Some History

I’m heading out soon for a 10 day vacation in the Rocky Mountains, blogging likely to change from sparse to non-existent for the next couple weeks. I’ve come across the following things that people with an interest in the recent history of mathematics may find worthwhile:

  • S. T. Yau over the past year has organized a series of talks on the recent history of mathematics, featuring prominent people in the subject giving expository talks on a topic, sometimes writing something up. The talks are available here, the write-ups here. I can especially recommend Nigel Hitchin’s detailed explanation of the work of Michael Atiyah relevant to physics, much of which he was personally involved in.
  • Lieven Le Bruyn at neverendingbooks points to some wonderful French math YouTube videos. Don’t miss Alain Connes interviewing Serre, with Serre explaining that he doesn’t know (or care) what a topos is.
  • For a good account of the fascinating life of Alexander Grothendieck, there’s Luca Signorelli’s The Man of the Circular Ruins. I hadn’t realized that some of the weirder writings from Grothendieck’s later life are now readily available, for instance La Clef des Songes.
  • For a long recent account by Langlands both of his recent ideas about geometric Langlands and his fascination with languages (including White Russian language instructors), see this letter to Yvan Saint-Aubin.
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5 Responses to Some History

  1. Richard Townsend says:

    The Grothendieck article is truly fascinating – what a story!

  2. I have been interested in Grothendieck for many reasons for many years. I think this article is absolutely the best introduction that I have read. Thanks!

  3. Hi Peter, thanks for the link! Grothendieck’s story, personality and achievements are indeed fascinating. It’s one of those historical figures that once approached “expand” in your field of view instead of fatally shrinking. No matter how complicated and often difficult to judge they may be.

  4. jsm says:

    The point of Langlands’s letter is to explain that to create a geometric Langlands theory in the style of Hecke, that is to say with eigenvalues and eigenfunctions, we will need a theory other than that of the Russians.

    To be more precise, the theory I described in my Russian article for the group GL(2) and elliptic curves begins with an article by Atiyah, an article that is in my opinion one of his best, and it remains, I believe, to do the same for any curve and any reductive group… It will be necessary to start by pursuing the ideas of Atiyah, but generalizing his article. But this will not, in my opinion, be easy. I haven’t tried it myself, especially I don’t know how to guess the structure of Bun(G) in general… The structure of Bun(G) being known, it will require an understanding of all the eigenfunctions attached to a given curve and a given reductive group. In general, this last question will be much more difficult than for elliptic curves, the case dealt with by Atiyah.

  5. Will Sawin says:

    @jsm,

    In connection with your summary it is probably worth mentioning that Pavel Etingof, Edward Frenkel, and David Khazdan recently (but earlier than the date appearing on the letter) formulated a version of geometric Langlands theory with eigenvalues and eigenfunctions, using the “Russian” theory, and without any kind of Atiyah-style explicit description of the points of Bun_G, in (https://arxiv.org/abs/1908.09677).

    They also handle the ramified case, at least for sufficiently mild (unipotent) ramification, which was something else that Langlands suggested as being interesting in this letter.

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