Recent estimates on exponential sums in higher dimensions

Abstract

One of the two important developments concerning the restriction theory in Fourier analysis during the last ten years was the development of the multi-linear restriction theory by Bennett, Carbery, and Tao in [2]. The second was the method introduced by Bourgain and Guth in [3] to convert multi-linear estimates to linear ones. In a recent paper [1], Bourgain used the method of [3] to obtain new results on exponential sums with frequencies E = DS for some hypersurface S with positive second fundamental form. Our thesis starts by explaining the Bourgain-Guth method in the plane and using it to give an alternative proof of the restriction theorem there. We then present the paper [1].