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The recent article on Quanta (by Natalie Wolchover) concerning $\aleph_1$ vs. $\aleph_2$ suggests that there is excitement within that community:

Juliette Kennedy: "It’s one of the most intellectually exciting, absolutely dramatic things that has ever happened in the history of mathematics."

Another instance is the Fargues/Scholze advances on "Geometrization of the local Langlands correspondence," which has the Langlands world excited:

Eva Viehmann: "It’s really changed everything. These last five or eight years, they have really changed the whole field."

This makes me wonder if there is something like a heat map for all of mathematics, which would show the areas with a lot of excitement. It seems difficult to capture this via arXiv postings, but that is an obvious starting point. Has anyone pursued this?

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    $\begingroup$ Not really an answer to your question, but places I personally look for nice broad overviews of recent developments in various mathematical subfields are: the ICM proceedings; the "Current Events Bulletin" series by the AMS; the Harvard/MIT "Current Developments in Mathematics" lecture series. $\endgroup$
    – Sam Hopkins
    Jul 24, 2021 at 22:03
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    $\begingroup$ In a sense, mathoverflow tries to be just that. You can take whatever ordering criterion of questions you prefer as your heat indicator. $\endgroup$
    – Alessandro Della Corte
    Jul 24, 2021 at 22:32
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    $\begingroup$ @AlessandroDellaCorte I'm not really sure this is true. In addition to the fact that MO is heavily biased towards certain fields (like my own), it's not really well-suited for discussing current research, which this question seems to be about. Usually people excited about current research who have questions about it will read the papers, attend seminar or conference talks about it, ask the authors directly, or organize learning seminars, not ask on MO. $\endgroup$
    – Will Sawin
    Jul 25, 2021 at 0:19
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    $\begingroup$ I'd suggest to use a robot counting superlatives. Who will win the race? $\endgroup$
    – YCor
    Jul 26, 2021 at 8:44

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https://paperscape.org/ is a 'heat map' of the arxiv if you color the graph by age. Unfortunately, its ability to detect links between mathematics papers is a bit lacking compared to physics papers for some reason, but it still gives a very interesting view of the subject.

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Not exactly the same (this image represents community detection more than intensity of activity), but Roja Bandari took all the programs run by the Institute for Pure and Applied Mathematics here at UCLA up to 2013, and formed a graph between them, with two programs close to each other if they had many common participants. Here is the outcome:

enter image description here

For instance, the very centrally located MGA2003 node (representing the 2003 workshop in Multiscale Geometry and Analysis) is where Emmanuel Candes and I met to work out some of the early results in the field now known as compressed sensing.

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  • $\begingroup$ For an institute with pure mathematics in the name, they don't half ignore all of pure mathematics. $\endgroup$
    – Ben McKay
    Jan 3, 2022 at 19:41
  • $\begingroup$ I view this as a byproduct of the existence of MSRI in the same institute space, as complementing either other's strengths rather than directly competing becomes the more optimal division of labor. Together, MSRI and IPAM collectively give quite a good coverage of all the mathematical sciences (both pure and applied). $\endgroup$
    – Terry Tao
    Jan 3, 2022 at 21:39
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A heatmap of highly cited current mathematics can be found on the arxiv explorer.

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