ABSTRACT: Some of the most important problems in combinatorial number theory ask for the size of the largest subset of the integers in an interval lacking points in a fixed arithmetically defined pattern. One example of such a problem is to prove the best possible bounds in Szemer\'edi's theorem on arithmetic progressions, i.e., to determine the size of the largest subset of {1,...,N} with no nontrivial k-term arithmetic progression x,x+y,...,x+(k-1)y. Gowers initiated the study of higher order Fourier analysis while seeking to answer this question, and used it to give the first reasonable upper bounds for arbitrary k. In this talk, I'll discuss recent progress on quantitative polynomial, multidimensional, and nonabelian variants of Szemer\'edi's theorem and on related problems in harmonic analysis and ergodic theory.
