theHigherGeometer

rimcræftiga |
bespoke constructions in categorified geometry since 2010 |
dude

Blog: https://thehighergeometer.wordpress.com/
Coffee fund: https://buymeacoffee.com/highergeometer
Location: Adelaide, Australia

Public Key

npub1d60hhaertjrlmhefe54s00hfyga38wangrqthcu3zdnnnfm3myase5mhwm

NIP-05 Address

[email protected]

Profile Code

nprofile1qqsxa8mm7u34eplamu5u62c8hm5jywcnhwe5ps9muwg3xeee5acajwcpz3mhxue69uhhyetvv9ujuerpd46hxtnfduqs6amnwvaz7tmwdaejumr0dspqzlvc

Publishing to

relay.damus.io

nos.lol

Author Public Key

npub1d60hhaertjrlmhefe54s00hfyga38wangrqthcu3zdnnnfm3myase5mhwm

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Last Notes

2026-02-06 04:18:39 GMT

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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 GMT

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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 10:01:29 GMT

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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 06:14:51 GMT

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.

2026-01-22 00:11:50 GMT

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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 03:26:21 GMT

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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 GMT

"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 06:54:46 GMT

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Good thing I'm keeping myself under that limit already....

2026-01-20 05:37:13 GMT

"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)

2025-12-16 00:01:23 GMT

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There plenty more instances of **phrase** in the article. There's also really badly made TikZ figures that I don't think any author would be happy to leave as-is.

2025-12-15 21:52:44 GMT

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ok, the paper has obvious signs of LLM-generation that the author hasn't proof-read and fixed. There's Markdown syntax mixed into the TeX in the image.... on MO this would be a fast close.
https://media.mathstodon.xyz/media_attachments/files/115/725/791/615/015/010/original/4e40df22ce7d3123.jpg

2025-12-15 21:39:36 GMT

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I wrote originally "if you mean Huawei, then note that they also fund research into topos theory and do things like this: https://www.ihes.fr/en/laurent-lafforgue-en/"
But it's not. That's not to make any judgement call on the paper (or the company) either way.

2025-11-28 04:21:09 GMT

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"one-hot" (sic) encoded, on top of using \(\aleph(i)\) for a set of nearest neighbours. Whyyyyyy. And that model. Wow. Very statistics.
The author being from https://www.college-cn.com/Anhui/1428/ doesn't fill me with confidence
cc [@infotainment.bsky.social](https://bsky.brid.gy/r/https://bsky.app/profile/infotainment.bsky.social )
https://media.mathstodon.xyz/media_attachments/files/115/625/373/708/714/817/original/8a32321b08948ac3.png
https://media.mathstodon.xyz/media_attachments/files/115/625/383/452/874/531/original/94a9ec272c5cfaf2.png

2025-11-28 00:42:50 GMT

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I agree about the super-general and -abstract bit. The key point is that one needs a) compactness of the base (so really some kind of finiteness condition) and b) the ample invertible sheaf, which is really about admitting an open immersion to some scheme of the form Proj(S).

2025-11-27 21:17:12 GMT

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The Brauer group can be defined in two ways: Azumaya algebra objects up to iso, where you say they are equivalent if there is a suitable bimodule, and Azumaya objects up to iso, where you mod out by the submonoid (under \(\otimes\) of things of the form \(End(E)\). It is the latter approach that this definition takes seriously.
I seem to recall that to go from the bimodule definition to the one I'm using you can take one of the vector bundles to be the underlying vector bundle of A, using the property \(End(A) \simeq A\otimes A^{opp}\) (and vice versa for \(B\), or else the underlying vector bundle of the bimodule. This was all 10+ years ago, but I know it works out.

2025-11-27 20:29:59 GMT

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https://mathstodon.xyz/@highergeometer/115618036587381377 has the definition in the picture with two slides. This easily implies the bimodule definition (there's a left A and right B action on A \otimes End(E)...) The other direction is less obvious. In any case, even if the definitions imply each other, the data is different.

2025-11-27 13:14:04 GMT

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"It actually looks the same to me: you talk about Morita equivalences of Azumaya bundles, but those should be the bimodules that are invertible (up to isomorphism)."
Aha, but I'm not using bimodules, and my 1-morphisms are not invertible even up to iso.
And the theorem about etale H^2 and Br is only about certain schemes: eg separated quasicompact schemes with ample line bundle

2025-11-27 11:53:09 GMT

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Cool, I looked it up, and some of it sounded familiar. It was this thread, right: https://categorytheory.zulipchat.com/#narrow/channel/266967-deprecated.3A-mathematics/topic/Azumaya.20algebras/near/368619176 ?
And for anyone watching, the pause-point for the n-category café posts was here, in part 6 : https://golem.ph.utexas.edu/category/2023/10/grothendieckgaloisbrauer_theor_5.html
I think I'm being much more low-brow, and taking a lot of algebraic ingredients for granted. As noted I'm deliberately using a *different* categorification of the Brauer group to a symm. mon. (2,1)-category, rather than the usual one with bimodules at 1-arrows. It's for no other reason but that I know how to adapt Gabber's proof using this 2-category, and not the bimodule one!
I can imagine that the construction I give generalises a lot, for instance, as you say, to the real and the super vector bundle setting. And even more generally, I'm sure. It felt to me like some kind of double comma construction at the time, and I wrote it as \(\mathbb{E} \downarrow \mathbb{E}\), which was on my office board for a while.
I'd be happy to talk more about this or your ideas.

2025-11-27 01:03:10 GMT

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A vector bundle where the fibres are Azymaya algebras. For complex vector bundles these are all kinda boring fibres (since they are all matrix algebras), but as a bundle they are nontrivial.

2025-11-26 23:26:34 GMT

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I should add the bits in little font in the theorem were conjectural as I hadn't sat down and proved those facts. This was where I stalled, but also my postdoc at the time finished and other still-ongoing projects were prioritised.

2025-11-26 21:07:12 GMT

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that was my original idea, about a decade ago, but sadly this project fell by the wayside. It very much feels like a double cat!
https://media.mathstodon.xyz/media_attachments/files/115/618/021/749/213/549/original/ebf8fc29779ff107.jpg
https://media.mathstodon.xyz/media_attachments/files/115/618/027/870/086/617/original/a2355cf553444605.jpg

2025-11-26 13:13:54 GMT

I had a thought that perhaps I should reframe my unpublished, descent-theoretic proof of the Grothendieck–Serre theorem (on the Brauer group of a compact 'nice' space) using symmetric monoidal \((\infty,2)\)-categories or something.

2025-11-26 02:08:55 GMT

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Does the department you are in have an admin phone number? If forced you could add that.

2025-11-24 00:43:48 GMT

Riemann Hypothesis proof claimers: here's my proof bro, all the mathematicians didn't see it because it's simple and they are blinded by BiG mAtH. It only took 5 pages!!
Actual analytic number theorist who proved the weak Goldbach conjecture: here's an explicit estimate that uses zeroes of the zeta function...
https://media.mathstodon.xyz/media_attachments/files/115/601/887/390/133/278/original/21b4c9bd16d00538.png

2025-11-20 13:36:07 GMT

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I think the result near the bottom here must be the result that Paseka–Vetterlein are using, though the proof is in the mystery 2003 paper (listed as [8] in the bibliography)...
Let me try cutting and pasting:
"Каждый оператор x ∈ B(H) представляется в виде конечной суммы x = ∑ x_k, где каждое x_k есть произведение не более чем двух проекторов при dim H = ∞ и не более чем трех проекторов при 2 ≤ dim H < ∞."
which machine translates to:
"Each operator x ∈ B(H) is represented as a finite sum x = ∑ x_k, where each x_k is a product of at most two projectors for dim H = ∞ and at most three projectors for 2 ≤ dim H < ∞"
Well, there you go. But we can't see the proof!
https://media.mathstodon.xyz/media_attachments/files/115/582/269/248/393/461/original/66e05ee037e0ba10.png

2025-11-20 13:28:26 GMT

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Hmmm. That 2005 paper has this citation:
Бикчентаев А. М. О представлении линейных операторов в гильбертовом пространстве в виде конечных сумм произведений проекторов // Докл. РАН.. 2003. Т. 393, № 4. С. 444–447.
https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=dan&paperid=1659&option_lang=eng
which *is* the citation in the Paseka–Vetterlein paper. It's listed as being in MathSciNet, but not @npub1w8x…d8qh
Annoyingly, you can't get a pdf!
@npub1nf4…nqe4

2025-11-20 13:10:37 GMT

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The closest paper I can find is this
A. M. Bikchentaev, “On representation of elements of a Von Neumann algebra in the form of finite sums of products of projections”, Sibirsk. Mat. Zh., 46:1 (2005), 32–45 mathnet mathscinet zmath; Siberian Math. J., 46:1 (2005), 24–34
https://www.mathnet.ru/eng/person18192
If any database would list this paper, it would be in math-net.ru
😠

2025-11-20 12:38:42 GMT

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HMMMMMMM
I'm not sure this paper exists.....
A. M. Bikchentaev, "On the representation of linear operators in a Hilbert space as finite sums of products of projections" (in Russian), Dokl. Ross. Akad. Nauk 393 (2003), 444–447.
Consulting @npub1w8x…d8qh I can't see a paper in that journal in that volume with those page ranges...

2025-11-20 12:30:54 GMT

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Yes, and they need to cite a non-obvious paper for that part!
A. M. Bikchentaev, On the representation of linear operators in a Hilbert space as finite sums of products of projections (in Russian), Dokl. Ross. Akad. Nauk 393 (2003), 444–447.
(this is not in zbMath with a few quick searches, and Google Scholar isn't helping. MathSciNet might have it, but I'm busy and not in the mood to look)

2025-11-20 09:55:12 GMT

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Oh, and also this one!
C. Heunen, A. Kornell, N. van der Schaaf, Axioms for the category of Hilbert spaces and linear contractions, Bulletin of the London Mathematical Society 56 (2024) https://arxiv.org/abs/2211.02688

2025-11-20 09:52:28 GMT

RE: https://mastoxiv.page/@arXiv_mathCT_bot/115581087904495544
A different setup and conclusion compared to:
C. Heunen, A. Kornell, Axioms for the category of Hilbert spaces, Proc.
Nat. Acad. Sciences 119 (2022) https://arxiv.org/abs/2109.07418
M. Di Meglio, C. Heunen, Dagger Categories and the Complex Numbers: Axioms for the Category of Finite-Dimensional Hilbert Spaces and Linear Contractions, Applied Categorical Structures 33 (2025) https://arxiv.org/abs/2401.06584
and
S. Lack, S. Tobin, A characterisation for the category of Hilbert spaces,
Appl. Categor. Struct. 33 (2025) https://arxiv.org/abs/2412.03776
#note1emm…ch76

2025-11-16 23:34:08 GMT

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I can't speak for @npub1nf4…nqe4 but here's a talk by a mathematical physicist that I know: https://researchers.ms.unimelb.edu.au/~dridout%40unimelb/seminars/120917.pdf

2025-11-12 11:18:07 GMT

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Oh, it was definitely pronounced as "th" in Old English, the writing at the time using ð and þ shows that. The -s ending was a Northern dialect thing (from the Old Norse influence), and it was eventually adopted further south. Shakespeare mixed the use of -s and -eth, so matters were definitely shifting strongly by that point at the latest. I saw one person claim in a discussion that by the late 1600s the -s/-es ending had mostly won out, except for certain common words like hath and doth.

2025-11-12 09:04:03 GMT

"There is an important passage in Richard Hodges’ Special Help to Orthography (1643) that deserves quoting in full, along with an extract from his homophone lists:
(a) . . . wee use to write thus, leadeth it, maketh it . . . &c Yet in our ordinary speech . . . we say leads it, makes it . . . Therefore, whensoever eth, cometh in the end of any word, wee may pronounce it sometimes as s and sometimes like z, as in . . . bolteth it and boldeth it, which are commonly pronoun’ct, as if they were written thus, bolts it, bolds it . . .
(b) cox, coks, cocketh; clause, claweth, claws; courses, courseth, corpses; fleas, fleaeth, flayeth; Mr Knox, he knocketh, many knocks; reasons, reasoneth, raisins"
I'd love to track down a scan of this 17thC book, when I'm at my computer...

2025-11-04 09:53:21 GMT

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probably not, which why I'm less confident. Maybe if I say something like the integral cohomology is torsion-free, maybe we get a projective Z[G]-module or whatnot.

2025-11-04 07:25:05 GMT

Hmm. I have a compact simple Lie group G, and some homogeneous space X for it. I need to know the G-invariants in H^k(X,Z). If one is calculating real cohomology then I could believe that because one can calculate de Rham cohomology using invariant forms (in this case), then the whole cohomology group is G-invariant. For integral cohomology it doesn't seem ludicrous, but I'd have to think about it.
Also I think I saw a claim that properties such as this group has imply taking G-invariants is an exact functor. I guess this follows from Schur's lemma...

2025-10-24 11:46:10 GMT

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Oh, good to know

2025-10-24 07:10:30 GMT

Hey @npub1qrh…usg2 this link gives me an internal server error: https://arxiv.org/abs/alg-geom/9306001 (is a link a search engine gave me) but https://arxiv.org/abs/alg-geom/9306001v1 works.

2025-10-23 22:54:06 GMT

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Thanks!
@npub1whr…qc2t

2025-10-23 22:53:14 GMT

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Indeed:
"Is this LLM-generated? "
"I'm not sure if this is real, but the abstract says machine-verified."
"It seems a pretty serious attempt though- it’s not just some random crank paper."
"The fact that they have formalized the proof should mean it will be quicker to verify whether or not this is indeed it."
@npub1nf4…nqe4

2025-10-23 21:47:37 GMT

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Oh dear, some people there don't seem to realise just how trash this is.

2025-10-23 20:16:29 GMT

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if it was on the actual site it's gone now, as far as I can tell.

2025-10-22 23:50:05 GMT

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These days, my mind is a bit split when I see the letter æ because for Old English it's the short a like in cat.
So my brain is suggesting at least three vowel sounds for the ash: short a, something like in 'day', and then the long e (as in the old encyclopædia).
@npub1whr…qc2t

2025-10-22 09:43:50 GMT

An image caption on a BBC article (about Philip Pullman and his comments on AI ingestion of authors' works):
"Dafne Keen, with her daemon Pan, played Lyra in the BBC's His Dark Materials series, which ran from 2019 to 2022"
I guess they do not really mean that Dafne Keen has a real-life daemon and together they played the character Lyra 🤣

2025-10-22 04:21:45 GMT

Oh dear. AI-generated Lean code and a document saturated with bullet-point lists
https://arxiv.org/abs/2510.17829
Don't bother. Cross-listed to math.CT, but should be in math.GM.

2025-10-20 23:02:48 GMT

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Oh, I can appreciate that, but in a context discussing physics and electromagnetism, I was being carefully pedantic to not be conflating the mathematical object with the physical concept!

2025-10-20 22:29:16 GMT

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The query on the entry to the right of "source" is tantalising. I'm sure Urs would have words to say for that lacuna! At the very least, it should be something like a de Rham current, I imagine.
@npub10mu…2h9c @npub18a6…szuj @npub1zz2…9mhx

2025-10-20 22:23:55 GMT

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It's notation from the original Wu–Yang paper. Amusingly, it seems to only be in this table, in the rest of the paper they write only \(U_1\) to match \(SU_2\). I suspect it might have been a conflation of the notations \(U_1\) and \(U(1)\)!
https://media.mathstodon.xyz/media_attachments/files/115/408/819/568/263/640/original/6f2a6e61e88a47e6.png

2025-10-20 00:10:35 GMT

In honour of the passing of Chen-Ning (Frank) Yang, here is the "dictionary" that he published with Tai Tsun Wu, that is an absolute foundation stone for the area of mathematical physics that I work adjacent to:
https://en.wikipedia.org/wiki/Wu%E2%80%93Yang_dictionary

2025-10-15 05:58:44 GMT

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Seems like a @npub12cu…d38p question

2025-10-13 23:21:21 GMT

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Since it seems not to be true, what implication does it have for the particle physics ideas?

2025-10-09 09:34:52 GMT

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90s anime fans will probably immediately think of this https://nz.pinterest.com/pin/336081190962419072/

2025-10-08 02:15:04 GMT

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From the point of view of planar ternary rings, I can understand the maps of shape x |-> ax+b, but I'm not sure where the division comes from. Hmm. But checking the Wikipedia page for PTR, multiplication is a loop, so you have (left and) right inverses, so there is an element 1/(cx+d). OK.
The nLab page says a projective line is called Desarguesian if you have even just a restricted version of that 3-transitivity on points, whereby you have a group that fixes two points (possibly the coinciding) and can move any point distinct from those (that) to any other. The grammar of the definition there is not clear if it means this has to be true for one case, or always. And you definitely get weirder examples than this. I wonder if OP^1 is a 'Desarguesian line'?

2025-10-07 21:09:14 GMT

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Regarding the projective plane/incidence geometry stuff, the condition analogous to Pappian and Desarguesian that gets you a Moufang plane (https://en.wikipedia.org/wiki/Moufang_plane, OP^2 is the prototypical example, I believe) is "little-Desarguesian". This holds iff the planar ternary ring is an "alternative division ring" (of which the octonions are an example).
Axiomatic/synthetic projective *lines* are less easy, and you get more exotic examples
https://ncatlab.org/nlab/show/projective+line#synthetic_projective_lines
analogous to how you get non-Desarguesian planes, but no non-"Desarguesian" projective n-space for n≥3.
I think Buekenhout's definition of projective line is instructive to consider for lines isomorphic to OP^1.
@npub1zz2…9mhx

2025-10-04 06:38:01 GMT

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I have no expectations! The argument that one shouldn't expect such a bundle from the screenshot seems to me insufficient: it just points out the elements of OP^2 aren't literally O-lines, so no obvious analogue of the canonical bundle on say CP^2 falls out immediately. But this isn't a proof that no nontrivial O-line bundle exists on OP^2!
The obvious thing to try to start is classify rank‐4 complex vector bundles on OP^2, which surely is possible, say by obstruction theory using the known cell-structure of the Cayley plane.
@npub1nf4…nqe4

2025-10-03 23:42:06 GMT

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on the nLab Urs pointed out a source (https://ncatlab.org/nlab/files/DrayManogue15-12-5.pdf#page=3) that says the literal elements of OP^2 are not octonionic lines, but with the recent discussion you had about octonionic things and Jordan algebras one might wonder if there is a better way to think about this.
Alternatively, one could ask if there is a map OP^2 -> BGL(2,H) that is not null-homotopic, hence classifying a nontrivial rank-2 quaternionic vector bundle, and if this has any structure that makes it "look like" an octonionic line bundle.
Of course, GL(1,O) doesn't really exist, so one needs to be creative....
@npub1zz2…9mhx
https://media.mathstodon.xyz/media_attachments/files/115/312/653/761/399/020/original/8ef091da66d1a6ba.png

2025-10-03 12:47:03 GMT

Is there a rank-8 real vector bundle (or rank 2 quaternionic vector bundle) on the octonionic projective plane, with any additional structure that makes it "octonionic"?
Cc @npub1nf4…nqe4 @npub1zz2…9mhx

2025-09-28 10:11:00 GMT

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My main difficulties are that the mindset of the area is very different to mine; the techniques are a mix of custom-tailored examples; sometimes hard measure theory, sometimes immediately restricting to considering special cases like separable or étale groupoids, or something technical about the interior of the bundle of automorphism groups, when the theorems available are more general, because the final intended result only works for that one case.
There's not a really principled category theoretic treatment, outside big theorems where it's still only implicit (like the C*-algebra really depending on the underlying topological stack, if one considers a bicategory of C*-algebras and Morita equivalences)

2025-09-28 10:04:53 GMT

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The term "reduced" is not about reducing the original C*-algebra, as you surmise, but it's because you do a completion of the more naive algebra of functions on the groupoid with a different norm to the one that gives the "full" C*-algebra of the groupoid.
I believe that the concept of Cartan subalgebra of a C*-algebra is what gives you a a locally compact topological groupoid, and such that the algebra is the full C*-algebra of that groupoid. And if you have a groupoid C*-algebra, then it has a Cartan subalgebra.
Such a subalgebra is a maximal abelian sub-C*-algebra with a retraction, satisfying properties. It's definitely possible to have many different Cartan subalgebras, including ones that give rise to very different groupoids.
It's not known how to tell if a given random C*-algebra has a Cartan subalgebra. I believe the known examples that fail to have a Cartan subalgebra are reduced C*-algebras of non-amenable loc.comp. topological groups.

2025-09-28 07:16:36 GMT

I had a random thought: given that at least some C*-algebras arise as that attached to a locally compact groupoid (which determines a topological stack), and the reduced C*-algebra is a quotient of that algebra, does that mean that it corresponds to something like a substack? And for those reduced C*-algebras that don't have a Cartan subalgebra (a necessary and sufficient condition to be a full C*-algebra of some loc.cpt groupoid), does this mean they are coming from something like a non-topological substack?

2025-09-27 12:55:25 GMT

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Yes. And I found an answer by Derek Holt on math.SE, who is a senior finite group theorist, who also says ATLAS introduced the notation, so I think I can take it as a fact.

2025-09-27 12:52:24 GMT

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I was under the impression this was called "ATLAS notation" in the literature. This author of MO question claims it is the source of the notation: https://mathoverflow.net/q/489747/
"The ATLAS of finite groups and representations thereof introduced the dot notation 𝐺=𝐴.𝐵 if 𝐴 can be embedded in 𝐺 such that 𝐺/𝐴=𝐵. Clearly 𝐴.𝐵 can mean many different non-isomorphic groups."
[@g_merzon](https://mathstodon.xyz/@g_merzon )

2025-09-27 10:13:42 GMT

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well, to me the ATLAS notation that uses . and ⋅ and : for what look like binary constructions on groups, but which are highly underdetermined is even weirder (using such notation for an arbitrary, unspecified normal extension of groups??). But that's a different Conway conversation.
@npub1nf4…nqe4

2025-09-27 10:07:57 GMT

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Looking just at the Conway–Smith proof of Theorem 14 there, I would have assumed it was a way of implicitly bracketing the product (to get around non-associativity, and needing to use lots of parentheses), and the snippet from the Russian edition seems to confirm that.

2025-09-24 08:15:21 GMT

https://travisdoesmath.github.io/s6/
funky visualisation of the nontrivial element in Out(S_6)

2025-09-24 02:47:59 GMT

@npub1255…9rve
https://web-cdn.bsky.app/profile/davidkbutler.bsky.social/post/3lyimmjpoxk2b
My honest answer:
It always made sense, from the moment I first saw it with @npub1nf4…nqe4 explaining it in a blog post. It was the coolest thing I had seen, to the point I physically printed out part of the webpage with the definition of category and functor (this was in 2001) and had it on hand to excitedly read it out loud to my friends while waiting outside the lecture theatre for Maths I to start.
What I didn't do, however (to the best I recall!), is get all smug about it and ask lecturers or tutors if such-and-such was a functor, as a way of showing off.
I realise I am atypical in these respects and I don't expect others to have the same experience.

2025-09-22 08:06:34 GMT

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Peter Scholze imagined them, and he and Clausen can apparently give the first Chern class of the underlying (perfectoid) vector bundle for these Kähler differentials, in the so-called twistor-\(\mathbb{P}^1\) formalism that is analogous to working on the Fargues–Fontaine curve in the non-Archimedean setting, but they don't have a formalism in which to construct the actual object.

2025-09-22 06:09:37 GMT

"So, Dustin, are these Kähler differentials on \(\mathbb{Q}\) over \(\mathbb{F}_1\) in the room with us now?"
🙃
https://media.mathstodon.xyz/media_attachments/files/115/246/453/160/114/125/original/867d6a9894bb0178.png

2025-09-21 10:54:58 GMT

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Strongly compact cardinals are listed here: https://neugierde.github.io/cantors-attic/Upper_attic and the page on them has lots of information, for people who can read serious set theory but haven't looked at them before.
Spoiler: they are waaaay above assuming there is a proper class of inaccessible cardinals (the Grothendieck–Tarski axiom of universes).

2025-09-21 10:48:06 GMT

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If there were genuine higher-geometric content—in the style I study—lurking in the standard model I would be ecstatic. In the meantime, I guess BRST quantisation (which secretly, or not, uses \(L_\infty\)-algebras) might count as close enough to being useful for SM physics?
@npub1zz2…9mhx

2025-09-21 08:17:36 GMT

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Given a map of semisimple Lie groups \(H \to G\), you can consider the \(\mathbb{Z}\)-linear map
\[ H^4(BG,\mathbb{Z}) \to H^4(BH,\mathbb{Z})\]
and that is what you define to be the index.
In the case that they are both simple and simply-connected, you get an injective map \(\mathbb{Z}\to \mathbb{Z}\). For \(G\) simple and \(H\) a semisimple subgroup (again simply-connected), you get a list of positive integers, indexed by the simple summands of the Lie algebra \(\mathfrak{h}\).
Because \(H^4(BU(1),\mathbb{Z})\simeq \mathbb{Z}\) you still get something meaningful even with \(U(1)\)-factors in \(H\).
That the subgroup of interest is quotient of \(SU(3)\times SU(2)\times U(1)\) (with \(H^4 \simeq \mathbb{Z}^3\) by \(\mathbb{Z}/6\) means there may well be some nonsense involving extra factors.

2025-09-20 09:46:53 GMT

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Thanks for keeping me in the loop!
@npub1zz2…9mhx might be able to guess a question that interests me off the bat: what are the Dynkin indices for the (Lie algebras of the) stabliser subgroups for the various F_4 homogeneous spaces in this story? Particularly, what is the index of su(3) + su(2)+ u(1) < f_f? (this is no longer a positive integer, since the subalgebra is not simple....and it's not obvious what "index" should be when there are u(1) summands. But from a Lie group perspective, you can think about the restriction of the 2-group extension String_G to a U(1) < G, and then see what 2-group extension of U(1) (what Ganter calls a "categorical circle") you get)
@npub10tf…rt7p

2025-09-15 09:25:15 GMT

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ah, ok, thanks for interpreting.
@npub10tf…rt7p @npub1zz2…9mhx

2025-09-15 09:14:24 GMT

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I can only guess the so(9) bivectors is somehow the tensor square of some so(9) representation, or perhaps the tensor square of so(9). I don't know what that works out to be...
I think it would help me if the answer were less telegraphic and didn't assume standard physics abuses of notation/terminology in representation theory. Also, writing not with unicode will help, because I suspect some symbols have been mangled.
@npub10tf…rt7p @npub1zz2…9mhx

2025-09-12 00:18:55 GMT

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Ah, was it this talk?
https://mathstodon.xyz/@johncarlosbaez/115174390161940101

2025-09-11 22:01:34 GMT

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you may well be asleep, but did you have slides online for this talk?
@npub1zz2…9mhx

2025-09-11 09:34:32 GMT

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it's in the paper you originally linked!

2025-09-11 08:59:04 GMT

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three. The other one doesn't contain a copy of SU(3)

2025-09-11 08:43:42 GMT

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Aha, looking at the table on page 219 I see they've done the trick of adding an extra node to the Dynkin diagram, and so the augmented so(9) = B_4 diagram fits inside the augmented F_4 diagram

2025-09-11 08:41:30 GMT

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BTW, here's the Borel–Siebenthal paper that is referenced for the classification of the maximal rank maximal subgroups: https://www.digizeitschriften.de/download/pdf/358147735_0023/log15.pdf

2025-09-11 08:38:37 GMT

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Seeing that SU(3) (and having a peek at (4.1) in the paper) makes me want to know what is the intersection of (SU(3)xSU(3))/Z_3 and G_2 (modulo of course the issues with which copies etc).
Presumably for the main problem one could do something like match up the embeddings of the copies of SU(2) corresponding to coroots in the Lie algebras. But looking at the Dynkin diagrams, it's not so clear how to do this mindlessly, putting so(2x4+1) inside f_4, unlike su(3)+su(3): I can imagine the two copies of su(3) Dynkin diagram corresponding to the two "arms" of the f_4 Dynkin diagram. This latter embedding (assuming it works) feels like a "canonical" way to get the (SU(3)xSU(3))/Z_3 subgroup in F_4.

2025-09-07 09:04:59 GMT

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I'll send you the pdf of the agreement so you can see all of it

2025-09-07 08:48:43 GMT

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I think the ACS editors should be made aware of the publishing agreement (maybe the guest editors of the volume) and asked if they are happy with that state of affairs. It would be nice if ACS declared independence from Springer over this and started afresh, say as a Centre Mersenne publication for instance.

2025-09-07 01:39:39 GMT

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If I'd known about the terms of this publishing agreement before I submitted I almost certainly wouldn't have.

2025-09-07 01:30:23 GMT

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I got a link to it when I received page proofs. I wanted to sort out those first before looking at the contract.

2025-09-05 08:58:11 GMT

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But I think you would know the names of some of the holders: Horace Lamb (famous for work on hydrodynamics), William Henry Bragg (Nobel Prize for x-ray crystollography), Ren Potts (mathematical physics), and Mathai Varghese, who I know you knew personally. He has just retired, btw.

2025-09-05 07:58:52 GMT

I learned today that the Elder Professorship of Mathematics (named for Sir Thomas Elder) was likely one of the two first professorships in The University of Adelaide. It was originally Mathematics and Natural Philosophy, before being split into two positions, one for Mathematics and one for Physics, after the first incumbent stepped down. The other foundation professorship was the Elder Professor of Natural Sciences (discontinued after one holder).
The Elder Professors of Music, Anatomy (and variant namings), Chemistry (now defunct) were all later.

2025-09-05 05:08:43 GMT

Springer now demand that your pre-acceptance version of a paper on the arXiv comes with a disclaimer:
“This preprint has not undergone peer review (when applicable) or any post-submission improvements or corrections. The Version of Record of this article is published in [insert journal title], and is available online at https://doi.org/[insert DOI]”."
and forbids you from putting the version accepted on the arXiv under an OA license:
"The rights granted to the Author with respect to the Accepted Manuscript are subject to the conditions that (i) the Accepted Manuscript is not enhanced or substantially reformatted by the Author or any third party, and (ii) the Author includes on the Accepted Manuscript an acknowledgement in the following form, together with a link to the published version on the publisher’s website: “This version of the article has been accepted for publication, after peer review (when applicable) but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/[insert DOI]. Use of this Accepted Version is subject to the publisher’s Accepted Manuscript terms of use https://www.springernature.com/gp/open-research/policies/accepted-manuscript-terms”. Under no circumstances may an Accepted Manuscript be shared or distributed under a Creative Commons or other form of open access licence."
So Springer is saying it has control over how people use papers on the arXiv: "Use of this Accepted Version is subject to the publisher’s Accepted Manuscript terms of use."
I knew none of this when I submitted my paper, nor when I uploaded my updated paper to the arXiv.

2025-08-30 10:09:59 GMT

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close:
"A surface is called \(S\) bielliptic if \(S \cong (E \times F) /G\), where \(E,F\) are elliptic curves and \(G\) is a finite group of translations of \(E\) acting on \(F\) such that \(F /G \cong \mathbb{P}^1\)."
https://mathoverflow.net/questions/85267/bielliptic-surfaces
@npub1j2r…lq0t

2025-08-30 09:02:44 GMT

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Just in case, flagging @npub1nf4…nqe4 who might have seen one (recall his blog 'visual insight' that had cool pictures of certain varieties at times)

2025-07-31 06:18:54 GMT

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Click through for the video of it being held by a human...

2025-07-31 06:16:47 GMT

https://www.abc.net.au/news/2025-07-31/giant-stick-insect-acrophylla-alta-wet-tropics-discovery/105596666
#straya #FNQ

2025-07-31 03:03:02 GMT

MathOverflow is now future-proofed against Google's immanent shuttering of the legacy goo.gl link-shortening service.
As far as is possible, all such links have been replaced by working links, or even the broken link targets, except in a very few cases where the target link was so long as to make insertion into the containing comment source Markdown impossible. User 'The Amplitwist' was diligent and noted all the working/broken/Wayback Machine-archived goo.gl targets in follow-up comments, and then I went and edited the old comments and cleaned up.

2025-07-28 08:50:26 GMT

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ah that clears that up, I didn't even think people would assume the tensor product was over the complex numbers.

2025-07-28 07:34:15 GMT

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ok, I'm being perhaps less explicit than I should be. ℂⁿ here (in fact ℂⁿ*, but I have an inner product an happy to identify things with the dual in a slightly sloppy way) is a *real* 2n-dim representation of SU(n). So the tensor product ℂⁿ⊗ℂⁿ is actually a tensor product over the reals! I'm working with real manifolds that is a homogeneous spaces for real Lie groups.... but everyone automatically jumps to complex things (or, because it's the interesting case, the split real form or something).
And ℂⁿ⊕ℝ is meant to be the tangent space of 𝑆²ⁿ⁺¹ ....
@npub1zz2…9mhx

2025-07-28 06:00:00 GMT

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Aha, this looks promising (from Fulton–Harris):
"More precisely, an irreducible representation of a compact group/Lie algebra will be real (resp. quaternionic) if and only if it has an invariant nondegenerate symmetric (resp. skew-symmetric) bilinear form. In other words, the classification ofan irreducible V is determined by whether
\[V\otimes V=\mathrm{Sym}^2 V\oplus \bigwedge^2 V\]
contains the trivial representation, and, if so, in which factor."
It's the very last section of the book before the appendices! And doesn't give much time explaining things...

2025-07-28 05:47:30 GMT

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Huh. Everyone refers to McKay and Patera, that seems like a scary book. I was thinking something like Fulton and Harris would be a reasonable place to at least see some basic standard results. They seem to basically talk about complex stuff, though, and the point is that I don't know this enough to be able to extract the compact real stuff!
However, I'm not sure that the real representation Λ²ℂⁿ is actually an irrep for SU(n). If this were true, the Schur tells is there are no nonzero SU(n)-invariant bilinear alternating forms on ℂⁿ, no? But \(\mathrm{Im}(v^\dagger w)\) is one such, if I haven't gotten my calculation completely wrong.
I've been thinking Λ²ℝⁿ is an irrep of SO(n), though, no?

2025-07-28 01:02:27 GMT

I spent time on the weekend thinking about calculating the adjoint to the Chevalley–Eilenberg differential on Ad-invariant k-forms on the reductive complement to an index 1 sub-Lie-algebra, in specific standard examples like those arising from homogeneous spheres. It's the first time I've really had to worry about representation theory! In particular, I don't know a reference that says how the alternating square of the defining representation of SU(n) splits into a sum of irreps (I suspect from various unproven statements it's a 1dim trivial rep plus other nontrivial stuff). I'll similarly need the analogous thing for Sp(n). What's a good reference?

2025-07-25 06:57:51 GMT

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SO(n+1)/SO(n) and Spin(n+1)/Spin(n). I was wondering if there were non-round homogeneous structures on S^8 coming from its structure as OP^1.

2025-07-25 06:21:16 GMT

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Both are round spheres though, right?