The Green-Tao Theorem and a Relative Szemerédi Theorem

Jacob Fox , MIT
Fine Hall 224

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 recent joint work with David Conlon and Yufei Zhao simplifying the proof. The main new ingredient in the proof of the Green-Tao theorem is a relative Szemerédi 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 Szemerédi theorem, showing that a much weaker pseudorandomness condition is sufficient. The key component in our proof is an extension of the regularity method to sparse hypergraphs.