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 some recent simplifications.
One of the main ingredients in the proof is a relative Szemeredi theorem, which says that any subset of a pseudorandom set of integers of positive relative density contains long arithmetic progressions. Our main advance is a simple proof of a strengthening of the relative Szemeredi theorem, showing that a much weaker pseudorandomness condition is sufficient.
The proof of the new relative Szemeredi theorem has three main ingredients: (1) a transference principle/dense model theorem of Green-Tao and Tao-Ziegler (with simplified proofs given later by Gowers, and independently, Reingold-Trevisan-Tulsiani-Vadhan) applied with a discrepancy/cut-type norm (instea d of a Gowers uniformity norm as it was applied in earlier works), (2) a new counting lemma, and (3) Szemeredi's theorem as a black box.
Based on joint work with David Conlon and Jacob Fox.