05C50 Online Seminar: Mike Tait

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.