Set, Cap Sets, Arithmetic Progressions, Polynomials by Prof. Jordan Ellenberg, University of Wisconsin, Madison, Math/Stats Kick-Off Colloquium

Set, Cap Sets, Arithmetic Progressions, Polynomials by Prof. Jordan Ellenberg, University of Wisconsin, Madison, Math/Stats Kick-Off Colloquium. Supported by the Class of 1960s Scholars Fund and the John and Louise Finnerty Class of 1971 Fund, Wednesday, September 26, 1-1:45 pm, Wege Auditorium.

Abstract:

Abstract: The game of Set is a popular game which is superficially about cards and pictures but is actually about the geometry of four-dimensional space over the field of three elements. I will explain this assertion. People who play the game generally start to wonder: how many cards do there need to be on the table before there’s guaranteed to be a legal play? And some of those people wonder, further: how does this change in higher-dimensional versions of the game? This ends up being a famous old problem in extremal combinatorics called “the cap set problem,” which is a cousin of the even older and more famous problem: how large a subset of the first N integers can you have if no three elements of the subset form an arithmetic progression? Many people, including me, worked very hard on the cap set problem with no success. But in 2016 the problem was solved using an argument only two pages long, via a technique called “the polynomial method.” I’ll talk a little bit about the method (which is basically “do algebra even if the problem doesn’t look like an algebra problem”) and what remains to be known (a lot.)

For people who want to read about the cap set problem, there’s a good general-audience treatment in Quanta magazine: https://www.quantamagazine.org/set-proof-stuns-mathematicians-20160531/

A more technical account can be found on Terry Tao’s blog: https://terrytao.wordpress.com/2016/05/18/a-symmetric-formulation-of-the-croot-lev-pach-ellenberg-gijswijt-capset-bound/

Or you could go to the source and read the original paper: https://arxiv.org/abs/1605.09223