for the lack of consensus concerning its answer: the largest known cap sets were of size $c^n$ for $c \sim 2.2$, while the best upper bounds on cap sets were of the form $n^{-1-\epsilon} 3^n$. Is the true upper bound close to the latter, or more like $C^n$ for some $C < 3$?

In 2016, work of Croot, Lev, Pach, myself, and Gijswijt showed the latter is the case, in a new application of the "polynomial method" from algebraic geometry. As is often the case with the polynomial method, the new proof is extremely short; so short I can, and will, explain it in an hour, and talk about some generalizations, too.

Event Sponsors

Atlantic Association for Research in the Mathematical Sciences Centre de recherches mathématiques The Fields Institute Pacific Institute for the Mathematical Sciences Canadian Mathematical Society Ryerson University Office of Naval Research Science and Technology