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

2025-04-24 00:37:54

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Thanks 🙂

2025-04-23 22:16:26

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can you link to the actual book? Or are you avoiding links to a certain publisher?

2025-04-10 00:36:02

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Certainly if you choose $\varepsilon \ast \varphi = 1$ the result is less impressive than it first looks.
Also there was much hay made in the video comments about the appearance of both "$\varepsilon < 0$" and "we set $\varepsilon = 4$" in the source document.

2025-03-31 23:52:34

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

2025-03-31 22:26:42

"There are certain emotional
difficulties which are intrinsic to the
mathematical life, and only a few
people are able to live with them all
their lives"
—Donald R. Weidman
I think this letter from 1965 is blind to all kinds of other difficulties people did and do face in becoming a mathematician (and the constant reference to the mathematician 'he' certainly betrays one of them)—but is still worth reading and reflecting on.
https://people.math.wisc.edu/~awmille1/old/perils.pdf

2025-03-31 02:12:49

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"But that's not why there were so many papers about it!"
The thing is, if you knew how to make a prediction different enough from everyone else's prediction, without too much effort, then if you turned out to be right the Nobel committee might well give you a call (and not all those other people making last-minute predictions!). Why not have a flutter and see if you can get something reasonable out of crunching some numbers? I'm not saying it's great for the field, but it's understandable.

2025-02-27 21:20:53

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https://ncatlab.org/nlab/show/Cartan+theorem+B

2025-02-27 20:29:42

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Quillen's Theorem A and B, and Cartan's Theorem B come to mind, though they aren't numbers

2025-02-24 02:06:50

Apropos @npub1xzw…l6gs 's video about Feynman not doing the writing process for books with his name on, it seems he probably didn't proof-read the text much either, in at least one case. The book is about 350 pages long, but the sample errors pointed out in this short paper are pretty bad.
Also, letting Feynman off being sloppy because he was "so smart"?? Give me a break. Private notes are one thing, but publishing a book with hundreds of very subtle errors from the typesetters doing their best with Hibbs doing his best with what he could glean from Feynman is pretty poor. It's not like the book was published by some cheap outfit or some overseas outsourced lowest-bidder contractor, like these days.
Did Hibbs proofread it, and not know that there were mistakes? Why did it take so long to produce this book and did somehow by the end they both didn't care and just wanted it off their plate?
https://www2.oberlin.edu/physics/dstyer/FeynmanHibbs/ErrorsOfFeynmanHibbsX.pdf
https://media.mathstodon.xyz/media_attachments/files/114/056/380/744/614/948/original/7d7230f5a5e339ce.png
https://media.mathstodon.xyz/media_attachments/files/114/056/380/764/856/811/original/5de93d1207519e42.png

2025-02-23 01:15:06

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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

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

2025-02-18 12:07:52

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what did you edit for this version?

2025-02-14 09:08:13

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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

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 21:48:17

"Arguably one of the gifts of being a beginner is being able to build your own personal mental model of how things hang together; and this subjective perspective likely plays a role in achieving breakthrough results.
...
One mustn’t just learn mathematics. One must build it from the ground up for herself. "
—commenter 'Deniz' on Peter Woit's blog.

2025-02-11 06:19:11

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This was sent to a pair of comic artists, can you believe it?

2025-02-11 06:07:36

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 12:21:08

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yes, I myself very much view isogenies as the morphisms between elliptic curves (and I guess more generally between abelian varieties, but I know even less in that case). I believe the point is that one gets a 'dual' isogeny [1, Theorem 1] in the other direction, so _existence_ of an isogeny is (I think) an equivalence relation. But really one should consider the actual maps...
[1] https://www.johannesloher.com/assets/files/the_dual_isogeny.pdf

2025-02-09 06:06:55

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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

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

2025-02-08 02:23:07

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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

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

2025-02-05 00:37:54

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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

2025-02-04 21:17:02

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Actually, I only care about "finite type over Z", so that rules out the example.

2025-02-04 21:12:34

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OK, I'm glad this is not completely trivial!

2025-02-04 02:24:25

Asked a question on MathOverflow ... and found it had already been asked 15 years ago. Means I can delete my question and I have my answer instantly.
At least, what I thought was the main part of my question. There's a small follow-up that now is really too small for MO, so I will ask the wisdom of the (small?) algebraic geometry crowd here: what's the definition of a finite-type algebraic space, if any? And given one, I'm hoping that the set of R-valued points, where R is a finite product of finite fields, is finite.
What say you?

2025-01-24 03:23:27

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yes, I guess it might just be opaque to mathematicians to be specific about the topic. Knowing that each Phys Rev has a specific topic might be enough.

2025-01-24 01:28:01

Igor Pak, "How Do Physicists Publish Papers? The Case of Physical Review E", Notices of the AMS Feb 2025
https://doi.org/10.1090/noti3069

2025-01-22 01:38:24

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The thing I'd be interested in, if I were to have time, is to compare the Lovelace-De Morgan correspondence
https://www.claymath.org/online-resources/ada-lovelaces-mathematical-papers/correspondence-with-de-morgan/
to how the books treat things, and also how these compare to the work of Lovelace's previous tutor, William Frend, who was so conservative he rejected negative solutions to equations in his 1796 book "Principles of Algebra"!
Imagine if Lovelace had started her education with someone more in touch with contemporary mathematics....

2025-01-21 23:31:55

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Aha, they are:
Lubbe, Samuel Ferdinand. "Traité De Calcul Différentiel et De Calcul Intégral", 1832. https://archive.org/details/traitdecalculdi00kartgoog
Boucharlat, Jean Louis. "Élémens De Calcul Différentiel et De Calcul Intégral", Fifth Edition, 1838. https://archive.org/details/bub_gb_QBsCHcT29asC/mode/2up
with the full list of books here: https://www.manhattanrarebooks.com/pages/books/2912/ada-lovelace-owned-by/du-systeme-social-et-des-lois-qui-les-regissent-elemens-de-calcul-differentiel-et-de-calcul

2025-01-21 23:27:15

You can see a small collection of books from Ada Lovelace's personal library in this video:
https://youtu.be/iCm5TPjmuKg?feature=shared&t=518
two books on calculus: "Traité de calcul différentiel" and "Elemens de calcul différentiel", but a bunch of other science books. I do wonder who wrote those calculus books, since unlike some of the other books, the author isn't on the spine!

2025-01-04 08:59:27

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I'm the same as you in that respect, but I was pleasantly surprised when I factorised 2025 in my head in an idle moment and found it was $3^4\cdot 5^2$

2024-12-22 01:20:21

Last night a friend of ours had a fantasy-themed fancy-dress birthday party. I went as the loremaster elf Pengolodh, but in a low-key way: conference badge and a copy of my manuscript
https://media.mathstodon.xyz/media_attachments/files/113/693/832/691/272/722/original/9d7bc4edd9844fbe.jpg

2024-12-19 00:08:00

Is Mochizuki implicitly comparing his IUT work to the pyramids of Ancient Egypt?? The following is missing all the usual emphatic typography, but it is a single sentence in Mochizuki's "Essential logical structure" paper, which I now find is a *hundred pages longer* than the version from 2021 I have saved on my computer:
"This approach of developmental reconstruction may be applied, for instance, to the task of evaluating the level of mathematical or scientific development of ancient civilizations, i.e., not via the direct study of detailed theoretical expositions [which are typically not readily available — cf. the discussion of §1.5!] of the mathematics or science understood by such an ancient civilization, but rather by observing what may be understood as the “fruits” of this mathematics or science, e.g., in the form of architectural achievements such as the famous
· pyramids of Egypt or
· Nazca lines and mysterious ruins of Puma Punku in South America."
This bit wasn't in the version I had saved and by now had put out of my mind because I can't read this prose any more that I already had done for reasons of trying to understand what on earth was going on.
I'm not going to link to the source, it's easily findable.

2024-12-18 11:53:47

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yes, that is very interesting

2024-12-18 04:04:32

This result of this paper seems to be slightly remarkable: Bott periodicity for GL(n) in the algebraic world, *over Z*.
https://arxiv.org/abs/2411.09122

2024-12-15 04:04:08

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particularly: "Wir betrachten zunächst zusammenhängende offene Teilmengen der komplexen Ebene, deren Komplemente in $\overline{\mathbb{C}}$ genau zwei Zusammenhangskomponenten haben. Solche Mengen heissen zweifach zusammenhängend."
@npub1nf4…nqe4 @npub1g40…vy7q

2024-12-15 04:00:50

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ok, see for example section 5.7 of https://people.math.ethz.ch/~salamon/PREPRINTS/cxana.pdf for what I think is a similar usage.

2024-12-15 03:45:51

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what's the reference?

2024-12-12 04:12:55

There are two irrational numbers $x$ and $y$ such that $x^y$ is rational.
Proof:
broke:
Either $\sqrt{2}^\sqrt{2}$ is rational, or if not, then $(\sqrt{2}^\sqrt{2})^\sqrt{2} =2$ is rational, but we don't know which case holds, hence one of $x=y=\sqrt{2}$ or $x=\sqrt{2}^\sqrt{2}$ and $y=\sqrt{2}$ gives what we want.
woke:
By the really hard Gelfond–Schneider theorem we know $x=\sqrt{2}^\sqrt{2}$ is irrational, and $y=\sqrt{2}$ is irrational by an elementary proof, and $x^y=2$, so these are the numbers we want.
bespoke:
By elementary proofs we know that both $x=\sqrt{2}$ and $y=\log_23$ are irrational, and $x^{2y}=3$.
#mtbos

2024-12-11 02:07:25

This is cool. Another categorical characterisation of the category of Hilbert spaces, this time in a way that doesn't assume the existence of a monoidal structure on the category, which means that it also is able to capture quaternionic Hilbert spaces. The real and complex Hilbert spaces are then the cases when there is a(n appropriate) monoidal structure.
Stephen Lack, Shay Tobin, "A characterisation for the category of Hilbert spaces"
Abstract: The categories of real and of complex Hilbert spaces with bounded linear maps have received purely categorical characterisations by Chris Heunen and Andre Kornell. These characterisations are achieved through Solèr's theorem, a result which shows that certain orthomodularity conditions on a Hermitian space over an involutive division ring result in a Hilbert space with the division ring being either the reals, complexes or quaternions. The characterisation by Heunen and Kornell makes use of a monoidal structure, which in turn excludes the category of quaternionic Hilbert spaces. We provide an alternative characterisation without the assumption of monoidal structure on the category. This new approach not only gives a new characterisation of the categories of real and of complex Hilbert spaces, but also the category of quaternionic Hilbert spaces.
https://arxiv.org/abs/2412.03776

2024-12-03 19:30:49

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As far as I can tell using GTranslate it's also French for 'The tribe'

2024-12-03 11:13:00

Is it just me, or did the Bourbaki Archives get a whole lot more issues of La Tribu?
I'm reading through and Grothendieck is in Bourbaki in the mid-60s pushing category theory for inclusion as chapter 5 of Theory of Sets.
Narrator: Theory of Sets was never published with more than four chapters.
cc @npub1j2f…937k

2024-12-02 03:02:58

Oh, wow. Just dropped: a 199-page preprint that is claiming a solution of the moving sofa problem!
https://arxiv.org/abs/2411.19826
cc @npub1nf4…nqe4

2024-12-01 11:46:46

Let me make good on my promise/threat to make a #Tolkien post or two.
I'm reading my way through volumes 6-9 of History of Middle-earth, the part that deals with the writing of The Lord of the Rings (covering 1937-49 or so). I'm just up to the part when Galadriel and Celeborn are invented, at the moment the company is travelling through Lothlórien. It's deeply ironic how much trouble Tolkien later gets into trying to figure out their backstory (see: The History of Galadriel and Celeborn in Unfinished Tales) when he realises these (and in particular Galadriel) are not just random characters our heroes happened to stop over with before breaking up and going their separate ways.
There was a random note that wondered out loud, what if there is no Lord X there with Galadriel? Maybe make her Elrond's wife? As Arwen is not invented until late in the writing process (hence her relatively non-existent role outside the Appendices), there was no Celebrían existing for Elrond to be already married to—we only had, more or less, Hobbit Elrond, who had recently gained the distinction of having been Gil-galad's herald a few chapters before, but not yet a wife. Only a fleeting thought, but kinda funny how Celeborn turned out to be much less consequential than Galadriel, even in the 1973 notes where he was so heroic as to ... make a boat to sail from Valinor to Middle-earth, for Galadriel to level-up, having learned all she could from the Valar?

2024-11-28 05:45:03

I don't post about politics, but this feels like a good step that transcends party politics, that I hope others take notice of:
"SA parliament has passed "world-leading" legislation banning most political donations.
"This reform is not in my government's immediate political interests. But it is the right thing to do," said Premier Peter Malinauskas.
The changes mean politicians, candidates and political parties can no longer accept financial donations, and caps party spending on election campaigns.
Third party groups — like unions and industry bodies — will also be limited to spending no more than $450,000 on a campaign."
https://www.abc.net.au/news/2024-11-28/south-australia-government-passes-reforms-on-political-donations/104658140
#auspol

2024-11-26 06:24:17

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I want to write the paper first!

2024-11-26 01:18:48

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I don't think so. If you can find the earliest instance of "canonical line bundle" that you can, and let me know, I'd be interested. I was tracing a particular usage/meaning backwards through collaborations, citations, etc, not just looking at dates. And I have a fairly good idea of the rational of how the usage shifted/was adopted to the new setting from the old, though there are multiple threads and I'm not pulling on them all...

2024-11-25 23:58:09

I may have discovered another link in the literature between the modern use of the word 'canonical' to describe certain privileged (iso)morphisms and the old use (that is, for a matrix/bilinear form).
There is a slight element of story-telling to the path from late 19th century usage through to what emerged at the end of the 30s, but then I'm not a professional historian.
But also, I can see from the citation chain, someone says the [data] of "the type just described will be called canonical", and he cites a paper to justify this data, and that paper uses the term "normal form" (it's specifically what we would call 'Smith normal form'). I then have a citation chain to the guy who introduced the term "canonical" to Bourbaki (attested in their drafts and by personal communication) 11 years later.
Now I just have to write the paper...

2024-11-11 06:41:42

From the categories mailing list
"To celebrate the 10th anniversary of the death of the great mathematician Alexander Grothendieck, occurred on 13 November 2014, the Grothendieck Institute organises a commemorative conference in his honour.
The event will be held online on Saturday the 16th of November 2024, from 2 to 6 p.m. CET"
https://igrothendieck.org/en/homage-to-alexander-grothendieck/

2024-11-09 23:28:13

On my to-read list is Turing's paper written as an undergraduate where he independently came up with a proof of the Central Limit Theorem, because he was not satisfied with the vague treatment he had been shown in lectures. He didn't know Lindeberg had already proved the result properly about a decade earlier.
https://turingarchive.kings.cam.ac.uk/unpublished-manuscripts-and-drafts-amtc/amt-c-28
You can read a 1995 paper in the American Mathematical Monthly discussing this, here: https://www.jstor.org/stable/2974762

2024-11-07 07:37:17

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My paper about bi-invariant 2-forms on infinite-dimensional Lie groups was the proof that the claim was never true.

2024-11-07 07:03:32

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The definition in those four lines (of a putative "action" of a group on a set) relied on a claim ("Recall that...") a bit further down that was just outright false. That's where the flippancy was.

2024-11-07 05:31:15

So far in this project I have used:
infinite-dimensional Lie groupoids
diffeological vector spaces as classifying spaces
bicategorical methods
formal anafunctors
differential form calculations
Hodge decomposition as a topological direct sum of TVSes
crossed modules
model categories
All to replace four flippant lines.

2024-11-06 07:41:15

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oh, not even there. I just need a temporary terminology for a lemma down in the guts of the paper. I don't think I'd refer to having a 2-fibration even in the intro.

2024-11-06 06:33:43

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I agree I'd like to ultimately say just "2-fibration", because at the end I want to compute the homotopy fibre using the strict fibre, and thereby show the homotopy fibres are 1-types not 2-types

2024-11-06 06:32:42

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It's just that the actual proof/calculation uses the co+op-version, because showing it the other way is much harder. And also one has to do work to show I've got bigroupoids. A priori it's a 2-functor with this property, and I can show separately they are bigroupoids.
So I could say it's a 2-functor between bicategories such that blah blah and oh by the way, they are (genuinely weak) bigroupoids.

2024-11-06 05:55:24

What's better for my 2-functor between bigroupoids, where I can lift 2-arrows with specified lifted target, and lift 1-arrows with specified lifted target. Additionally it is surjective on objects and the strict fibres are 1-groupoids.
Or comment if you think that '2-fibration' is sufficient, because everything is invertible, at least up to isomorphism.

2024-10-28 00:56:36

Michael Harris will be teaching a course "Mathematics and the Humanities: Mathematics and Philosophy" the content of which looks very interesting:
https://www.math.columbia.edu/~harris/website/courses1/mathematics-gu4200-spring-2025
Even just looking at the class readings shows up stuff that would be fun to read through on one's own.

2024-10-23 00:37:55

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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 00:32:22

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Yes, that's a bit annoying.
Other people's (rather later) responses are here:
Scott Aaronson "Death of proof greatly exaggerated" https://scottaaronson.blog/?p=4133
Krantz, S.G. (2011). John Horgan and “The Death of Proof?”. In: The Proof is in the Pudding. Springer, New York, NY. https://doi.org/10.1007/978-0-387-48744-1_12
Steven Krantz article in particular highlights Horgan thinking not like a mathematician, and I think he would have done well to talk to people that know about history and philosophy of mathematical practice, rather than extrapolating outside his realm of expertise.

2024-10-23 00:02:19

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

2024-10-22 01:10:41

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I said in my response that I decline to do free work for Elsevier, though i didn't make any points about the publisher except for pointing out the Cost of Knowledge pledge. For the sake of the author, I mentioned that the paper is a good one, but one cannot put a recommendation in a rejection to review 🙂

2024-10-21 23:59:03

Just had to decline a request to do a referee report for the Elsevier-published journal Journal of Pure and Applied Algebra, which is de facto the oldest category-theory journal (even if never explicitly labelled as such).
Not just http://thecostofknowledge.com/ but also because I don't have time these days, and I like to write extremely detailed reports.

2024-10-13 05:49:10

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the issue is that sometimes Mastodon populates a little preview that includes a page title, and sometimes it doesn't. And it doesn't happen immediately, so I sometimes try to check after a short delay, and update with some info if the link is wholly cryptic.
@npub1eng…zqfn

2024-10-11 02:59:46

Oh, this looks fun "A Categorical Approach to Forcing?"
https://www.saattrupdan.com/posts/2018-09-20-a-categorical-approach-to-forcing
"As a finishing comment we could ask if we gain anything by attempting to construct the “true” forcing category; i.e. a category consisting of cbas in which isomorphism in this category corresponds exactly to forcing equivalence. Since forcing equivalence is not the same as strong forcing equivalence there will be more isomorphisms in this category. But how can we construct such a category? One such way could be to somehow use Solovay’s characterisation of forcing equivalence to extract the weaker notion of a “forcing equivalence morphism” from this and then localise the category by these maps."
I'll have to come back to this to digest it...

2024-10-04 02:56:54

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It's trivial(isable) for n=1, and I'm pretty sure for 3, but I can't double check that right now. This is because what you've written down is the oriented frame bundle of the sphere, and the 1- and 3-spheres, being Lie groups, have trivialisable frame bundles.
For n=2, the LHS has trivial pi_2, using the L.E.S. of Z/2 -> SU(2)=S^3 -> SO(3) (double covering group), and the RHS has a Z summand, from the S^2.

2024-09-23 07:29:11

"Generative AI has polluted the data. I don’t think anyone has reliable information about post-2021 language usage by humans."
https://pivot-to-ai.com/2024/09/19/word-frequency-tool-wordfreq-stops-updates-overwhelmed-by-ai-spam/

2024-09-18 02:36:21

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How much does one's weight drop up at peak flying altitude?

2024-09-13 02:56:07

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Not exactly an easy thing to search for! Not crucial, it was only for amusement purposes.

2024-09-13 00:41:28

I have a vague memory that @npub1nf4…nqe4 once made a joke about rgs, which are, like rigs and rngs, are like rings with less assumptions about their structure, namely no (multiplicative) identity *or* negatives. I can't remember where...