theHigherGeometer

rimcræftiga |
bespoke constructions in categorified geometry since 2010 |
dude

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nprofile1qqsq4q2u0gf4fy9egh0yzczz2pf0knxnu6p7tf5jj5rjdga0crkrrucpz4mhxue69uhhyetvv9ujuerfw36x7tnsw43q43e2da

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npub1p2q4c7sn2jgtj3w7g9syy5zjldxd8e5ruknf99g8y636ls8vx8esq5c0lr

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Last Notes

2026-02-06 13:31:26 UTC

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@nprofile…xuzd @nprofile…47dl I personally would have said 'injections' rather than bijections. For instance injecting some sensible uncountable set of functions to the reals. Proving Baire space or Cantor space is (classically) in bijection with |R is a pain. Not saying the lazier statement is trivial, as the reals are anything but trivial, but it would illustrate a useful idea: anything containing an uncountable set is uncountable. I'd hate to accidentally let a student walk away thinking that to prove something is uncountable it needs to be by constructing a bijection with P(|N)....

2026-02-06 10:43:59 UTC

A phrase a colleague used today describing some work this week: "embarrassingly large speed-up factor".
I then realised tonight that I can embarrass myself even harder because of an earlier quick-and-dirty implementation, for the purposes of getting a single component working in a sophisticated algorithm, coming back to bite me when trying to run a moderately realistic sized example. Good to catch these things on toy models, but a little sad that past-me had left it as a trap for now-me.

2026-02-06 09:29:43 UTC

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@nprofile…u3x7 I like the fact that these questions are not "please calculate a number out of this tower of abstractions", but rather a realistic answer that mathematicians in less number-focused fields actually encounter in their work.

2026-02-06 04:18:39 UTC

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@nprofile…t5a9 If you had to ask me, the key idea is that there are two ways of measuring the topology of certain higher-dimensional shapes described by polynomial equations: one that involves calculus (so something vaguely analogous to how electromagnetic fields are described), and one that only involves cutting out curved "slices" of the shape using more polynomial equations. The conjecture is that the latter, which can on the face of it measure at least *some* of the features that the calculus-based approach can see, actually captures everything.

2026-02-06 04:14:00 UTC

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@nprofile…t5a9 In Devlin's book The Millennium Problems he spends nearly three pages at the start of the chapter on the Hodge conjecture explaining that it's really difficult to explain. He says "To the layperson, the very inaccessibility of the problem is perhaps its most interesting feature." And he had a lot of experience explaining mathematics to the general public, but nonetheless resorted to something halfway between understandable and accurate: so more or less neither.

2026-02-05 23:45:58 UTC

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@nprofile…kys3 Working link for the published article's mentioned doi:
Niyousha Najmaei, Niels van der Weide, Benedikt Ahrens, and Paige Randall North. 2026. "From Semantics to Syntax: A Type Theory for Comprehension Categories" Proc. ACM Program. Lang. 10, POPL, Article 83 (January 2026), 30 pages. https://doi.org/10.1145/3776725
@nprofile…htrj @nprofile…f7m9

2026-02-05 10:01:29 UTC

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@nprofile…4hp5 I've been keeping half an eye on what he has been doing with his collaborators over the past decade or so, it's definitely fun to watch. Having a sketch of Connes' current ideas at the level of 19th century tech (sweeping a few things under the rug that got proved in the early 20th century) was a fascinating exercise. It reminds me of when arithmetic geometers give talks that give an on-ramp to a key modern conjecture in terms of some really concrete classical problem, like the congruent number problem leading to elliptic curves and then Tunnell's theorem, which is conditional on the BS-D conjecture.

2026-02-05 06:18:01 UTC

https://doi.org/10.26379/CZ2024.2_006 This seems tricky to get
#icanhazpdf

2026-02-05 06:14:51 UTC

New from Alain Connes. His "Letter to Riemann" is fun.
https://arxiv.org/abs/2602.04022
He is honest about the fact we cannot prove the RH, and that it's not clear if his preferred approach will be the way, but also:
"Our discovery of a large class
of functions directly related to the Weil quadratic form and with zeros provably on the critical line, combined with the extraordinary numerical evidence linking truncated Euler products to the actual zeros of zeta, suggests that Riemann’s original insights may contain more power than previously realized. The accuracy achieved using only primes less than 13—with errors as small as \(2.6 \times 10^{-55}\)—cannot be dismissed as coincidence"
Apparent numerical coincidences in number theory like this at very minimum demand an explanation.

2026-02-04 23:56:43 UTC

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@nprofile…sqfw I hope I don't regret this

2026-02-04 13:24:49 UTC

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@nprofile…sqfw "I'm also going to be straight in case anyone else is reading this in the future: this offer is not something anyone else can expect. I have better things to do with my life than check LLM output, like my own research, for which I get precious little time to do."

2026-02-03 10:53:08 UTC

Whee, another indirect influence on the number theory literature ^_^
https://arxiv.org/abs/2602.01001
(they cite the paper that following a blog post of mine introduced the term in the title, because of the meme format)

2026-02-02 06:01:00 UTC

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@nprofile…hh7n I can't say I'm needing to temper my excitement.

2026-02-02 03:39:33 UTC

I heard a merger fact today that requires me to actively make myself calm down.
I don't think I can share it publicly, so I won't be.

2026-02-02 03:39:17 UTC

I heard a merger fact today that requires me to actively make myself calm down.
I don't think I can share it publicly, so I won't be.

2026-02-01 10:50:37 UTC

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@nprofile…dn5h Miss R has the same one! Her name here is Mimi, after a grey cat in an old kids book we have read.

2026-01-30 00:14:34 UTC

"The proof will not exist if the reader does not do half the work".

2026-01-30 00:13:34 UTC

I really enjoyed this one. Lots to think about from a pedagogical point of view.

2026-01-29 22:43:41 UTC

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@nprofile…lfhd Thanks!

2026-01-29 22:23:19 UTC

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@nprofile…lfhd On the sentence "Cet exposé fait suite à celui d’il y a deux ans (25 janvier 2024)," the link on celui is broken!
https://webusers.imj-prg.fr/~antoine.chambert-loir/exposes/tb-20240125.pdf

2026-01-29 00:48:48 UTC

I've learned recently from snooping on discussions about formalising explicit estimates in number theory that integration by parts
\[ \int_a^b f'(x)g(x)dx = \big[f(b)g(b) - f(a)b(a)\big] - \int_a^b f(x)dg(x)\]
can be done with a pair of functions \(f,g\) (on an interval in the reals, with real values) when one of them (say \(f\)) is continuously differentiable (sure) and the other is merely of bounded variation (!). The point is that you take the \(dg\) in the naive undergraduate formula seriously, and use the Riemann–Stieltjes integral, rather than taking \(dg(x) = g'(x)dx\) as we teach students. And BV functions are precisely those where the R–S integral of a continuous function with respect to them makes sense.

2026-01-28 23:06:35 UTC

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@nprofile…g0zc Maybe not a topos-theoretic account, but presumably a Markov-category-theoretic one?

2026-01-28 23:01:26 UTC

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@nprofile…xuzd Thanks to this post, I looked and discovered the macOS Mail app can be configured to the layout I remember from the mid-00s, as you described in 1.

2026-01-28 22:13:58 UTC

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@nprofile…lfhd To think that Ehresmann was working (for himself) on "combinatorial topology" at the time Bourbaki started and it took 80 years to be Bourbakified into a book.

2026-01-28 22:11:59 UTC

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@nprofile…lfhd Thanks! I've seen that you are scrupulous about posting slides on your site.
@nprofile…fws0

2026-01-28 21:15:03 UTC

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@nprofile…xek8 ? DiCaprio?

2026-01-28 10:55:03 UTC

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@nprofile…fws0 Awesome, thank you!

2026-01-28 10:16:34 UTC

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@nprofile…fws0 Thanks :-) now that things are in full swing, are participants likely to make their slides available on the conference site?

2026-01-28 10:02:19 UTC

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@nprofile…fws0 How's this morning's talk on Chevalley?
I ask because Serre once answered a question of mine, and noted of him "c'était le plus puriste des membres de Bourbaki."

2026-01-28 08:37:26 UTC

I remember the original tweet thread by @nprofile…cu8d, even though it was 4-5 years ago!
https://davidkbutler.xyz/2026/01/28/its-just-subbing-into-formulas/
There's a whole 'nother such discussion to be had around teaching pure mathematics, where students "just have to check the hypotheses in the definition/theorem".
#mtbos

2026-01-28 03:47:49 UTC

Really rubbing it in now.
https://media.mathstodon.xyz/media_attachments/files/115/970/665/240/455/454/original/bab54a05c67e8149.png

2026-01-28 01:57:20 UTC

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@nprofile…7jv6 Well done, authors.

2026-01-27 23:54:47 UTC

Oops.
https://media.mathstodon.xyz/media_attachments/files/115/969/753/170/360/914/original/696bcdcb5d494db0.png

2026-01-27 20:36:32 UTC

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@nprofile…udvu I agree it made the expert testimony of the engineers nigh impossible to ignore, so it wasn't for nothing.

2026-01-27 20:10:56 UTC

Amazing, no need for a stunt by a certain physicist to magically discover and reveal why...
https://www.abc.net.au/news/science/2026-01-28/space-shuttle-challenger-disaster-nasa-explosion-video-engineers/106162902

2026-01-27 01:38:49 UTC

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@nprofile…wncd @nprofile…4hp5 Just commenting so I can listen in

2026-01-26 23:42:48 UTC

From July 2017, despite the upload date, the Big Proof conference (which got @nprofile…pm2x into Lean)
https://www.youtube.com/watch?v=M5LQNJ6kXDE

2026-01-26 22:43:16 UTC

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Some fun anecdotes of Dirac's visit to Adelaide, nearly a decade before I was born, but involving at least one person (Prof. Angus Hurst) I knew later, when I was at uni.
http://www.neilgunther.com/Dirac/dirac.html

2026-01-26 22:34:48 UTC

While I can't compare myself to Dirac, I feel some kindred motivation here.

2026-01-26 19:34:05 UTC

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@nprofile…r0xq I was just thinking of asking locally if the @nprofile…6gum had been in discussions about such a thing

2026-01-26 09:15:33 UTC

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I fear "Congregati ex toto orbe mathematici" might ring a little hollow this year...

2026-01-26 08:37:31 UTC

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@nprofile…7jv6 Well, it has been shown that an online ICM can be done, and with less lead-up time than if a similar decision were made this month.

2026-01-26 08:35:01 UTC

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@nprofile…7jv6 Well, it has been shown that an online ICM can be done.

2026-01-26 08:24:00 UTC

I say the following reluctantly, since I leave the online discussion of politics to others better versed and placed to do so. My voice is not that strong, and does not count for much, but I do wish the IMU would publicly address what is going to happen to the ICM this year. The leadership would, I hope, either make a hard call sooner rather than later to shift venues, or explain why not. Or, at absolute minimum, let everyone know if there are ongoing discussions the outcomes of which will be announced in due course. I cannot imagine such discussions aren't happening.
#ICM2026

2026-01-22 00:11:50 UTC

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@nprofile…4hp5 I don't read the last one as 'quintessentially American'. The internet tells me the earliest reference recorded by the OED is rather old:
"The OED's earliest citation for touched as an adjective, meaning slightly crazy, is from John Humfrey's 1672 book, Authority of Magistrate of Religion: 'I count this man now as one in a Fever, that is, touched in his head, and who can help such a conceit?'"

2026-01-21 22:22:03 UTC

This is not for lack of staff working their butts off trying to get things to work
https://www.abc.net.au/news/2026-01-22/adelaide-university-merger-leave-students-stressed/106252460

2026-01-21 06:03:16 UTC

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@nprofile…wkcu 1915 is too old to really talk about mathematical isomorphism in the abstract, but one can see that there isn't even the usual symbol(s) that we have today
https://archive.org/details/vademecumdutypog00dumo/page/180/mode/2up
https://media.mathstodon.xyz/media_attachments/files/115/931/557/624/011/855/original/de91876b06a65c59.png

2026-01-21 03:26:21 UTC

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@nprofile…4hp5 No, he's opening up his book by description of natural numbers and then different kinds of numbers that arise in solving problems (squares, 'roots' etc)
https://archive.org/details/algebraofmohamme00khuwuoft/page/5/mode/1up

2026-01-21 03:08:21 UTC

"When I consider what people generally want in calculating, I found that it always is a number."
—Abu Ja'far Muhammad ibn Musa Al-Khwarizmi, https://en.wikipedia.org/wiki/Al-Jabr ca.820
TIL this is the opening sentence of the book proper (after the author's preface).

2026-01-20 23:29:35 UTC

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@nprofile…cytr I guess ?tela aranea means 'spider's web'? (Latin, no?)

2026-01-20 23:26:19 UTC

"For these reasons, it would be wrong to treat agent-led shopping as value-neutral. Systems designed to increase loyalty and revenue should not automatically be assumed to act in the best interests of consumers, even when they deliver genuine convenience."
https://theconversation.com/do-woolworths-shoppers-want-google-ai-adding-items-to-buy-well-soon-find-out-273342
hmmmmmmmmm

2026-01-20 23:25:30 UTC

"For these reasons, it would be wrong to treat agent-led shopping as value-neutral. Systems designed to increase loyalty and revenue should not automatically be assumed to act in the best interests of consumers, even when they deliver genuine convenience."
https://www.abc.net.au/news/2026-01-20/woolworths-google-ai-items-shopping-groceries/106243642
hmmmmmmmmm

2026-01-20 09:51:43 UTC

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@nprofile…wkcu Thanks. I'm interested in a very specific item: the use of = instead of a symbol for isomorphism (≃ or ≅ etc), which someone once suggested to me might have been a shortcut (before widespread computer typesetting, perhaps driven by typewriter usage) that people at the time could have used with full knowledge that it was really an isomorphism.

2026-01-20 09:47:35 UTC

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@nprofile…fws0 Well, that's that! If there are not abstracts I can't justify asking for them :-) Thanks for letting me know.

2026-01-20 06:54:46 UTC

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@nprofile…4hp5 Good thing I'm keeping myself under that limit already....

2026-01-20 05:37:13 UTC

"I’ve decided to push back on the pressure to publish by making a rule for myself: I will no longer publish more than seven papers per year. I’ve published a median of 15 papers a year for the past five years, so that means I will have to halve my output. "
🤯
https://doi.org/10.1038/d41586-025-04061-w
(From a member of the research committee of Australia's NHMRC, the main medical research funding body here)

2026-01-20 03:12:56 UTC

"Hegelian philosophy permeated large domains of German culture and the
Hegelian lexicon penetrated even mathematics — for instance Dedekind’s
habilitation lecture (1854) and 'Was sind und was sollen die Zahlen?'." (https://www.cirm-math.fr/RepOrga/3330/Slides/Benis-Sinaceur-Bost-Cavailles.pdf)
Oh, interesting! I wonder if people can point out how

2026-01-20 02:58:09 UTC

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@nprofile…lfhd Awesome! Bravo

2026-01-20 02:57:10 UTC

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@nprofile…wkcu I wonder if there is an analogue of this for mid-century *French* mathematical printing.

2026-01-20 02:15:53 UTC

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@nprofile…fws0 Hmm, looking at a different conference on the same platform, I see that it has a similar setup, but the abstracts are linked from the schedule in that one. I don't want to make more work for anyone, but I hope that perhaps the abstracts will land at some point (not to mention slides etc after the fact).

2026-01-20 02:11:40 UTC

@nprofile…fws0 sorry to be a pain, but the link that says 'abstracts' on https://conferences.cirm-math.fr/3523.html points instead to a differently-formatted version of the schedule instead. Is this intentional?

2026-01-20 02:11:12 UTC

2026-01-18 03:48:19 UTC

"while that gives you science, it doesn't give you press releases."
Precisely

2026-01-17 23:30:08 UTC

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@nprofile…u3x7 Since trying an AI system on solving mathematics problems is essentially experimental science, one might reasonably demand https://en.wikipedia.org/wiki/Preregistration_(science) as a standard for the field, like in other experimental sciences

2026-01-16 06:01:58 UTC

If you see a paper on the arXiv claiming a proof that Reinhardt cardinals are inconsistent with ZF, note that the author already claimed this in a prior arXiv preprint and retracted it. And also claimed a proof of P ≠ NP.
I cannot make an informed judgement about the new paper, but I would await experts' assessment before leaping to conclusions.

2026-01-16 00:21:24 UTC

https://www.abc.net.au/news/2026-01-16/qld-woman-wakes-to-find-python-on-top-of-her-snake-season/106233916
#straya

2026-01-15 23:30:41 UTC

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And here's a write-up on @nprofile…cu8d 's blog
https://davidkbutler.xyz/2016/09/20/sqwigles/

2026-01-15 21:57:07 UTC

From @nprofile…cu8d, nearly 9 years ago.
https://www.youtube.com/watch?v=u3v_7G5D5xI

2026-01-15 06:35:16 UTC

> Stephen explained this to me by saying that you cannot make jokes about failure in the States. It’s like cancer, it just isn’t funny at any level. In England, though, for some reason it’s the thing we love most.
This explains a lot, including LinkedIn.

2026-01-14 23:06:28 UTC

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@nprofile…rk9k I like the idea of trying novel visualisations here, I might have a think and give you some feedback

2026-01-14 08:24:16 UTC

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@nprofile…l3hr I saw 'theory type theory' recently in a block of text and it looked like a typo!
TIL a new thing,
Two references I found
https://www.carmin.tv/en/video/towards-synthetic-locale-theory
https://topos.institute/blog/2025-07-13-liberating-synthetic-quasi-coherence-from-forcing/

2026-01-13 23:31:38 UTC

Interval arithmetic implemented in Lean, to get verified numerical computation:
https://github.com/alerad/leanbound

2026-01-13 21:47:37 UTC

There's a multi-site drama going on involving BSky and meta.MO right now. I'm not going to add to it here, and don't wish to spread the muck, but anyone who is seeing it please be aware the board and the mods are aware and discussing.

2026-01-13 21:45:58 UTC

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@nprofile…hh7n Oh, cool, thanks! I recognised some editor names, so my originally drafted comment was much softened to the above.
@nprofile…sqfw

2026-01-13 11:29:35 UTC

A great report of an experiment by Asaf Karagila with LLMs (not just ChatGPT), asking them about his own (public) work. Spoiler: results are very patchy at absolute best
https://karagila.org/2026/chatgpt/

2026-01-13 07:58:16 UTC

I don't think this old question is quite resolved
https://mathoverflow.net/questions/281660/stacks-algebraic-over-a-given-stack

2026-01-13 07:08:15 UTC

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@nprofile…sqfw The publisher seems a little niche (speaking generously) but the paper seems serious.

2026-01-13 04:39:44 UTC

The following article claims that the logic in Kant's "Critique of Pure Reason" is _geometric logic_, and not some incomplete, too-weak system as had apparently been the consensus view.
Achourioti, T., & van Lambalgen, M. (2017). "Kant's logic revisited". The IfCoLog Journal of Logics and their Applications, 4(4), 845-865.
https://pure.uva.nl/ws/files/26494036/Kants_logic_revisited.pdf

2026-01-13 04:38:56 UTC

The following article claims that the logic in Kant's "Critique of Pure Reason" is _geometric logic_, and not some incomplete, too-weak system as had apparently been the consensus view.
Achourioti, T., & van Lambalgen, M. (2017). "Kant's logic revisited".
The IfCoLog Journal of
Logics and their Applications,
4(4), 845-865.
http://www.collegepublications.co.uk/ifcolog/?00013
(https://pure.uva.nl/ws/files/26494036/Kants_logic_revisited.pdf)

2026-01-12 23:01:45 UTC

Last night I had a Japanese-themed beef-topped pizza. It had pickled ginger, sesame seeds, Kewpie mayonnaise etc. It was really nice!

2026-01-12 22:59:15 UTC

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@nprofile…7jv6 oh, that's interesting. It perhaps give a psychological release valve to mark something on the page rather than do nothing and feel guilty about it.
@nprofile…t6k2

2026-01-12 22:59:06 UTC

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@nprofile…7jv6 oh, that's interesting. It perhaps give a psychological release valve to mark something on the page rather than do nothing and feel guilty about i.
@nprofile…t6k2

2026-01-12 11:28:36 UTC

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@nprofile…lfhd The slides link on your site is broken! https://webusers.imj-prg.fr/~antoine.chambert-loir/exposes/20260111-cpp2026

2026-01-12 10:01:25 UTC

Very nice looking proof!

2026-01-11 06:20:06 UTC

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@nprofile…j36t Fair enough.

2026-01-11 06:12:17 UTC

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@nprofile…g0zc And of course: https://www.speedrun.com/upc people are still setting world-record times

2026-01-11 06:08:52 UTC

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@nprofile…g0zc The numbers of pages on the Wiki are small now, but just you wait!

2026-01-11 06:05:03 UTC

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@nprofile…j36t Do they not have an option of giving your talk remotely? Or even just a prerecorded talk? In today's world this seems an absolute no-brainer, since people might not travel for all kinds of reasons: parenting, caring responsibilities, environmental concerns, visas being denied on 'curious' grounds etc.

2026-01-10 23:15:29 UTC

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Here's the original paper: https://arxiv.org/abs/dg-ga/9407015

2026-01-10 08:28:36 UTC

https://doi.org/10.1112/jlms.70374
Not every day someone's paper gets a review 30 years later by a leading figure in the same (top-notch) journal. Michael invented the main definition in his now-classic paper while waiting for Brylinski's then-recent book on gerbes to be shipped to Australia, based on the tiny blurb in a publisher's ad for it.