Erica Klarreich, writing for Quanta Magazine:
It remains to be seen how much more can be wrung out of Zhang’s and Maynard’s methods. Prior to Maynard’s work, the best-case scenario seemed to be that the bound on prime gaps could be pushed down to 16, the theoretical limit of the GPY approach. Maynard’s refinements push this theoretical limit down to 12. Conceivably, Maynard said, someone with a clever sieve idea could push this limit as low as 6. But it’s unlikely, he said, that anyone could use these ideas to get all the way down to a prime gap of 2 to prove the twin primes conjecture.
“I feel that we still need some very large conceptual breakthrough to handle the twin primes case,” Maynard said.
Tao, Maynard and the Polymath participants may eventually get an influx of new ideas from Zhang himself. It has taken the jet-setting mathematician a while to master the art of thinking about mathematics on airplanes, but he has now started working on a new problem, about which he declined to say more than that it is “important.” While he isn’t currently working on the twin primes problem, he said, he has a “secret weapon” in reserve — a technique to reduce the bound that he developed before his result went public. He omitted this technique from his paper because it is so technical and difficult, he said, adding that he may publish it next year.
I love mathematicians in the way that only a son of mathematicians can.