## 05C50 Online Seminar: Mike Tait

- Date: 01/13/2023
- Time: 08:00

Online

Proving algebraic results using only graph theory and linear algebra

We discuss how to use spectral graph theory to count subgraphs of graphs where the subgraph counted is motivated by finite field versions of questions in geometric measure theory. One representative question is the following:

Let E be a set in F_q^d, and \alpha and \beta be elements in F_q^*. How large does E need to be to guarantee that there are four points w, x, y, and z in E such that they form a rectangle of side lengths \alpha and \beta, i.e.

( w - x ) \cdot ( x - y ) = 0

( x - y ) \cdot ( y - z ) = 0

( y - z ) \cdot ( z - w ) = 0

( z - w ) \cdot ( w - x ) = 0

and

|| w - x || = || y - z || = \alpha and || x - y || = || z - w || = \beta.

We provide a general framework which answers this and other similar questions as a corollary. This is joint work with Thang Pham, Steven Senger, and Vu Thi Huong Thu.

Stephen Kirkland, University of Manitoba

Hermie Monterde, University of Manitoba

The 05C50 Online is an international seminar about graphs and matrices held twice a month on Fridays.

**Time: 8AM Pacific/10AM Central **

For more information, visit https://sites.google.com/view/05c50online/home.

If you would like to attend, please register using this form to receive the zoom links: https://docs.google.com/forms/d/e/1FAIpQLSdQ98fh58cgeSWzbFe3t77i28FXDck1gYuX9jv_qd4kEf5l_Q/viewform?usp=sf_link