# Windows into new mathematical worlds [closed]

Yitang Zhang's Annals of Mathematics primes-gap result opened a new window, which Polymath's reduction from $70\times 10^6$ to $246$ attests. Perhaps Harald Helfgott's celebrated proof of the odd Goldbach's conjecture has not similarly open new avenues—or at least not yet. Certainly the Green-Tao theorem has opened new windows. Perhaps the Guth-Katz breakthrough on the Erdős distance problem has opened new windows.

My question is,

Q. Which results in the recent past (~last decade+) have opened significant windows into new mathematics?

I realize this is quite subjective, but it requires a high-level view of fields of mathematics to notice this while it is happening, in a way that others (like me) without that expertise cannot discern. It would be educational to learn of expert opinions, without diminishing the significance of any particular result. Rather I am hoping for a celebration of those results which seem not to be the end of a line of investigation, but rather a new beginning.

• Clearly this should be CW. – Todd Trimble Sep 7 '15 at 1:10
• If Zhang himself had proved the 246 bound, would it still have opened new avenues? – Lennart Meier Sep 7 '15 at 7:49
• I'm voting to close because the question seems too broad and vague. I don't even understand the stated example of Zhang versus Helfgott. Does it just have to do with whether the paper triggers a minor industry of followup papers? This happens all the time. – Timothy Chow Sep 7 '15 at 21:44
• (I have asked the moderators to close the question.) – Joseph O'Rourke Sep 7 '15 at 22:16
• With some slight reluctance, I'll grant Joseph's request. The idea behind the question seems to me to have merit, so perhaps tweaking is all that's needed? – Todd Trimble Sep 8 '15 at 0:24