## Introduction to decoupling

Abstract: In the last few years, Jean Bourgain and Ciprian Demeter have proven a variety of striking “decoupling” theorems in Fourier analysis. The method has applications both in PDE and in number theory. In particular, it resolved a question about diophantine equations raised by Vinogradov in the 1930s.

The proof of decoupling uses geometric ideas. In particular, it gets a lot of mileage out of studying the problem at many different scales.

In this lecture, I will introduce decoupling and explain how it connects to number theory, and then I will sketch some of the ideas of the proof.