Fourier Restriction and Exponential Sum Estimates

In this talk we will discuss some results in discrete Fourier restriction estimates, a type of exponential sum estimates. The discrete Fourier restriction problem comes from the study of nonlinear Schrödinger equations on tori, and the corresponding estimate (also called Strichartz estimate in PDEs) is used to control the nonlinear term. To prove the estimate, we follow an approach combining both number theory and harmonic analysis. We will also give some recent progress of the similar topics.