PIMS/WMAX Postdoctoral Colloquium: Roth's theorem in the primes

In 1953, K. Roth proved that any subset of positive integers of positive density contains infinitely many non-trivial three-term arithmetic progressions. (By a non-trivial arithmetic progression we mean one of the form (a, a+d, a+2d) with d > 0.) First, I shall explain the main ideas of the proof of Roth's theorem.The second part of my talk will be devoted to Roth's theorem in the primes. I shall explain how B. Green proved that a subset of primes of positive relative density must contain some non-trivial 3-term arithmetic progressions and how H. Helfgott and I sharpened his quantitative result.