Decoupling and Vinogradov's mean value problem

Abstract: Decoupling is a recent development in Fourier analysis that has solved several longstanding problems.  We will give an introduction to decoupling where we describe one of these problems and then describe some of the ideas of the proof.

Vinogradov's mean value problem is about the number of solutions of a certain system of diophantine equations, which can be rewritten as the moment of a certain trigonometric polynomial.  It was raised in the 1930s and resolved in the last decade.

The main new idea in the proof of decoupling is to combine estimates from many different scales.  We will try to describe this process and why it is helpful.