GAAG seminar - Gowers uniformity of the primes in short intervals

The Gowers norms, introduced by Tim Gowers, are a way to measure the pseudorandomness properties of a set. In 2012, Green, Tao and Ziegler showed that the set of prime numbers is Gowers uniform, in the sense that a suitably normalised version of it has small Gowers norms. I will discuss a recent work where we show that the primes are Gowers uniform even when restricted to almost all short intervals of the form [x, x+x^{1/3+\epsilon}]. I will also discuss an application of such results to an averaged version of the Hardy--Littlewood prime tuples conjecture. This is based on joint work with Kaisa Matomäki, Maksym Radziwiłł, Xuancheng Shao and Terence Tao.