Yormark Distinguished Lecture: Dominique Maldague

Title: Cubic Weyl sums in 2-D from the Fourier restriction perspective

Abstract: We consider cubic Weyl sums of the form \sum_{n=N}^{2N} e(x\cdot(n,n^3)), where x lies in the unit square [0,1]^2. It is expected that for many x, such sums exhibit square root cancellation behavior, but this remains far out of reach: the classical argument for quadratic Gauss sums does not adapt to the cubic setting. Even the average L^p behavior of cubic Weyl sums is not fully understood. The current best L^p estimates for variable coefficient sums \sum_{n=N}^{2N}b_n e(x\cdot(n,n^3)), where b_n are complex numbers, or the constant coefficient case are due to Hughes-Wooley and Wooley, respectively, using a number theory approach. A major application of the 2016 decoupling theorem of Jean Bourgain, Ciprian Demeter, and Larry Guth was the resolution of the Vinogradov Mean Value conjecture, which in particular yields sharp L^p bounds for cubic Weyl sums in three dimensions. Motivated by recent developments in decoupling and Fourier restriction theory, including the dramatic resolution of the Kakeya set conjecture in R^3 by Hong Wang and Josh Zahl, we develop a new Fourier approach to bounding L^p estimates for 2-D cubic Weyl sums. This is ongoing work that recovers and in some cases generalizes Wooley's results on the constant coefficient problem.

You can learn more about Assistant Professor Dominique Maldague here.