Bourgain Lecture

In 1975, Szemer\'edi proved that any subset of the natural numbers with positive upper density must contain arbitrarily long finite arithmetic progressions. Szemer\'edi's original argument was purely combinatorial, and then Furstenberg gave an alternative proof using ergodic theory a couple of years later. Objects called "nonconventional ergodic averages" appeared for the first time in Furstenberg's proof, and understanding the limiting behavior of these averages became an important problem in ergodic theory. After breakthrough work of Bourgain in the late 1980s and early 1990s, no further progress had been made on proving pointwise almost everywhere convergence of nonconventional ergodic averages until very recently. I will report on this progress, along with some of the key inputs from additive combinatorics.