SPEAKER:     Terence Tao

TITLE:     Long arithmetic progressions in the primes


ABSTRACT:

We present some joint work with Ben Green on counting the number of arithmetic progressions of length k in the integer primes, and in particular showing that for each k there are infinitely many of these progressions. One key idea is to view the primes not as a sparse subset of Z, but rather as a dense subset of "almost primes", which (after some local obstructions from small divisors are eliminated) are distributed "pseudorandomly" in the integers, thanks to some recent work of Goldston and Yildirim. One can then transfer Szemeredi's theorem (dense subsets of integers contain arbitrarily long progressions) to this setting and obtain arbitrarily long progressions of primes. More recently, we have been able to transfer the more Fourier analytic analysis of Gowers concerning the k=4 case of Szemeredi's theorem, and thus obtain asymptotics for the number of progressions of primes for length 4. (The case k=3 was already known to van der Corput).