Tristan Freiberg: Strings of congruent primes in short intervals

We will discuss a proof that there are infinitely many pairs of consecutive primes p_n < p_{n+1} such that (1) p_{n+1} - p_n < epsilon*log p_n and (2) p_n = p_{n+1} = a mod q both hold simultaneously, where epsilon is a fixed, arbitrarily small positive number, and q and a are a given pair of coprime integers. The proof combines the ideas of Goldston, Pintz and Yildirim, who proved that (1) holds infinitely often (a conjecture of Hardy and Littlewood), and of Shiu, who proved that (2) holds infinitely often (a conjecture of Chowla). This will be a colloquium style talk aimed at a general audience.