The Fourier restriction and Kakeya conjectures: some recent developments

In recent years there have been a number of remarkable developments at the interface of harmonic analysis, geometric analysis, and PDE theory, which has in turn led to some truly astonishing breakthroughs in areas such as analytic number theory, combinatorics, geometric measure theory, and spectral theory. At the heart of these developments one finds the restriction theory for the Fourier transform and its connections to the Kakeya conjecture. It is here that we begin our discussion. After introducing some of the classical aspects of the theory, we highlight some of the more recent developments which have arisen from a certain multilinear viewpoint.