Higher-order Fourier analysis and applications

Abstract: Higher-order Fourier analysis is a tool developed by several mathematicians over the last couple of decades in order to solve, primarily, systems of linear equations in sets of integers. I will talk about the state of the theory now, what applications it has, and what open questions remain. As a particular application I hope to discuss recent joint work with Ford, Konyagin and Tao on large gaps between primes.