The following excerpt from Wikipedia made me laugh out loud:

The large power output of the Sun is mainly due to the huge size and density of its core (compared to Earth and objects on Earth), with only a fairly small amount of power being generated per cubic metre. Theoretical models of the Sun’s interior indicate a maximum power density, or energy production, of approximately 276.5 watts per cubic metre at the center of the core,[76] which is about the same rate of power production as takes place in reptile metabolism or a compost pile.

Takeaway: If the sun were a giant ball of lizards, nothing would change.

Things to do during a pandemic

  • Try and fail to buy toilet paper at Edeka.
  • Make ketchup from scratch.
  • Watch 30 episodes of Iron Chef.
  • Annoy your wife by following her around the apartment, or by listening to old blues songs too loudly.
  • Annoy VP* with basic questions about the p-adic Langlands program.
  • Try and fail to buy toilet paper at Aldi’s and at Lidl.
  • Meet with your masters and PhD students on Zoom.
  • Buy a car.
  • Gain a proper appreciation for Auslander-Gorenstein rings.
  • Buy 72 rolls of Polish toilet paper on Amazon and pay extra for it to be absurdly shipped from Britain, because that’s the only option for some reason.
  • Start a cool new joint project with BB and spend way too much time thinking about it (more on this soon!).
  • Promise CJ you’ll finish writing the proof of a certain result in a certain nearly final version of a certain paper and then somehow don’t finish doing it (yet). See previous item.
  • Move one kilometer to a much cheaper and much nicer apartment.
  • Write silly blog posts that (hopefully) no one will read.

*The Lithuanian VP, not the French VP.

spectral spaces; snark

(Update 12/24: Unfortunately the argument below doesn’t work as written. I overlooked the following detail in my “proof” that X^{wl} \to X is open: if S \to S' is a surjective continuous map of finite T_0 spaces, it’s not necessarily true that S^Z \to S'^Z is surjective. For instance, if S'= \{ \eta \rightsquigarrow s \} is the spectrum of a DVR and S=\{ \eta, s \} is S' equipped with the discrete topology, then S'^Z has three points but S=S^Z has two.

For the argument below to work, it would be enough to know that for any open subset V \subset X^{wl}, its image in X contains an open subset. Is this true?

In any case, the Corollary is still true, although by a totally different argument.)


I spent about six hours yesterday and today proving the following thing.

Lemma. Let X be a spectral space, and let Z be a closed generalizing nowhere-dense subset of X. Then Z is nowhere-dense for the constructible topology on X, i.e. Z doesn’t contain any nonempty constructible subset of X.

This has the following concrete consequence, which is what I really needed.

Corollary. Let (A,A^+) be some Tate-Huber pair with adic spectrum X=\mathrm{Spa}(A,A^+), and let Z \subset X be a Zariski-closed nowhere-dense subset. Suppose U_1 and U_2 are quasicompact open subsets of X such that U_1 \cap (X - Z) = U_2 \cap (X - Z). Then U_1 = U_2.

Proof of Corollary. We need to check that V= (U_1 \cup U_2 ) - (U_1 \cap U_2) is empty. But V is a constructible subset of X contained in Z, so this is immediate from the lemma.

Amusingly, even though this corollary is pretty down-to-earth, I only managed to prove it by proving the lemma, and I only managed to prove the lemma by exploiting the structure of the w-localization X^{wl} of X. Is there a more direct approach? Am I missing something obvious?

(Sketch of actual argument: the profinite set of closed points X^{wl}_{c} maps homeomorphically onto X equipped with the constructible topology, so if V \subset X is constructible it is clopen when viewed as a subset of X^{wl}_{c}. The key point is then to check that Z is nowhere-dense when viewed as a subset of X^{wl}_c. This can be done, using that the natural surjection t: X^{wl} \to X is open and that t^{-1}(Z) (which is then closed, generalizing and nowhere-dense in X^{wl}, the last point by openness of t) is the preimage of its image in \pi_0(X^{wl}) \cong X^{wl}_c.

The openness of t doesn’t seem to be stated in the literature, but it can be deduced from the proof of Lemma 2.1.10 in Bhatt-Scholze, using the fact that it’s obviously true for finite T_0 spaces.)

You may have noticed that RIMS is hosting a series of four workshops next year under the umbrella of a “RIMS Research Project” entitled Expanding Horizons of Inter-universal Teichmuller Theory.  The first of the workshops looks pretty reasonable, the other three not so much.  In case you’re wondering (as I did) how much money RIMS is ponying up for this, it seems to be capped at 5 million yen, or about $41k (according to e.g. this document). This doesn’t seem like very much money to support four workshops; I guess some funding is also coming from that infamous EPSRC grant.

Anyway, when you’re inside a black hole, your horizons might seem quite expansive indeed, but I doubt you’ll have much luck convincing others to join you.

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. …”

Modern fictions

On the website for the journal Algebra and Number Theory, one finds the following remarkable statement:

ANT publishes high-quality articles of interest to a broad readership, at a level surpassing all but the top four or five mathematics journals.

I have no problem with ANT, but shoving this self-assessment down people’s throats strikes me as deeply silly. We can probably all agree that the top five journals are (in some order) JAMS, Annals, Inventiones, Publ. IHES, and Acta Mathematica. So ANT is claiming to surpass Compositio, Duke, Ann. Sci. ENS, Cambridge J. Math, etc. Hmm…

Thanks to [redacted] for pointing this out to me.

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):

Arithmetische geometrie

Just attended a week-long meeting at Oberwolfach on arithmetic geometry.

  • “So did you do this computation like Gauss, or did you use a computer?” – Gabber to Katz


  • “Let the indices work it out themselves!” – Janssen


  • “Shouwu, either you’re going to answer my question, or I’m going to hand you over to Ofer!” – Kisin


  • Katz (telling a story at the beginning of his talk): “… So anyway, after Spencer returned to Princeton, this is how he described the math department at Stanford [where he had just been a professor for a couple years]: ‘At Stanford, they’re still studying the topology of the unit disk!’ ”
    Conrad (from the audience): “Those days are over.”


  • “We use what I wrote.” – Janssen reassuring Gabber


  • “So Peter, why did you turn down the breakthrough prize? [pause] I’m only asking because I’m drunk!”


  • Anon.: “So Ofer, do you come here much?”
    Gabber: [looks down at table, silently moves his finger across it in stepwise motion for 30 seconds] “Seventeen times.”


  • Two common referees for technical papers on Shimura varieties: Frobenius and Verschiebung.


  • Me (after writing down the “new” definition of a diamond): “Is that OK, Peter?”
    Scholze (from the back row): “Looks good!”


  • Zhang: “So Mochizuki is like the Buddha.  He writes his ideas.  He is satisfied.  If you want to understand them, you visit him, you ask him questions, he gives you a little idea, you go away and study.  You have to be a monk.  Have a monk’s approach.”
    Anon.: “Unfortunately, there aren’t very many good monks.”


  • A “symplectic lifting whatever shit”. Apparently they’re defined in Kai-Wen Lan’s thesis?


  • Gabber was NOT happy when he heard about Mochizuki’s Gaussian integral analogy.


  • While eating the horrible bread casserole thing, which Kedlaya, Lieblich and I had mangled pretty badly while serving ourselves:
    Lieblich:”What is this supposed to BE?”
    Kedlaya: “Some kind of croque madame?”
    Nizioł: “Yes, a croque madame.  But I think you guys croqued it.”

Autocorrect doesn’t know math

The autocorrect feature in Gmail has the unfortunate but hilarious habit of vigorously changing standard math terms into free-associative nonsense.  Here are some highlights (and I might add to this list from time to time):

  • “Igusa varieties” -> “Iguana varieties”
  • “Zariski topology” -> “Czarist topology”
  • “Gelfand spectrum” -> “Gelatin spectrum”
  • “cokernel” -> “cockerel” (my favorite so far)