Decoupling Theory and Exponential Sum Estimates

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

This project concerns the study of decoupling inequalities in harmonic analysis. Such inequalities measure the oscillation and cancellation in the Fourier transform of various curved geometric surfaces such as the paraboloid, cone, or moment curve. These inequalities arise from studying partial differential equations such as the Schrödinger equation or wave equation and also from number theory through exponential sums, which can be thought of as Fourier series that encode certain arithmetic data. Over the years, tools in these two areas have developed somewhat independently from each other. One aim of this project is to bring more tools from number theory into Fourier analysis. The connections between these two areas will be studied with hopes of proving decoupling inequalities for a wider and more general class of surfaces and improving upon quantitative versions of these inequalities. Activities will further include organizing an online seminar, several undergraduate activities, and even presentations motivated by the research that are accessible to the public. In 2015, Bourgain, Demeter, and Guth were able to prove a decoupling theorem for the moment curve from which the Main Conjecture in Vinogradov's Mean Value Theorem (VMVT), a longstanding open question from 1935, followed as a corollary. Their method was purely Fourier analytic. At roughly at the same time, Wooley used his method of efficient congruencing to give a purely number theoretic proof of the VMVT. Decoupling and efficient congruencing developed separately and independently of each other. One objective of this project is to further study connections between them. Previous attempts at interpreting ideas from efficient congruencing from the perspective of decoupling have yielded fresh insights and new points of views. The project includes a continued study of progress on VMVT in hopes of uncovering new tools in harmonic analysis to use them to prove decoupling estimates for a more general class of surfaces. Other goals of this project are to obtain improved quantitative estimates for VMVT via decoupling over local fields which has applications, for example, to the Riemann zeta function. Additionally, work will be done on proving decoupling estimates for different norms and surfaces and more refined situations where more information is known than what is typically given. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.