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.

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    $\begingroup$ Clearly this should be CW. $\endgroup$ – Todd Trimble Sep 7 '15 at 1:10
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    $\begingroup$ If Zhang himself had proved the 246 bound, would it still have opened new avenues? $\endgroup$ – Lennart Meier Sep 7 '15 at 7:49
  • $\begingroup$ 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. $\endgroup$ – Timothy Chow Sep 7 '15 at 21:44
  • $\begingroup$ (I have asked the moderators to close the question.) $\endgroup$ – Joseph O'Rourke Sep 7 '15 at 22:16
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    $\begingroup$ 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? $\endgroup$ – Todd Trimble Sep 8 '15 at 0:24

The work of Lurie, and others (for instance, Rezk, Joyal, and also older work of Toen, Vezzosi, Simpson, ...) on Higher Topos Theory and Higher Algebra, leading to e.g. a proof of the cobordism hypothesis (see also Freed's article) and the proof of the Weil conjecture on Tamagawa numbers for function fields over finite fields.

I'm making this community wiki so others can add things.

  • $\begingroup$ I've added a link to Lurie's Higher Algebra. I think derived algebraic geometry should also be mentioned. $\endgroup$ – user62675 Sep 7 '15 at 1:39
  • $\begingroup$ I agree, but I was trying to think what it has been used for: clearly thinking about eg equivariant elliptic cohomology $\endgroup$ – David Roberts Sep 7 '15 at 1:54

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