Finding arithmetic progressions in dense sets of integers

One of the central problems in additive combinatorics is to determine how large a subset of the first $N$ integers can be before it is forced to contain $k$ elements forming an arithmetic progression. Around 25 years ago, Gowers proved the first reasonable upper bounds in this problem for progressions of length four and longer. In this work, Gowers initiated the study of "higher-order Fourier analysis", which has developed over the past couple of decades into a rich theory with numerous other combinatorial applications. I will report on some very recent progress in higher-order Fourier analysis and how it has led to the first ever quantitative improvement on Gowers's upper bounds when $k\geq 5$.