Description: In an unsuccessful attempt to prove Fermat’s last theorem, Schur showed that every finite coloring of the integers contains a monochromatic solution to x + y = z, an early result in Ramsey theory. Professor Zhao begins the course with a proof of Schur’s theorem via graph theory and how it led to the modern development of additive combinatorics. He then takes the class on a tour of modern highlights of the field: Roth’s theorem, Szemerédi’s theorem, and the Green–Tao theorem.

Instructor: Prof. Yufei Zhao