Mathematical Physics Seminar

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
other tools such as quadratic form theory and sum-product/incidence
estimates) to establish the first finite field restriction estimates for
the paraboloid beyond the Stein-Tomas exponent over an arbitrary finite
field.