The Kakeya and restriction conjectures over finite fields 

The Kakeya and restriction conjectures are two of the central open problems in Euclidean
Fourier analysis (with the second logically implying the first, and progress on the first
often implying progress on the second). Both of these problems have formulations over finite
fields. In 2008, Dvir completely settled the finite field Kakeya conjecture, while the finite
field restriction conjecture seems far from complete resolution. It seemed to have been widely
believed, in contrast to the Euclidean situation, that a formal connection between the two
problems did not exist in the finite field setting. In this talk, I will discuss recent work
that establishes a formal connection between these two problems. Most significantly, we are
able to use finite field Kakeya estimates (alongside tools from quadratic form theory,
additive combinatorics and incidence geometry) to establish the first finite field restriction
estimates for the paraboloid beyond the Stein-Tomas exponent over an arbitrary finite field.