The next lecture in this series is here.
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. Prof. 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.
These videos are of a lecture course by Yufei Zhao at the Massachusetts Institute of Technology in 2019, and made available as part of its OpenCourseWare initiative. The website for the course may be found here.