Finite Field Restriction Estimates

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 typically implying progress on the second). Both of these have formulations over finite fields. In 2008 Dvir completely settled the finite field Kakeya conjecture, however neither his result nor the proof method have yet yielded any progress on the finite field restriction conjecture. In this talk I will describe some recent progress on the finite field restriction conjecture, improving the the exponents of Mockenhaupt and Tao. The key new ingredient is the use of incidence / sum-product estimates. [This will be a longer version of the talk I gave in the Princeton CCI seminar on 9/13/2013]