The discrete restriction problem for the parabola studies the L^6 norm of \sum_{n = 1}^{N}a_n e^{2\pi i (nx + n^2 t)} subject to the {a_n} having l^2 norm 1. Bourgain initially studied this problem due to its connections to Strichartz estimates for the Schrodinger equation on the torus. The best-known upper bounds are given by the speaker in joint work with Shaoming Guo and Po-Lam Yung and make use of the high-low method, brought into Fourier decoupling theory by Larry Guth, Dominique Maldague, and Hong Wang. This talk will discuss the discrete restriction problem for the parabola, the high-low method (à la Guth-Maldague-Wang), various optimizations to obtain the current best-known upper bounds, and some open questions.

Speaker: Zane Li, North Carolina State University