Georgia Institute of Technology College of Sciences

The celebrated Green-Tao theorem states that there are arbitrarily long arithmetic progressions in the primes. In this talk, I will explain the ideas of the proof and discuss our recent simplifications. One of the main ingredients in the proof is a relative Szemerédi theorem, which says that every relatively dense subset of a pseudorandom set of integers contains long arithmetic progressions. Our main advance is both a simplification and a strengthening of the relative Szemerédi theorem, showing that a much weaker pseudorandomness condition suffices. I will explain the transference principle strategy used in the proof. Also see our recent exposition of the Green-Tao theorem: http://arxiv.org/abs/1403.2957 Based on joint work with David Conlon and Jacob Fox.