*k*-plane Furstenberg sets over finite fields

## Rutgers University

## Date and time: 2pm, Friday, October 18, 2019

## Place: CUNY Graduate Center, Rm 5383

An important family of incidence problems are discrete analogs of deep
questions in geometric measure theory. Perhaps the most famous example
of this is the finite field Kakeya conjecture, proved by Dvir in
2008. This proof introduced the polynomial method to incidence
geometry, which led to the solution to many long-standing problems in
the area. I will talk about a generalization of the Kakeya conjecture
posed by Ellenberg, Oberlin, and Tao.

A $(k,m)$-Furstenberg set $S$ in $\mathbb{F}_q^n$ has the property that, parallel to
every $k$-plane $V$ there is a $k$-plane that intersects $S$ in at least $m$
points. Using a substantial amount of algebraic geometry, Ellenberg
and Erman showed that $|S| > c m^{n/k}$, for a constant $c$ depending on $n$
and $k$. In recent joint work with Manik Dhar and Zeev Dvir, we improve
their bound, using much simpler proofs. For example, if $m>2^{n+7}q$,
then $|S|=(1-o(1))mq^{n-k}$, where the function $o(1)$ depends on $q$. I
will discuss this work, its limitations, and why the proofs of
Ellenberg and Erman are still worth learning.

This is joint work with Manik Dhar and Zeev Dvir.