The GreenTao Theorem and a Relative Szemerédi Theorem
The GreenTao Theorem and a Relative Szemerédi Theorem

Jacob Fox , MIT
Fine Hall 224
The celebrated GreenTao 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 GreenTao 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.