So as we head towards the organizational meeting in December, there is rapid-fire development on the problem of bounded gaps. As mentioned just a few days ago in a previous post (https://theconsciousmathematician.wordpress.com/2013/11/18/a-new-result-concerning-bounded-gaps-between-primes/), James Maynard was able to obtain a substantial improvement in several directions. Firstly, Maynard was able to improve the current record for the bound on gaps substantially; down to . More importantly, Maynard demonstrated that one does not need to obtain a level of distribution (of primes) above 1/2 to obtain bounded gaps (Goldston, Pintz, and Yildirim proved that a level of distribution above 1/2 would imply bounded gaps, and Zhang did just that, but with some restrictions on the range of primes considered). Indeed, the main ‘work horse’ of Maynard’s argument is the Bombieri-Vinogradov theorem. Finally, Maynard was the first to obtain a finite bound for the gap for .
Maynard’s paper can be found here: http://arxiv.org/abs/1311.4600
Terry Tao’s blog post on the matter can be found here: http://terrytao.wordpress.com/