The twin prime conjecture asserts that there are infinitely many primes
p for which p + 2 is also prime.
This conjecture appears far out of reach of current mathematical techniques.
However, in 2013 Zhang
achieved a breakthrough, showing that there exists some positive
integer h for which p and p + h are
both prime infinitely often. Equidistribution estimates for primes in
arithmetic progressions to smooth
moduli were a key ingredient of his work. In this talk, I will sketch what
role these estimates play in
proofs of bounded gaps between primes. I will also show how a refinement
of the q-van der Corput
method can be used to improve on equidistribution estimates of the
Polymath project for primes in
APs to smooth moduli.