In the last few years, Jean Bourgain and Ciprian Demeter have proven a variety of striking ``decoupling'' theorems in Fourier analysis. I think this is an important development in Fourier analysis. As a corollary, they were able to give very sharp estimates for the L^p norms of various trigonometric sums. These sums appear in PDE when one studies the Schrodinger equation on a torus, and they appear in analytic number theory in connection with the circle method. In this first lecture, I will explain what decoupling theorems say, look at some examples, and discuss applications. I will try to describe why I think the theorems are important, and to say something about what makes the problems difficult. In the next two lectures, we will discuss how to prove decoupling theorems. We will focus on the simplest decoupling theorem: decoupling for the parabola in the plane.

The proof of decoupling for the parabola In this lecture, we will give a detailed sketch of the proof of the decoupling theorem for the parabola, combining the ingredients from Lecture 2