An introduction to Fourier decoupling theory

Tuesday, 21-May-2024 

Abstract

Fourier decoupling theory was first introduced by Thomas Wolff in 2000. Since the proof of decoupling for the paraboloid by Bourgain and Demeter in 2014, decoupling has had a wide range of applications in analytic number theory, geometric measure theory, and PDE. For example, in 2015, the long standing main conjecture in Vinogradov's Mean Value Theorem, a conjecture about the number of integer solutions to a particular system of equations, was proven by Bourgain, Demeter, and Guth as a corollary of Fourier decoupling for the moment curve. In this talk, I will try to explain important features that make decoupling effective and explain various tools and techniques used when trying to prove and think about decoupling estimates.