Some historical snippets

Probably everyone is familiar with the MacTutor History of Mathematics website. While browsing through their additional material, I came across some wonderful things:

  • Hardy writing to Veblen from Princeton, ca. 1928:
    “…However, I suppose my present passion for the soda fountain will abate by degrees.” … “I do find myself regretting that Wiener’s not here: but no doubt if he were I should very quickly revise my opinion.”
  • Pedoe on Hodge: “One fine morning Hodge and I were inside the grounds of Pembroke College when we met J A Todd, an excellent geometer, the author of a fine textbook on projective geometry, a University lecturer – and a pipe smoker who spent more time striking matches than actually smoking. As we stood talking, Todd struck match after match and dropped them on the ground at Hodge’s feet. Hodge, as the Acting Bursar, was responsible for the proper maintenance of the grounds of Pembroke, so as Todd dropped each match, Hodge bent down to pick it up. Todd, who wore eyeglasses with strong lenses, was completely unaware of what was going on. The spectacle of the very thin Todd unconsciously dropping matches, and the rotund Hodge bending down every few seconds – while becoming more and more exasperated – is one I shall never forget.”

    “Hodge became Master of Pembroke and President of the Royal Society. He was very shrewd and usually tactful, but had definite ideas on certain matters. He thought, for example, that a mathematical paper should be just that, with no embellishment. When Patrick Du Val, a contemporary of Donald Coxeter, a good geometer, and a lover of the arts, submitted a paper to the Cambridge Philosophical Society for publication, with a quotation from Dante following its title, Hodge suggested that this was not “appropriate.” He was badly flustered when a furious Du Val withdrew the paper.”
  • Schwartz on Maurice Audin’s thesis
  • Hermann Weyl’s speech at Emmy Noether’s funeral
  • Hardy again, this time on Waring’s problem
  • Dedekind attending a lecture course by Gauss in the winter of 1850: “…The lecture room, separated from Gauss’ office by an anteroom, was quite small. We sat at a table which had room for three people comfortably at each side, but not for four. Gauss sat opposite the door at the top end, at a reasonable distance from the table, and when we were all present, the two who came in last had to sit quite close to him with their notebooks on their laps. Gauss wore a lightweight black cap, a rather long brown coat and grey trousers. He usually sat in a comfortable attitude, looking down, slightly stooped, with his hands folded above his lap. He spoke quite freely, very clearly, simply and plainly; but when he wanted to emphasise a new point of view, for which he used a particularly characteristic word, then he would raise his head, turn to one of those sitting beside him, and gazed at the student with his beautiful, penetrating blue eyes during his emphatic speech. That was unforgettable. …”
  • Thue on mathematics in Berlin in 1891-92: “Fuchs, whom I heard lecture on analytical mechanics, did not at first make much of an impression on me. The concepts he employed were, as far as I could see, surrounded by a mist of vagueness. When I heard him in a seminar, however, I got a strong presentiment that he can excel when he wants to do so. He lectures with his eyes shut and looks thoroughly tired and peevish. He can also be rather absent-minded. I remember how he was once talking about differentials, and quite unconsciously he picked up a handful of bits of chalk which he waved in illustration before our wondering eyes. Afterwards he carefully laid his differentials down again on his desk, with his eyes still closed. Professor Fuchs, like Kronecker, is a very prepossessing man, but not overly talkative. I was at a ball at his home this winter. It was a delightful affair. We danced so energetically that the floor cracked in a couple of places.”

    “The mathematical seminar down here functions in much the same way as yours does in Oslo. It is an established university institution. Fuchs and Kronecker preside in turn. Meetings are held between 5 and 7. No report is circulated. I have requested Kronecker to permit my highly attractive voice to be heard at the aforementioned place, but so far he hasn’t paid any attention. …”

Some disconnected thoughts

Sorry for the silence here, the last six months have been pretty busy. Anyway, here are some small and miscellaneous thoughts.

\bullet Dan Abramovich is a really excellent and entertaining writer.  I especially recommend this BAMS review of books by Cutkosky and Kollar, and this survey article for the 2018 ICM; after reading these, you’ll feel like you really understand what “resolution of singularities” looks like as a working field.  His mathscinet review of Fesenko’s IUTT survey is also deeply funny – let us hope that the phrase “resounding partial success” enters common usage.

\bullet Speaking of the 2018 ICM, Scholze’s article is also now available here. I was amused to see a reference to “prismatic cohomology” in the last paragraph – when I visited Bhargav in October, this object had a truly horrible and useless provisional name, so it’s great to see that name replaced by such a perfect and suggestive one.  (This is a new cohomology theory which is closely related to crystalline cohomology, and is also morally related to diamonds – cf. these notes for the actual definition.)

\bullet Recently I had need of the following result:
Lemma. Let S=\mathrm{Spec}\,A be the spectrum of a valuation ring, with generic point \eta \in S. Let X \to S be a separated and finite type map of schemes, and let x: \eta \to X_{\eta} be a section over the generic point of S, with scheme-theoretic image Z \subset X. Then the induced map Z \to S is an open immersion.

Note that if X \to S is proper, then Z \to S is an isomorphism. Anyway, I ended up working out two proofs: a short argument relying on Nagata compactification, and a longer argument which avoided this result but used some other pretty tricky stuff, including Zariski’s main theorem and Raynaud-Gruson’s magic theorem that if R is a domain, then any flat finite type R-algebra is finitely presented. If someone knows a truly elementary proof of this lemma, I’d be interested to see it.

Next time I’ll say something about what I needed this for…

Report from Tucson

Just back from the 2017 Arizona Winter School on perfectoid spaces.  First of all, I should say that everything was impressively well-organized, and that the lecturers did a fantastic job, especially considering the technical weight of this material. (Watch the videos if you don’t believe me.)  Jared Weinstein, in particular, has an almost supernatural ability to make a lecture on some technical thing feel comforting.

Now to the jokes.

  • In his opening lecture, Scholze called perfectoid spaces a “failed theory”, on account of his inability to completely settle weight-monodromy. “You see, I’m Prussian, and when a Prussian says he wants to do something, he really feels responsible for doing it.”
  • Audience member: “Why are they called diamonds?”
    Scholze: “[oral explanation of the picture on p. 63 of the Berkeley notes]”
    Weinstein: “Also, diamonds are hard.”
  • Anon.: “When you’re organizing a conference, the important thing is not to give in and be the first one who actually does stuff.  Because then you’ll end up doing everything!  Don’t do that!  Don’t be the dumb one!”
    Me: “Didn’t you organize [redacted] a couple of years ago?”
    Anon.: “Yeah… It turned out that Guido Kings was the dumb one.”
  • Mazur: “It just feels like the foundations of this area aren’t yet… hmm…”
    Me: “Definitive?”
    Mazur: “Yes, exactly.  I mean, if Grothendieck were here, he would be screaming.”
  • “Do you ever need more than two legs?”
  • During the hike, someone sat on a cactus.
  • Finally, here is a late night cartoon of what a universal cohomology theory over \mathbb{Z} might look like (no prizes for guessing who drew this):
    cartoon