Small and large gaps between the primes

There are many questions about the gaps between consecutive
prime numbers which are not completely solved, even after decades of
effort. For instance, the twin prime conjecture, which asserts that
the gap between primes can equal 2 infinitely often, remains open.
However, there has been recent progress in understanding both very
small and very large gaps between primes. Last year, in a
breakthrough work of Yitang Zhang, it was shown that there were
infinitely many gaps between primes of bounded size; Zhang’s original
bound here was 70 million, but it has since been cut down to 246
thanks to the efforts of James Maynard and an online collaborative
Polymath project. In the opposite direction, recent work of Kevin
Ford, Ben Green, Sergei Konyagin, and myself have shown the existence
of prime gaps significantly larger than their average spacing,
improving upon earlier work of Rankin, Pintz, and others. The two
proofs are somewhat different in nature; the first argument relies on
sieve theory methods, whereas the second argument uses probabilistic
arguments combined with the results of Ben Green and myself in
arithmetic progressions in the primes. We present the main ideas of
both of these results in this lecture.