theHigherGeometer

rimcræftiga |
bespoke constructions in categorified geometry since 2010 |
dude

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

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highergeometer_at_mathstodon.xyz@momostr.pink

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Last Notes

2025-09-21 10:54:58 UTC

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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).

2025-09-11 22:01:34 UTC

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

2025-09-07 01:30:23 UTC

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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.

2025-07-25 00:00:03 UTC

I'm sure I've seen a description of $\mathbb{OP}^1$ as $G/H$ for suitable Lie groups, probably exceptional. But I can't find it. Was I dreaming?

2025-07-24 10:22:47 UTC

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well, we've slowly built the theory of manifolds from the literal ground up this year, so that we have proved O(n) is a manifold from things like the inverse function theorem, or constructed charts on what was essentially the Stiefel manifold (non-square matrices of a suitable rank), so that the quotient down to the Grasmannian gives you a manifold down there. And have seen how the tangent spaces of classical groups are certain subspaces of all matrices etc. We did the example of the irrational slope lines inside a torus this week, and I really want to get the closed subgroup theorem at least for matrix groups. It rules out things like the universal cover of SL_2(R), but that's ok.

2025-07-24 03:23:45 UTC

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Yes, I've done the matrix exponential, and I have found notes that work through what I need (the closed subgroup theorem) in the matrix group case.

2025-05-06 07:36:39 UTC

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Looking at the references in the paper, it cites an 1855 self-published book ("Joseph B Mott solves a general polynomial equation using undetermined coefficients in his remarkable self-published book [20]") as being close to what they do
"...Mott [20,21] and Lettl [33] come closest to our results"
Here's the "remarkable" book https://archive.org/details/mathematicalkey00mottgoog/page/n1/mode/1up
It reads like a crank who doesn't believe in Galois theory or calculus, but then uses power series methods.
"As it respects the general root theorem, I claim both the discovery and invention of it as my own. If any other person has made a similar discovery, it must be a late occurrence, and is wholly unknown to me. Davies and other late authors say expressly that no direct formula has appeared for the solution of equations of a degree higher than the fourth, except when the roots are part or all of them rational ; but it will be seen in the following pages that equations of all degrees, even many cases of infinity, have a direct solution by one general rule, that is less tedious when applied to numerical equations than any rule of approximation, as well as more perfect."
and
"I should prefer a resort to algebra rather than to the differential and integral calculus ; though it is true I have not had time to examine all the mysterious developments that are reported in the higher mathematics, so as to see whether they can be referred to algebraical methods or not. Nevertheless I have not a doubt but that the blind methods of reasoning (so at least they appear to many minds) introduced in the calculus might be wholly dispensed with."
However, the power series solutions for really simple quadratics (like x^2+x-1=0) can be divergent!
https://media.mathstodon.xyz/media_attachments/files/114/459/699/300/097/779/original/038892c4ca98be47.png

2025-05-06 03:45:50 UTC

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This is my main gripe about his (ultra)finitist/constructivist claims: I have seen no evidence (and I'm not going to sit through hours of YouTube videos, I've just seen their titles, and read a little of his writing) that he has engaged with a century's worth of constructivist/intuitionist thinking. Doron Zeilberger might not believe in arbitrarily large integers, but he solves actual problems that are interesting with software, and proves results that for him are "formal symbol manipulation"—power series results—but which for everyone else are just nice facts about generating functions or whatnot.

2025-05-04 10:42:58 UTC

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I was also annoyed by this, but the paper, in AMM (and open access), is thankfully not that bad. No bad takes on ultrafinitism, though it does take a particular angle that I would be fine reading from Doron Zeilberger.
https://doi.org/10.1080/00029890.2025.2460966
You can be sure the AMM is not going to publish nonsense like "irrational numbers don't exist". It's more like "here's this combinatorial construction that gives a power series that generalises things like Eistenstein's power series for the Bring radical to arbitrary degrees".

2025-03-31 23:52:34 UTC

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I also thought the 'if you get a null result at least you got something' extremely naive.

2025-02-23 01:15:06 UTC

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ok, I wondered if it did (I didn't get a chance to look at the paper) because that's the big result in finite groups that gives you an exhaustive case breakdown.

2025-02-22 22:31:22 UTC

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Link to actual paper: https://arxiv.org/abs/2410.20392

2025-02-14 09:08:13 UTC

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in a blog post, titled "The theologico-teleological apology", which I'm sure people can find if they want. Though he didn't use the word 'privilege', rather the word 'charisma', which he seems to use to denote a combination of privilege and social capital.

2025-02-14 03:58:02 UTC

Michael Harris once claimed I had mathematical privilege, when I pointed out in a review that his writings come across as *incredibly* privileged (being invited to govt soirées in Paris, Ivy league background and position etc). I of course have only an adjunct academic position and have never lived outside Australia. My work is pretty obscure in a niche topic and I trouble envisioning my papers ending up in top journals as a result (I don't do mainstream differential geometry, or number theory, or PDEs or whatever is the thing that carries the most weight in the big-name general journals).
However, somehow, I have acquired privilege, despite all that. It struck me that I could email a number of leading figures in the mathematics community who through accidents of time know me, and expect a decent answer. I also have a relatively large platform in mathematical social media, I guess merely out of sheer persistence, but not because I aimed to.
I'm grateful for this privilege and the responsibility it gives me is something that only sometimes comes to mind. I know there are many people out there that do not have what I have, in this regard, and I shouldn't forget it.
So yes, I have a form of privilege. I cannot envy Michael Harris, it doesn't get me anywhere. The joy I get from doing mathematics and talking about it doesn't depend on me being an Ivy League professor.

2025-02-11 06:07:36 UTC

This was not sent to me, but I would be pretty annoyed if it were (but with 'content' -> mathematics).
"I’m writing to you on behalf of Muse. Muse is a new creative platform that can create your own AI-generated series so you can dive into a new world of storytelling without the need for personal content creation."
😠
"For example, with Muse you can: make your own series with your script/storyline bring your book characters to life promote your new project/book by creating a mini-series help create content by creating other worlds make your story come alive for your audience by creating a series about it. Please let me know if you would like to discuss collaborating and becoming one of the first creators to gain an early advantage in the realm of content creation."

2025-02-09 06:06:55 UTC

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Did the idea of considering what isogenies do in relation to the Hasse–Weil species [1] go anywhere?
[1] https://golem.ph.utexas.edu/category/2024/04/the_modularity_theorem_as_a_bi.html#c063103

2025-02-08 06:10:32 UTC

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I just did, so you don't have to.

2025-02-08 02:23:07 UTC

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What would be cool is if Quanta linked to your article from their article. They are not averse to linking to research papers, so maybe you could drop a hint that you have indeed more to say on the matter, and a large number of readers would like to know more—and can actually appreciate a technical exposition.

2025-02-05 01:33:30 UTC

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cool, thanks!
@npub1nf4…nqe4 phew, from me too.

2025-02-05 00:37:54 UTC

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Hmm. So here's where I'm confused: one can define the arithmetic zeta function of a finite type scheme over Z, and allegedly such schemes have the property I mentioned: for each R that is a finite product of finite fields, there are only finitely many R-points.
In the case this is not always true, the latter claim needs another adjective. Something like quasi-compact, then? Or something weaker is ok?
cc @npub1nf4…nqe4

2024-11-26 06:24:17 UTC

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I want to write the paper first!

2024-10-23 00:37:55 UTC

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is "online people" a euphemism for Peter Woit and Sabine Hossenfelder?
String theory is perfectly good mathematics, when done well, and as mathematical physics provides insights into certain types of QFTs. But the "string theory will be tested at the LHC" hype that, when SUSY is not found, this is somehow irrelevant....?
My mathematical work in geometry is largely inspired by string theory, and I feel no shade whatever from Woit et al, because I don't claim I'm doing physics.
Also, I truly think that the work of Sati and Schreiber trying to put M-theory on a solid theoretical foundation is 100% worth doing, even aside from physics considerations.
@npub1nf4…nqe4

2024-10-23 00:02:19 UTC

https://johnhorgan.org/cross-check/the-horgan-surface-and-the-death-of-proof
I think I'll have to go read Horgan's 'The Death of Proof' now:
https://www.jstor.org/stable/24941653