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 question is, “What is 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 good upper bounds for arbitrary k. In this talk, I will discuss recent progress on polynomial and multidimensional variants of this question and on related problems in additive combinatorics, harmonic analysis, and ergodic theory, and explain what higher order Fourier analysis is and why it is relevant to the study of certain arithmetic patterns.