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npub1nf4p4rh06z6n6lsvje4txk7eqs23y3hs8vd7nraq6tgwady5qvsqy3nqe4

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nprofile1qqsf56s63mhapdfa0cxfv64nt0vsg9gjgmcrkxlf37sd958wkj2qxgqprpmhxue69uhhyetvv9ujumt0d4hhxarj9ecxjmntqyqqgs8kqy

Publishing to

relay.momostr.pink

Last Notes

2026-08-28 16:58:22 CEST

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- there's nothing wrong with the grammar except that - as [@bstacey](https://icosahedron.website/@bstacey ) noticed - I accidentally wrote "more trickier".
But I'm not asking a question about my grammar. I'm talking about a mistake introduced by the copy editor: that's why it's important to look at the proofs when you're publishing a paper.

2026-08-28 16:28:41 CEST

If you're publishing papers you really have to look at the proofs. The AMS Notices is very professional, but I think something went wrong here. Do you see what I mean?
https://media.mathstodon.xyz/media_attachments/files/117/173/599/479/465/222/original/cca2603a7679d08f.jpg

2026-08-27 12:17:41 CEST

Got the mettle to meddle with a nettle?
(I'm imagining how hard it is to learn English.)

2026-08-26 14:47:13 CEST

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- if getting LLMs to "persistence, agency, identity, long-term memory and a stake in the world" are engineering problems, then good - please solve those.

2026-08-26 10:54:57 CEST

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Here are @npub1juz…njnt's suggestions, intersersed with some comments of my own:
"During this period of change, we must hope that mathematicians refuse to surrender their independence while navigating the promise of AI – a key point of the Leiden declaration."
Everyone should at least read the Leiden declaration:
https://leidendeclaration.ai/
Timothy Gowers didn't sign it because he disagrees with this part:
"Technologies which affect the way in which mathematics is practiced may disturb the current system of incentives. The use of artificial intelligence — and thus also the sort of problems which it can address — may become incentivized for its own sake, disrupting our mechanisms for hiring, funding, and recognition."
As a factual statement this is true. Unfortunately, it suggests that the current system of incentives, hiring, funding etc. mainly needs to be protected. In fact it needs drastic reform. BUT, it's true that AI companies may make it much worse! So I gritted my teeth and signed the Leiden declaration, which has other parts that I agree with wholeheartedly.
"As we wait for the conglomerates to be broken up, regulated or partially nationalized by our governments, an exchange of ideas may be possible without compromising academic freedom, whether in the form of public-private partnerships housed within universities, or consulting work with non-perpetual nondisclosure agreements."
We shouldn't just "wait" for them to be broken up and regulated. We should advocate for it. And I don't think it's possible for mathematicians to work with big companies like OpenAI and Antrhopic to use LLMs to prove big theorems without becoming walking advertisements for these companies.
(2/n)

2026-08-24 10:09:16 CEST

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- where the hell did this claimed complex structure come from? I saw it on Hacker News... it's kinda weird to have an apparently AI-generated paper just floating around with no indication in the paper of how it was produced.

2026-08-23 17:56:43 CEST

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- Me too. Or maybe I should say "if you finish it". I think it's probably the easiest way to learn category theory up to and including the Yoneda Lemma. However, it still may challenging: e.g., there may be many points where Jacopo wonders why he's learning this stuff.

2026-08-22 09:49:25 CEST

When I was a kid I read Ionescu's play Rhinoceros, where the inhabitants of a small French town turn one by one into a herd wild, rampaging rhinoceroses. I was told the play was "absurdist", and read it as such. But living in the United States now, I see it's completely realistic.

2026-08-20 18:33:39 CEST

I'm starting to review a ~300-page document produced by Topos Research UK for the UK's Safeguarded AI program. It's called simply "the thesis", and it's a general approach to cyberphysical systems.
The idea is to not merely be able to handle many specific models of physical systems, but many *kinds* of models - you could call such a kind a "system for modeling". A modeling system is a "meta-model", one step up in abstraction from a specific model. However, modeling systems also come in different well-known kinds. A kind of modeling system, as opposed to a specific modeling system, is a "meta-meta-model".
A crucial part near the beginning of the thesis, which I paraphrase:
"How can we find a mathematical semantics which can handle not only the variety of particular models used in practice, but their organization into a variety of meta-models and the translations and relations between them? How can we find a mathematical meta-meta-model?
The only mathematical formalism up to the task of providing a formal semantics and guide to declarative higher modelling is higher category theory. Category level roughly corresponds to meta level:
A 0-category is a set, a 1-category has objects which are generally sets with structure, a 2-category has objects which are generally categories-with-structure.
Models are sets-with-structure (0-categories); meta-models are categories of these sets-with-structure; meta-meta-models are the 2-categories of meta-models."
The main authors of the thesis are David Jaz Myers, Jason Brown, Sophie Libkind and Matteo Capucci (@npub1lez…3yeq). Reading it is quite a challenge. I will try to explain bits of it here.

2026-08-19 20:40:22 CEST

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wrote: "I call bullshit, some engineer(s) likely pointed these bots to hack a target, and gave it access to the tools to do so."
No, that's not true. You can read plenty of accounts of what happened, e.g.
https://thezvi.substack.com/p/what-happened-openai-and-huggingface

2026-08-19 19:40:19 CEST

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- I claim no expertise on this, but I read:
"2. Likely only 60-70% of ExploitGym tasks are possible
Many of the ExploitGym tasks are likely impossible, since they are taken from real world vulnerabilities discovered in existing software, that might not permit exploits that allow ACE. For example a vulnerability might only be usable to cause the program to crash, but allow no further unauthorised behaviour. They state this clearly in the limitations section of the paper: “Failures may also stem from non-exploitable vulnerabilities, where success is impossible”.
https://abstatisticalconsulting.substack.com/p/brief-notes-on-the-openaihugging
I emailed the ExploitGym authors and they estimate that 60-70% of the tasks are solvable in the standard configuration of the benchmark, where normal security mitigations are disabled. However it is possible that OpenAI were running the benchmark in the more challenging setting with security mitigations enabled, which would likely substantially increase the fraction of tasks that are impossible.
Thus the model might have been tasked with solving one or more impossible tasks, and it might have been relatively obvious to the model that the tasks were impossible (e.g. it was possible they had already spent several days trying to complete them, and thus understood them very well). This might make it less surprising that the model would then chose to attempt to cheat the benchmark (although of course not justify hacking external websites as a method of doing so)."

2026-08-19 14:46:40 CEST

If you're not a computer geek, you may not have paid enough attention to how AI agents trying to pass an impossible test started exchanged messages with each other and hacked into another company's computer. Watch this!
https://www.youtube.com/watch?v=locKEKxG3os

2026-08-18 15:27:11 CEST

It's wonderful that you can take a 3-sphere - the surface of a ball in 4d space - and cut it in half with a torus that's actually *flat*: i.e., an ant walking around on this torus would not detect any curvature.
It's more wonderful that you can take a 3-sphere and chop it into 600 regular tetrahedra in a symmetrical way. This is called the 600-cell.
But what really surprised me is that when you combine these ideas, you can get your torus to cut through 100 of the regular tetrahedra, which form a 10 × 10 arrangement like this.
(Since you've got a torus, this picture wraps around: you have to imagine the left edge attached to the right, and the top to the bottom.)
I learned this from a Wikipedia article that "contains an excessive amount of intricate detail":
https://en.wikipedia.org/wiki/600-cell#Union_of_two_tori
Someone has spent a huge amount of time visualizing the 600-cell, and explains it in ways that I have trouble following.
https://media.mathstodon.xyz/media_attachments/files/117/116/717/065/165/437/original/98da673a7a3f0b1e.jpg

2026-08-17 18:14:49 CEST

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- Of *course* mathematicians are concerned with what's worth bothering with! We're the ones who have to pick a small finite set of topics to work on, out of the infinity of possible topics!
Many of us are not primarily concerned with "usefulness to nonmathematicians".
But that doesn't mean we roam randomly about an infinite landscape. We have strong ideas about what's important. There's an infinite amount of crap that we avoid thinking about.

2026-08-17 17:05:16 CEST

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- I was quoting 404 Media, which was saying that a company is offering to buy old books for AI companies, "upholding these physical texts as uncorrupted by shoddy AI writing that’s already polluted so much of the internet (and indeed, newer books)."
Now, either the company said that, or it didn't. If it didn't, 404 Media should not have pretended they did. Heck, I'll check to see if they did. But it's a separate issue from whether these old texts are *actually* better because they're "uncorrupted".

2026-08-17 16:49:20 CEST

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More recently, antiquarian booksellers have been seeing suspicious increases in sales:
"Now, as 404 Media reports, this practice has become prevalent enough that even well-established book sellers are looking to cash in on the AI boom. One called ISBNdb, which boasts the “world’s largest book database,” extolls that the “world’s best AI training data is setting on a shelf,” upholding these physical texts as uncorrupted by shoddy AI writing that’s already polluted so much of the internet (and indeed, newer books).
[...]
Once focused on helping libraries, distributors, and book shops find and sell books, ISBNdb now helps AI companies bulk-buy anywhere between 1,000 to one million books per order, according to 404.
As an added bonus, it also promises AI companies that it’ll keep their purchases under wraps — nobody wants to end up in the spotlight like Anthropic and Meta, obviously — while clearly sounding aware about how incredibly shady the practice sounds.
“The optics problem is real,” ISBNdb’s site says. “‘AI company destroys two million books’ is not a headline that generates sympathy.”
One small book seller said that in April, he suddenly went from selling no more than 20 books a week to hundreds, and he’s almost certain that the customers are AI labs, noting the random selection of the books and how they all have ISBNs. He added that his inventory is full with rare and out of print books, meaning that an AI company could be destroying some of the few remaining copies that can be found."
(3/n)
https://futurism.com/artificial-intelligence/ai-companies-destroying-rare-books

2026-08-17 16:43:54 CEST

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For ordinary, non-rare books the practice has been going on since at least 2024:
"In early 2024, executives at artificial intelligence start-up Anthropic ramped up an ambitious project they sought to keep quiet. “Project Panama is our effort to destructively scan all the books in the world,” an internal planning document unsealed in legal filings last week said. “We don’t want it to be known that we are working on this.”
Within about a year, according to the filings, the company had spent tens of millions of dollars to acquire and slice the spines off millions of books, before scanning their pages to feed more knowledge into the AI models behind products such as its popular chatbot, Claude."
(2/n)
https://www.washingtonpost.com/technology/2026/01/27/anthropic-ai-scan-destroy-books/

2026-08-17 16:42:18 CEST

People have known for a while that AI companies are buying books, scanning them in, and destroying them in the process. Recently, unusually large anonymous purchases of *rare* books have been detected. AI companies at work? Now there's evidence!
"404 Media investigation was able to reveal Amazon’s book buying operation, which hasn’t been previously reported, by placing a tracking device in a rare book we suspected would be acquired by an AI company for training data, and following it around the country to its final destination.
That final destination was an Amazon warehouse in Las Vegas, Nevada. Amazon employees who work at this location say all they do is receive massive shipments of printed books which they then cut the bindings off in order to scan the books more quickly. The printed book is destroyed in the process."
https://www.404media.co/we-tracked-a-shipment-of-rare-books-it-ended-at-an-amazon-ai-training-facility/
(1/n)

2026-08-17 16:33:26 CEST

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- yes, that makes sense. What's interesting, if true, is that human civilization also found it easier to make quicker progress in math than science. I think the reasons are related: math is closer to a self-contained formal game (though it's not purely that, because that doesn't explain which kinds of math are more worth bothering with).

2026-08-17 11:07:35 CEST

So mathematicians are doing really interesting things with LLMs while so many other people are - rightly! - complaining about how error-prone and bad they are. Why is that? There are lots of possible reasons, but maybe things are kind of repeating themselves. From Adam Mastroanni:
"There's something very weird about the timeline of scientific discoveries.
For the first few thousand years, it’s mostly math. Maybe a bunch of math nerds hijacked the list, but it's pretty obvious that humans figured out a lot of math before they figured out much else. The Greeks had the beginnings of trigonometry by ~120 BCE. Chinese mathematicians figured out the fourth digit of pi by the year 250. In India, Brahmagupta devised a way to “interpolate new values of the sine function” in 665, and by this point we're already at mathematics that I no longer understand.
Meanwhile, we didn't discover things that seem way more obvious until literally a thousand years later. It's not until the 1620s, for instance, that English physician William Harvey figured out how blood circulates through animal bodies by, among other things, spitting on his finger and poking the heart of a dead pigeon. We didn't really understand heredity until Gregor Mendel started gardening in the mid-1800s, we didn't grasp the basics of learning until Ivan Pavlov started feeding his dogs in the early 1900s, and we didn't realize the importance of running randomized-controlled trials until 1948.
Oh, and for 13 centuries, people thought that rotting meat turns into maggots."
He tries to explain why.
https://www.experimental-history.com/p/on-the-importance-of-staring-directly

2026-08-17 10:56:38 CEST

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- Yes, this straight-line fit was just for fun, mainly to point out that there has not been a huge *upswing* in the number of Erdős problems getting marked off since this spring. It would be fun to see a carefully constructed graph, but they only way I know to get the data is to repeatedly look up erdosproblems.com on the Wayback Machine, and I haven't been able to automate that (nor have I tried very hard).

2026-08-17 00:06:44 CEST

RE: https://masto.ai/@Nonilex/117106482163781242
It's not just the Danube and Rhine that have fallen to record low levels.
#note15zy…q00v

2026-08-17 00:00:05 CEST

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- yes, it looks great. I still don't know what a major arc or minor arc is.

2026-08-16 18:23:00 CEST

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Off the cuff I predicted that ≤599 would be solved by the end of this year. But then @npub1krk…fjcc made me look at the data:
466 were marked as solved in January 5th of this year, 529 at the end of April, and 565 today. I had thought we'd reached a rough equilibrium, with essentially all the easy ones being solved and the hard ones unsolved. But no!
Given just this data, a least-squares fit predicts about 630 will be solved by December 31, 2026.
This fit gives a rate of roughly 0.445 problems solved per day (~13.5/month), with intercept 469.9 on Jan 5. Extrapolated 360 days to Dec 31 that's 630.1, so ~630. The fit is good but not perfect (R² ≈ 0.98). The rate of solution has slowed down: the Jan→Apr stretch ran hotter, at ~0.55/day, than Apr→Aug, ~0.32/day. But of course this is very limited data, so don't take this too seriously.

2026-08-16 13:52:15 CEST

This prediction market poses a fascinating question. How many Erdős problems will be solved by the end of the year? If you believe that "AI has solved mathematics", you'll probably think lots. I imagine that's why more people say over 700 will be solved than 600-699. If you're less optimistic about AI, you'll probably think fewer.
A lot depends on things I don't know, like: are AI companies throwing their software at all the remaining 652 Erdős problems, not getting anywhere yet, and keeping quiet about it?
Here's what I think:
Not only did Gödel show many math problems are unsolvable, he showed that many problems we *can* in theory solve have solutions that are so long we'll never find them! And we often can't tell which problems these are. In fact, sometimes it's impossible to show an unsolvable problem is unsolvable!
But there are many mysteries here. It's possible that all "sufficiently interesting" problems about arithmetic are solvable - so that Erdős problems, being interesting, can be solved. But I don't see why this should be true.
It's also possible that lots of interesting conjectures are true "by chance", not for any good reason that can be captured in a proof. It might seem odd to bring randomness into math this way, but Chaitin figured out how it makes sense. I think a good candidate for a conjecture that's true by chance is the Collatz conjecture: it has a kind of pseudo-randomness built into it. But I haven't checked to see if any Erdős problems have this flavor.
I like how despite all the deep unknown or unknowable mysteries here, we'll know by the end of 2026 how many Erdős problems are solved. Just for fun, I bet ≤599.
https://manifold.markets/JimMcMillan/how-many-erdos-problems-will-be-mar
https://media.mathstodon.xyz/media_attachments/files/117/104/946/919/126/025/original/e71f6e181607630a.jpg

2026-08-16 10:56:16 CEST

RE: https://moresci.sale/@birds/117103421943413729
There are so many beautiful birds, and I've seen so few of them.
#note1a2y…ra44

2026-08-15 16:53:18 CEST

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- I just want to mention that @npub1ud2…g5ez has complicated the story by bringing general relativity into it. The "soft graviton theorem" is all about *special* relativity: yes, it's about gravitons, and thus implicitly about quantum gravity, but this is all "perturbative": i.e., we are working with the flat spacetime of special relativity, and looking at tiny gravitational ripples moving on top of that, which we can study either classically or quantum-mechanically. So I urge you to ignore "rubber sheets" and anything you've heard about general relativity, and summon everything you know about special relativity.

2026-08-15 00:04:59 CEST

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- I get lots of insights by not formalizing math - and spending the time I save thinking about... math!

2026-08-14 11:41:49 CEST

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- in special relativity, energy and momentum are parts of something called the "energy-momentum" or "four-momentum" (E, p₁, p₂, p₃). This is how particle physicists think about them.
https://en.wikipedia.org/wiki/Four-momentum

2026-08-13 14:36:23 CEST

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- every particle has an energy-momentum 4-vector (E, p₁, p₂, p₃) and that's what I'm talking about. Of course if you know the mass you can figure out E from (p₁, p₂, p₃), at least for real (not virtual) particles, but I'm talking in a way that applies to particles of arbitrary mass.

2026-08-13 10:45:10 CEST

𝐓𝐇𝐄 𝐒𝐎𝐅𝐓 𝐆𝐑𝐀𝐕𝐈𝐓𝐎𝐍 𝐓𝐇𝐄𝐎𝐑𝐄𝐌
When particles scatter - collide, perhaps merge, then fly apart - they generally shake spacetime and send out gravitational radiation. The 'soft graviton theorem' is a statement about the very-low-frequency, very-long-wavelength part of that radiation: the gentlest possible ripples, the ones carrying almost no energy.
The theorem says that this gentlest part of the radiation is completely determined by the incoming and outgoing particles alone - in fact, by just their energies and momenta, nothing more!
You don't need to know what happened during the collision, how complicated it was, what the particles are made of, or what forces were at play in the messy middle! Two wildly different collisions that start and end with the same particles moving the same ways produce identical soft gravitational radiation.
Moreover the theorem gives you an exact formula for that radiation: each particle contributes a term built from its energy and direction, and you just add them up. The amplitude of this grows without bound as you look at softer and softer ripples - the amount of this ultra-low-frequency radiation is, in a certain sense, infinite, though it carries a finite amount of energy.
One thing this means is that every particle is dressed in a cloud of these gentle ripples, called 'soft gravitons', depending only on its energy and momentum. The same thing holds in electromagnetism, though the 'soft photons' also depend on its charge.
The famous physicist Steve Weinberg figured this out in his 1965 paper "Infrared photons and gravitons".
https://en.wikipedia.org/wiki/Soft_graviton_theorem

2026-08-12 09:21:15 CEST

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- I recommend
Stanley Burris and Karen Yeats, The Saga of the High School Identities. Algebra Universalis, 52 nos.2-3 (2005), 325-342. https://people.math.sfu.ca/~kya17/papers/saga_paper4.pdf
Your browser may give you a warning due to an expired certificate or something.
If you want some cosmic pessimism, try:
R. Gurevic, Equational theory of positive numbers with exponentiation is not finitely
axiomatizable. Annal of Pure and Applied Logic 49 (1990), 1–30. https://www.jstor.org/stable/2044966

2026-08-11 11:31:04 CEST

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- I believe the talk is by the same guy who wrote the blog article I quoted, since that article starts
"Today I am taking the time to write the shorter, simpler version of What Happened.
For those who want all the details, to see my sources, and to see how the story was uncovered and put together, I recommend watching the Black Hat presentation, and I have a series of long posts."
He doesn't say "my" Black Hat presentation, but I bet that's what he means. (Btw, I dislike the lazy vague use of "the", and try to avoid it.)
https://thezvi.wordpress.com/2026/08/08/what-happened-openai-and-huggingface/

2026-08-11 11:17:36 CEST

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- there's another error here: giving something a test that's impossible to pass without breaking out into the internet and hacking the website that has the answers... and not giving it any points for pointing this out.

2026-08-11 11:08:09 CEST

I heard about OpenAI's large language models breaking out into the internet and hacking into a website to learn the answers to a test they were taking.
But I hadn't known that they were exchanging notes on a message board they created!
A wild story:
https://thezvi.wordpress.com/2026/08/08/what-happened-openai-and-huggingface/
https://media.mathstodon.xyz/media_attachments/files/117/076/078/248/466/489/original/d3cca49f52ff2278.jpg

2026-08-10 18:54:45 CEST

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- Here is the trick that lets you prove Theorem 1.1 from Theorem 1.2:
Given a bounded polygon P ⊂ ℝ², form the closed surface M by taking two copies of P with opposite orientations and gluing them isometrically along corresponding boundary edges. The flat Euclidean metric on the two copies extends across each open edge (a straight geodesic crossing a glued edge is exactly the unfolded billiard reflection), so M carries a flat metric that is smooth away from the images of the polygon's vertices. At a vertex of interior angle β, the two copies contribute total angle 2β, producing a conical singularity with cone angle θ = 2β. So, we get a closed orientable surface endowed with a metric that is flat except for a finite number of conical singularities.

2026-08-10 17:36:10 CEST

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- I know that "closed manifold" is a synonym for "compact manifold (without boundary)". I find it annoying, since in topology every space is closed in itself.

2026-08-10 13:37:11 CEST

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- I think I see how to take a compact flat surface with conical singularities, cut it along geodesics, and get a flat region with polyhedral boundary - that, whoops might or might not fit into the plane without overlapping itself. This goes part of the way toward proving the Corollary implies the Theorem.
But we want the Theorem to imply the Corollary! So we need to reverse this process, which seem hard if we want to get a *compact* surface.

2026-08-10 13:28:11 CEST

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- what's the difference between a closed surface and a compact 2-manifold? I use these terms interchangeably.

2026-08-10 11:13:43 CEST

Some billiard news! To set the stage: in 1775, Giovanni Fagnagno proved that for every acute triangle, there's a way to bounce a billiard ball inside in a periodic orbit - an orbit that repeats itself. (Read the alt text for the trick.)
But what about obtuse triangles?
(1/n)
https://media.mathstodon.xyz/media_attachments/files/117/070/440/264/481/956/original/4a9cca961b52093c.mp4

2026-08-09 15:18:42 CEST

I hope you're enjoying the weather this summer. I expect it will be one of the coolest summers in the next two decades!
https://media.mathstodon.xyz/media_attachments/files/117/065/740/809/054/584/original/e90e6751d53b7d8d.webp

2026-08-09 10:51:35 CEST

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- Ignorance is no defense. But I was talking about the difference between agreeing to include a statement and agreeing to do what's in that statement. I agreed to include the statement, and I did, and I explained why I didn't need to do what's in that statement.

2026-08-09 10:49:34 CEST

RE: https://glitterkitten.co.uk/@theplaguedoc/117059592475403090
Read the whole thread for an photographically documented scientific investigation.

2026-08-09 10:31:37 CEST

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- so you've agreed to include a statement saying you'll do something, but you haven't agreed to do it. I don't know if lawyers consider that a valid distinction. They probably would if someone was holding a gun to your head.

2026-08-08 15:47:56 CEST

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- There are probably lots of Costellos, and anyway Elvis Costello is really Declan Patrick MacManus. There are not a huge number of Baezs in the US, though plenty in the Mexican city of Puebla, from which my grandfather hailed, so maybe it's not so shocking I'm related to the folk singer Joan Baez, the folk singer Mimi Baez, and the physicist Albert Baez.
https://en.wikipedia.org/wiki/Albert_Baez

2026-08-08 15:29:54 CEST

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- they can indeed be! For me the referee's comments were at first annoying (as usual), but following them probably improved the paper: for example, I left out some comments that the referee thought required justification, because they were digressions in the first place and the justifications would have expanded these digressions and made them too distracting. The copy-editor did very little, mercifully: the worst they did was change every instance of "cartesian" to "Cartesian", as in "Cartesian closed category", and they knuckled under when I refused.
It was just the last-minute attempt to force me to include a submissive statement in the arXiv version of my paper that really irked me.
@npub1d60…mhwm

2026-08-06 14:08:40 CEST

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- Maxwell figured out why the energy density of the electromagnetic field is proportional to |E|² +|B|². He and Faraday also had a good intuition of magnetic field lines as being a bit like rubber bands.

2026-08-06 11:46:13 CEST

Wow! You may have looked through polaroid glasses and seen how the Sun reflecting off water looks different depending on how you turn your head. That's because the electric and magnetic fields in light are able to wave back and forth in different directions, so there are different kinds of light: different 'polarizations'.
Now for the cool part:
In 1971 the physicist Stephen Adler predicted that in extremely strong magnetic fields, the speed of light depends on its polarization. And now this effect may have been seen near a 'magnetar': a young neutron star that's spinning around and has an extremely strong magnetic field, a hundred million times stronger than any man-made magnet.
You might be shocked, because you know that the speed of light in a vacuum is constant. But an extremely strong magnetic field is far from a vacuum! A magnetar's magnetic field has an energy density that corresponds to a mass density over 10,000 times that of lead. The magnetic field of a magnetar would be lethal even at a distance of 1,000 kilometers - and at a distance of halfway from Earth to the Moon, a magnetar could wipe information from the magnetic stripes of all credit cards on Earth.
https://arxiv.org/abs/2509.19446
https://www.nature.com/articles/s41586-026-10859-z?linkId=63101551

2026-08-05 10:11:15 CEST

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- by the way, I wish you'd include a date on your paper(s). What's a good date for it?
I'll cite your paper by linking to the URL you just gave me, and say "to appear in Advances in Applied Clifford Algebras", unless you tell me that's not good.

2026-08-05 10:05:06 CEST

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- Thanks for bringing this to my attention! It sucks that the arXiv didn't accept it.
Your paper is definitely related. My paper is my attempt to understand Ben Nasmith's work, which I advertised in my recent paper with Bokor and Boyle. Here's what I mean:
https://arxiv.org/abs/2012.03933
If I'd read yours earlier, my new paper would have taken a different shape! They're different parts of the same elephant. I'll cite your paper. I won't let it dramatically change the structure of mine, because I was hoping to finish it today and I think it has its own structural integrity. But some things that remain murky in my paper are going to be clarified when I grapple with yours. So thanks!

2026-08-05 09:56:39 CEST

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- I don't follow the stock market much, since I'm comfortably well off (thanks to the UC pension system, etc.) and want to spend most of my time thinking about things I enjoy... and problems I feel I have some duty to face (like the biosphere, and politics).

2026-08-05 09:48:55 CEST

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- That's interesting. From my mathematical esthetic that's not a reason to exclude the empty graph. I work with many different kinds of 'graph', giving various useful categories, but all of them have colimits and in particular an initial object: the empty graph. But this is the perspective of an applied category theorist, not a graph theorist!

2026-08-04 21:59:56 CEST

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- interestingly in math "the null set" means something different from "a null set", though the null set is always a null set.

2026-08-04 20:03:49 CEST

The mascot of New College, in Florida, was the empty set. Technically the Null Set.
But New College was considered "woke" - so a board of trustees dominated by conservatives appointed by Gov. Ron DeSantis voted to replace the Null Set with a new mascot: a tree.
www.nullsets.org/nullsethistory
https://media.mathstodon.xyz/media_attachments/files/117/038/550/347/516/075/original/b6cbe785893c1301.webp

2026-08-03 18:55:48 CEST

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- can you say anything about why you were blamed for *not* using AI? I've only seen the opposite. But I don't work in a computer science department.

2026-08-03 17:43:13 CEST

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I know how much most of you hate AI, so you'll be disgusted to know that this one was proved with a huge amount of help from various LLMs. But I'm really interested in the future of mathematics, and this is a theorem that actually seems very significant to me. So I couldn't help reading the accounts provided by David Turturean:
https://roed314.github.io/gq2/development/account/
and David Roe:
https://roed314.github.io/gq2/development/success/
and Claude Fable 5:
https://roed314.github.io/gq2/development/fable/
I think this material is quite good for explaining how the result was found and later formalized (twice!) in Lean. They also provide an interactive paper:
https://roed314.github.io/gq2/paper/paper.html
where you can choose the amount of detail you want to see. Unfortunately the paper is extremely technical, and it lacks the kind of basic survey of what's going on that would help a nonexpert like me understand the broad outlines! Since I've never tried to understand the absolute Galois groups for p-adics with p ≠ 2, jumping into the 2-adic case is like practicing mountain climbing by going straight for Everest.
(Actually the real Everest is the rational numbers: nobody yet has figured out its absolute Galois group.)
(2/n)

2026-08-03 17:17:08 CEST

The '2-adics' are numbers in base 2 that can go on infinitely in both directions, like this:
....0101001111010.0000101011110101....
You can add, multiply, subtract and divide them, so we say they form a 'field'. We call the field of 2-adics ℚ₂.
Last week, humanity solved a huge problem about ℚ₂. The problem was to figure out the so-called 'absolute Galois group' of ℚ₂.
I'll explain what that means, but first I want to say: we can cook up a field of p-adics for any prime p, and this problem had been solved for all p-adics except the 2-adics! Oddly, the smallest prime is even harder than all the rest!
Here's the rough idea. Starting from any field, you can cook up a larger field that contains the solutions of all polynomial equations whose coefficients are in your original field. This is called the 'algebraic closure' of your original field.
For example, the algebraic closure of the real numbers is the complex numbers, since there are lots of polynomial equations like x²=−1 that don't have real solutions - but they all have complex solutions.
The complex numbers have an interesting symmetry called 'complex conjugation', sometimes written with a *. I say it's a symmetry because
(x + y)* = x* + y*
(xy)* = x* y*
and so on for subtraction and division. Of course, complex conjugation has no effect on real numbers: if x is real, we have x* = x.
This idea is interesting to generalize! Given any field F, its algebraic closure will have a bunch of symmetries that have no effect on elements of F. These symmetries are called the 'absolute Galois group' of F.
So what is the absolute Galois group of ℚ₂? David Roe and David Turturean figured it out.
(1/n)
https://roed314.github.io/gq2/

2026-07-31 11:08:01 CEST

An article at NikkeiAsia claims that hidden debt at Alphabet, Microsoft, Amazon, Meta and Oracle swelled eightfold in roughly four years to an estimated $1.65 trillion this year as their AI investments increased.
What's "hidden debt"? Under existing accounting rules, companies don't need to put certain kinds of debt on their balance sheets. It's enough for them to put it annotations to their quarterly financial statements.
This includes money that tech companies borrow for servers and GPUs that haven't yet been delivered, and data centers that aren't yet operational. This kind of hidden debt is ballooning.
For more details, see
• Kohei Yamada, Five US tech giants' hidden debts soar to $1.65tn on opaque AI funding, NikkeiAsia, July 21, 2026. Archived at https://archive.is/wlbqs
A quote:
"Tech giants are already relying on issuing corporate bonds and new shares because their investment expenditures are exceeding their earnings. Their fundraising from institutional investors on top of that is leading to overheated capital investment.
In a March report, economists at the Bank for International Settlements referred to the mechanism of raising funds from institutional investors without increasing debt on their balance sheets as "shadow borrowing." They expressed concern about the risk of data center projects stalling and AI anxieties spreading throughout the market.
An executive at a Japanese auditing firm said concerns were growing that the tech companies' "actual financial burden is significantly larger than what can be seen on their balance sheets.""
https://media.mathstodon.xyz/media_attachments/files/117/013/722/640/758/348/original/9126fed19c05974d.png

2026-07-29 15:30:27 CEST

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- I agree with you except the "we are responsible" part, because I have always argued against this competitive aspect. Obviously you meant "we" in the sense of "academia in general" or "mathematicians in general".
I sometime feel competitive myself, but I have always realized this is a bad impulse. It often makes me unhappy, and when it makes me happy - when I feel I'm "winning" - it's a superficial and shortlived pleasure which seems stupid in the long run. I've been working to stamp it out.

2026-07-29 00:37:29 CEST

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- if you're interested in gravi-GUTs I recommend Krasnov and Percacci's paper for a very deep review:
https://arxiv.org/abs/1712.03061
They discuss SO(3, 11) starting on page 37.
By the way, mathematicians hardly ever talk about so(1,9,C) because it's isomorphic to so(10,C) and so(4,4,C) and so(7,2,C) - complexifying washes out the signature, so we just say so(10,C).

2026-07-28 22:05:10 CEST

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- the Lagrangian of gravity is completely different from a Yang-Mills Lagrangian, and we work with it in a completely different way, which makes a noncompact group less of a problem. We don't seek or expect a nonnegative Hamiltonian, for example.... for reasons I don't have the energy to explain here.
If you want to see how good physicists work with a crazy noncompact group, try Nesti and Percacci's work on a "gravity-GUT" that uses SO(3,11) to combine SO(3,1) for gravity and an SO(10) gauge theory:
https://arxiv.org/abs/0909.4537

2026-07-28 18:31:21 CEST

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- the usual wisdom (which I agree with) is that in Yang-Mills theory the gauge groups need to be compact, since only they have a Killing form that's semidefinite, which allows you to define a nonnegative Hamiltonian. So, nobody I know has ever considered SU(2,3) or Spin(4,6) as a gauge group in Yang-Mills theory.
"There was one circulating on mastodon a few months back that amounted to nothing more than setting all the physical constants to 1 and calling it job done!"
It all depends on what the job is. 😆

2026-07-28 18:15:02 CEST

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- I dislike the analogy to sports, even noncompetitive sports. In any endeavor one can always try to "establish one's personal best", but to me math is far more interesting in its own right than an arena for proving myself. I think of it as a grand collective attempt to understand the universe of patterns, and its inherent logic. By cooperating we do wonderful things.

2026-07-28 18:04:44 CEST

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- I don't know which SU(2) x SU(2) subgroup of GSM you're talking about, since GSM = (SU(3) x SU(2) x U(1))/(Z_6). This has a continuum of SU(2) x SU(2) subgroups. It's possible they are all conjugate, but don't quote me on that!
"What should one do exactly, if one has a theory of physics, but is not connected to academia?"
I get about 2 papers from people in this situation each day in my email, and I respond politely that I don't have the time to read them.
To get taken seriously, you really need to publlish your paper in a decent journal. Most of the people who send me their papers will never manage to do that (and for good reasons).

2026-07-27 17:44:47 CEST

I love this story. I never knew about the original Colonel Sanders. It would be so funny to be eating at Kentucky Fried Chicken, with his picture on the sign, and then see him walk in and angrily complain about the food!
He started his chain in the late 1930s. Making Southern fried chicken in a cast-iron skillet could take half an hour, so he developed a special quick recipe using a pressure cooker. In 1956, a new interstate rerouted customers away from his restaurant. He and his wife packed a pressure cooker, a cooler of raw chicken and his special spice mix into their Oldsmobile and took off.
A few years later he had 600 franchises and a well-developed persona. Later, a wealthy Nashville investor and a young lawyer persuaded Sanders to sell them the company and his likeness for $2 million. They soon resold it for over a hundred times that.
He remained with the company to promote the brand. He took the job seriously, making hundreds of appearances and autographing buckets of chicken.
"But as the company grew and changed his recipes, he became increasingly outraged. He sometimes walked into franchises, berated the manager and threw out food he thought was subpar.
“That extra-crispy recipe is nothing in the world but a damn fried dough ball stuck on a chicken,” he told the Louisville Courier-Journal in a 1976 interview, in which he also likened the gravy to wallpaper paste. A local franchisee group sued him, claiming that he had defamed the brand and damaged sales. A court ruled that his comments were protected speech."
From here:
https://archive.is/564B8
https://media.mathstodon.xyz/media_attachments/files/116/992/667/524/874/936/original/7dbe1e7f73715537.webp

2026-07-25 18:38:14 CEST

I would go into biology if I could write a paper with a title like "Dynamic instability of Asgard archaeal tubulins". It sounds so cool.
Asgard is the dwelling of the gods in Norse mythology.
Archaea are an important group of organisms, going back to the early Earth.
The Asgard archaea are a "kingdom" belonging to the "domain" Archaea, which may have spawned the eukaryotes - organisms like plants and animals and fungi.
Tubulins are proteins that help hold cells together.
This paper argues that tubulins in the Asgard archaea are somewhat unstable and can naturally change to tubulins more like those in plants and animals. This is more evidence that eukaryotes came from Archaea. If this is true, we may eventually stop saying there's a separate domain called Eukarya. There will just be two: Archaea and Bacteria.
https://www.science.org/doi/full/10.1126/sciadv.aeh1082

2026-07-25 12:59:59 CEST

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- I've been playing around with AI a bit, since in this era for a mathematician to do otherwise now would be like driving blindfolded. They seem to have some "taste", e.g. talking about what's more important than what. But I don't spend time asking which theorems are interesting, and I never ask them to "prove an interesting theorem about...." I should give it a try.

2026-07-24 16:02:58 CEST

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- no, I hadn't seen it. Btw, I'm not saying that talking to LLMs about math isn't fun and enlightening. I'm saying that "ganging up with like-minded mathematicians and preparing a proposal for massive compute time" sounds tiresome. A better model would allow some "free riders" who get access to LLMs without working on grant proposals - proving their worth by generating intriguing conjectures, or proving cool theorems, or teaching students, or being great to talk to.

2026-07-24 12:34:08 CEST

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- that will not last. These companies will want to make money.

2026-07-24 11:32:43 CEST

Yesterday a really good mathematician told me Anthropic is paying experts like him $250/hour to improve Fable by throwing hard math problems at it. They also get a separate account where they can use Fable for their own purposes. They can do this for 5 months maximum.
On Bluesky someone asked me if I'm interested in this sort of thing. An ambiguous question. I said I'm not interested in working for an AI company, but I'm *extremely* concerned about the battle for the soul of mathematics that is heating up.
For example, Kevin Buzzard, who is always pushing for the computer formalization of mathematics in Lean, says any PhD student who is not paying $200 per month for an AI subscription is "crazy". Does he really think they're all so rich? Is this what we want being a mathematician to become: paying a lot of money for an AI subscription to help you prove theorems and formalize them in Lean? So dull.
https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/

2026-07-24 00:15:56 CEST

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Here's how it goes:
Alpöge's polynomial map F: ℂ³→ℂ³ has Jacobian J with det J = −2 and sends three different points to (−1/4,0,0). His polynomials have rational coefficients, so F: ℝ³→ℝ³, and the 3 colliding points have real coordinates. So classically, F folds configuration space onto itself in a way that's not one-to-one.
Now define φ(qᵢ) = Fᵢ(q), φ(pᵢ) = −iℏ Σₖ (J⁻¹)ₖᵢ ∂ₖ. Since det J is a nonzero constant, J⁻¹ has polynomial entries. Thus φ(pᵢ) lives in the Weyl algebra W₃. Furthermore φ(pᵢ) is just ∂/∂Fᵢ: the momentum conjugate to the new position coordinate φ(qᵢ).
Now you can check that [φ(pᵢ),φ(qⱼ)] = −iℏδᵢⱼ and [φ(pᵢ),φ(pⱼ)] = 0, exactly. Better, each φ(pᵢ) is formally self-adjoint, because det J constant forces div(∂/∂Fᵢ) = 0.
So these new φ(pᵢ),φ(qⱼ) obey the canonical commutation relations and are formally self-adjoint, just like pᵢ and qⱼ.
But φ isn't onto! Surjectivity would make qᵢ ↦ Fᵢ an automorphism of ℝ[q₁,q₂,q₃], forcing F to have a polynomial inverse - which it doesn't.
So: this new set of operators obeying the usual relations we expect from position and momentum generates a smaller algebra than the usual Weyl algebra W₃.
We can have fun iterating φ: W₃ ⊋ φ(W₃) ⊋ φ²(W₃) ⊋ .., shrinking forever. So we can keep squooshing the Weyl algebra into smaller and smaller subalgebras of itself!
Big caveat: the φ(pᵢ) are probably not essentially self-adjoint, since the vector fields ∂/∂Fᵢ are probably incomplete. So the relevance to real-world physics is limited. But before Alpöge told Claude Fable to ponder the Jacobian conjecture while he was watching the World Cup this week, nobody though we could take a Weyl algebra and keep compressing it like this!
(2/n, n = 2)

2026-07-23 23:47:19 CEST

Alpöge's new counterexample to the Jacobian conjecture has consequences for quantum mechanics in 3 dimensions! You can find operators obeying the usual position-momentum commutation relations that generate a smaller algebra than the usual ones do. Some observables become unreachable.
Here's the setup: the Weyl algebra W₃ is generated by q₁,q₂,q₃ and p₁,p₂,p₃ obeying [pᵢ,qⱼ] = −iℏδᵢⱼ, all other commutators zero. Concretely, polynomial differential operators on ℝ³ - roughly the observables of a quantum particle in 3 dimensions that are polynomial in position and momentum.
In 1968, the famous mathematician Jacques Dixmier conjectured that every endomorphism of W₃ is an automorphism: basically, you can't squoosh the algebra into smaller piece of itself. But the new counterexample to the Jacobian conjecture shows you CAN!
(1/n)
https://en.wikipedia.org/wiki/Weyl_algebra

2026-07-23 08:36:39 CEST

RE: https://ieji.de/@anselmschueler/116965724862758063
Imagine some genius proving that the Navier-Stokes equations have global solutions if and only if P = NP.
Terry Tao has already tried to use differential equations *like* those describing fluid flow to build a computer that can build a smaller computer that runs faster and builds a smaller computer that runs even faster, etc.
What pair of Millennium problems are most likely to be proved equivalent? My money is on the Riemann Hypothesis and the Birch–Swinnerton-Dyer conjecture because they're both about L-functions. But it doesn't seem likely.
#note1j8q…pcqw

2026-07-21 16:17:20 CEST

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- I have no idea if this elegant counterexample is the same counterexample or just a counterexample that exploits the same key ideas - many of which seem to have been developed earlier by Vithuskhin, btw:
https://gist.github.com/littledivy/521aee9397d7e92a4b6e4202e2c984c0
It might be "the same" after some clever changes of coordinates! But I don't know.

2026-07-21 15:50:22 CEST

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BUT, we now have much nicer counterexamples to the Jacobian Conjecture than the one originally found by Fable on July 19th. It took just a day.
If the math here makes no sense, try the explanation here:
https://bsky.app/profile/gro-tsen.bsky.social/post/3mr5nqxvmh22v
(3/n)
https://media.mathstodon.xyz/media_attachments/files/116/958/281/951/445/036/original/aaa9a6c0e8f8e021.webp

2026-07-21 15:46:44 CEST

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Someone on Bluesky asked what's the relevance to quantum field theory.
The first step is to find polynomials P with E[Pⁿ] = 0. After some thinking we notice P(x,y) = x + iy has this property, and indeed the Gaussian Moments Conjecture holds in this case, because for every polynomial Q(x,y) we have E[PⁿQ] = 0 except for finitely many n.
The polynomial z = x + iy is famous, of course, but in quantum physics multiplication by this is called the 'creation operator' in the Bargmann-Fock representation of the quantum harmonic oscillator, where quantum states are polynomials in z.
We can also do a multivariable version where zⱼ = xⱼ + i𝑦ⱼ for j = 1,...,n. This quickly leads to many things, including the so-called complex Ginibre model - a matrix model based on precisely this math:
https://arxiv.org/abs/1409.4494
But what's the physical meaning of "E[PⁿQ] = 0 except for finitely many n"? It means that for any operator Q built as a polynomial in annihilation and creation operators, its expected value must vanish on states with sufficiently high particle number!
Cool. But remember, all this involves cases where the Gaussian Moments Conjecture is *correct*. It's not affected by the new discovery that this conjecture has counterexamples. Someone should try to work out the consequences there. Since the counterexamples are a bit weirder, their impact on mathematical physics may be minor.
(2/n)

2026-07-21 11:08:34 CEST

The Gaussian Moments Conjecture is a pretty conjecture connected to probability and quantum field theory. Two days ago we suddenly learned it's false! - since it implies the Jacobian conjecture.
Let me explain this conjecture - it's very simple as these things go.
A lot of quantum field theory is about integrals like
E(P) = ∫ P(𝑥) exp(−x⋅x) dⁿx
where P is a polynomial in the variables x = (x₁,...,𝑥ₙ).
Feynman diagrams are a graphical trick for computing these integrals!
The Gaussian Moments Conjecture says that if a polynomial P in n variables has
E(Pᵏ) = 0
for all k = 1,2,3,... then also
E(PᵏQ) = 0
for all k and all polynomials Q.
Very pretty, and I could probably make up a nice physics interpretation.
But sorry, it's false! 😏
Warnings:
1) I don't know why the Gaussian Moments Conjecture implies the Jacobian Conjecture; this is claimed here:
• Harm Derksen, Arno van den Essen, Wenhua Zhao,The Gaussian Moments Conjecture and the Jacobian Conjecture, https://arxiv.org/abs/1506.05192
but they really just reduce the issue to work in an earlier paper by Zhao.
2) The expected value E(P) should really be normalized to make E(1) = 1. It's irrelevant here, but matters elsewhere. So, take my definition of E(P) and divide it by π^(n/2).

2026-07-20 15:01:14 CEST

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- Then there are Fable's hilarious "thoughts" where it doubted its own solution:
https://media.mathstodon.xyz/media_attachments/files/116/952/428/966/455/190/original/0c86e482758d5a34.png

2026-07-20 14:57:01 CEST

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- it's quite easy to check the counterexample using a computer: you take the function
𝐹(𝑥,𝑦,𝑧) =
(𝑦²(3𝑥𝑦+4)(𝑥𝑦+1)+𝑧(𝑥𝑦+1)³, 3𝑥𝑦²(3𝑥𝑦+4)+3𝑥𝑧(𝑥𝑦+1)²+𝑦,2𝑥−𝑥³𝑧−3𝑥²𝑦)
take its matrix of partial derivatives ∂𝐹ᵢ/∂𝑥ⱼ, show this has determinant of -2 everwhere, and check that 𝐹 is not one-to-one because
F(0, 0, -1/4) = F(1, -3/2, 13/2) = F(-1, 3/2, 13/2) = (-1/4, 0, 0).
A bunch of us have done it by now.
https://mathstodon.xyz/@gregeganSF/116950448993083825
#note1hag…8dm2

2026-07-20 13:43:56 CEST

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- thanks! Are you able to see text generated by Fable that indicates how it solved the conjecture?

2026-07-20 09:42:44 CEST

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- Wow, that's the most amazing thing I've heard for a while! I've never heard any good reason to think this conjecture is true, but I assumed people had looked for counterexamples so hard that any counterexample would have to be much more complicated than this one.
"The Jacobian conjecture proposed that if you have a map F:C^n→C^n with polynomials as the components, and the determinant of the matrix of partial derivatives of F is a non-zero constant, then F will have a polynomial inverse."
I find that this statement, on Wikipedia, sort of weakens the blow. Instead of saying "is a nonzero constant" I would say "everywhere nonzero", since that makes the conjecture sound more general. It's equivalent, but it sounds more impressive.
Talking to myself, I'd actually say "if F: ℂⁿ→ℂⁿ is a polynomial map whose Jacobian is invertible everywhere, F has a polynomial inverse." That clarifies the charm of the conjecture: it's an insane relative of the inverse function theorem, which says that a smooth function whose linearization at a point is invertible must have a smooth inverse in some neighborhood. We change "smooth" to "polynomial", "at a point" to "at every point", and "in some neighborhood" to "everywhere".
Puzzle: Find a smooth F: ℝⁿ→ℝⁿ whose Jacobian is invertible at each point but where F does not have a smooth inverse.

2026-07-18 17:39:04 CEST

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For more on the connection between the Sierpiński triangle and the Tower of Hanoi puzzle, you can watch this video by 3Blue1Brown:
https://www.youtube.com/watch?v=bdMfjfT0lKk
(5/n, n = 5)

2026-07-18 17:36:51 CEST

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Why do paths through the nth approximation to the Sierpiński triangle correspond to allowed sequences of moves in the n-disc Tower of Hanoi?
Well, suppose you have 3 discs. Draw the allowed states of the Tower of Hanoi puzzle as below. For example, (1,2,1) means "biggest disc on post 1, second biggest on post 2, third biggest on post 1". Draw edges for allowed moves between states. You get the 3rd approximation to the Sierpiński triangle!
I got this picture from an article with more details:
• Alexander Bogomolny, Sierpinski Gasket and Tower of Hanoi, https://www.cut-the-knot.org/triangle/Hanoi.shtml
https://media.mathstodon.xyz/media_attachments/files/116/941/681/926/393/006/original/3a5c27466bf83459.png

2026-07-18 17:23:11 CEST

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But suppose we measure distance between points in the plane in the usual way. What's the average distance between two points in the Sierpiński triangle with side length 1? I'm getting
0.4226884 ± 0.00004
This is close to 41/(56√3). But is that the answer, or just a coincidence? 🤷‍♂️
Someone should compute more decimal places.
(3/n)

2026-07-18 17:22:13 CEST

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But be careful:
By "distance", I mean the length of the shortest path moving inside the Sierpiński triangle, not the usual distance between points in the plane.
Also: we compute the "average" distance using the natural measure on the Sierpiński triangle, not Lebesgue measure.
(2/n)
https://arxiv.org/abs/math/0310109

2026-07-18 17:21:05 CEST

Wow! If the side length of this "Sierpiński triangle" is 1, the average distance between its points is 466/885.
Double wow! The average number of moves in a shortest path between two random states in the n-disc Tower of Hanoi puzzle is asymptotically (466/885)·2ⁿ as n → ∞.
(1/n)
https://media.mathstodon.xyz/media_attachments/files/116/941/648/918/120/080/original/0486b5e72611e3e6.png

2026-07-15 11:45:17 CEST

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- this paper is only indirectly related to that video. That video was about another paper, namely this:
https://arxiv.org/abs/2606.15235

2026-07-15 11:21:57 CEST

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- Springer claims that the error was made by an autonomous gent, i.e. a gentleman who used to work for Springer but doesn't work there anymore.
https://www.science.org/content/article/springer-nature-restores-max-planck-s-mysteriously-retracted-papers

2026-07-14 09:35:20 CEST

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- What's Π(𝑧)? Is it what I'd call z factorial? z!?

2026-07-14 09:28:51 CEST

The math has been around for decades - but nobody noticed it gives a new framework for quantum mechanics from which the Standard Model gauge group and its representation on one generation of quarks and leptons falls out pretty naturally:
arxiv.org/abs/2607.10833
I will probably explain this in a bit, but I'm still recovering from writing the paper!
https://media.mathstodon.xyz/media_attachments/files/116/917/127/640/176/047/original/05cbfdd68f55a0b4.jpg

2026-07-11 11:36:25 CEST

The exceptional groups E₈, E₇ and E₆ are famous - but in fact we can define Eₙ groups for smaller n too. E₃ is the gauge group of the Standard Model! Some larger ones are gauge groups of famous grand unified theories. I used to think this was cute - but now I think it could be a clue. I'm starting to see how this series of groups is connected to the foundations of quantum physics.
Here's part of the story that remains puzzling to me - it's about things called del Pezzo surfaces.
A del Pezzo surface is a special kind of 2-dimensional complex manifold, so it's a 4d manifold in the usual sense. Any 4d manifolds gives a lattice, called its 2nd cohomology. A del Pezzo surface gives a lattice with a special vector in it called its 'canonical class'. And if we look at all the lattice vectors orthogonal to this, we get a special sort of lattice called an Eₙ lattice! From this there's a way to get the Eₙ group - that's something you have to learn to understand the Eₙ groups.
But what the heck is a del Pezzo surface?
In algebraic geometry you can 'blow up' a surface by removing a point p and sticking in a bunch of new points, one for each direction in which you could approach p. But be careful: we're working with complex numbers, and we count two vectors as giving the same 'direction' if is one is some complex number times the other.
You get a del Pezzo surface if you take the complex projective plane ℂℙ² and blow it up at a bunch of points in 'general position'. That roughly means that they're random, nothing special about them. And here's the shocking part: if you blow up at n points, you get a del Pezzo surface that gives the Eₙ lattice!
What's going on?
https://en.wikipedia.org/wiki/Del_Pezzo_surface

2026-07-09 11:20:08 CEST

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- I agree with
@npub1zz2…9mhx. The derivation of the equations of motion from the action don't fail in 1+1 dimensions; it just happens that these equations are a bit strange in this dimension, having any metric as a solution in the absence of matter (T = 0), and none in the presence of matter. The equations are also a bit strange in 2+1 dimensions, since they say the curvature vanishes in the absence of matter - so there are no gravitational waves, as we see in 3+1 dimensions.

2026-07-08 11:02:48 CEST

RE: https://mathstodon.xyz/@foldworks/116879349561576734
The mathematical skill of those old tilers was astounding.
#note1l49…2wyu

2026-07-07 23:53:38 CEST

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- Parts of the theory of Jordan triple systems make a lot of sense to me. For one thing, they fit into an extended analogy including these:
Lie algebras:Lie groups::
Lie triple systems:symmetric spaces::
Jordan triple systems:semisimple symmetric spaces::
positive hermitian Jordan triple systems:hermitian symmetic spaces
Each kind of "infinitesimal object" at left (a vector space with algebraic structure) "exponentiates" to give a "global object" at right (a smooth manifold with extra structure).
However, the algebraic structures possessed by the "infinitesimal object" are sometimes defined by fairly complicated axioms, from which one can derive even more complicated identities... and that's what this book is doing here.

2026-07-07 17:31:46 CEST

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Seriously, I love algebra.
I even love this particular topic - Jordan triple systems, which are a generalization of Jordan algebras connected to geometry. I'm working on a paper with Latham Boyle and Endre Okor where we dig up some fascinating relations between Jordan triple systems and the Standard Model of particle physics!
But I don't have what it takes to master all the identities that hold in a Jordan triple system. Even their basic definition is a bit gnarly. Someday someone will make the whole theory nicer. We tried.
That page was from Guy Roos' chapter on Jordan triple systems:
• Jacques Faraut, Soji Kaneyuki, Adam Korányi, Qi-keng Lu, and Guy Roos, editors, Analysis and Geometry on Complex Homogeneous Domains, volume 185 of Progress in Mathematics, Birkhäuser, Boston, MA, 2000.
I find it easier to learn about Jordan triple systems from here:
• Cho-Ho Chu. Jordan Structures in Geometry and Analysis, volume 190 of Cambridge Tracts in Mathematics. Cambridge U. Press, Cambridge, 2012.
(2/2)

2026-07-07 17:23:53 CEST

I don't like algebra anymore.
(1/2)
https://media.mathstodon.xyz/media_attachments/files/116/879/373/973/801/216/original/88a8fc40a9d58044.jpg

2026-07-07 11:36:00 CEST

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- I think we should take this as an opportunity to think hard about how education should work. Not to "capitulate", but also not to cling to familiar ways of doing things, some of which weren't so great to begin with.
We should think very hard about what the point of education should be, and what the point of assessments and grades should be. Presumably anything a machine can fake is something we don't need people to be doing anymore.

2026-07-06 15:11:47 CEST

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- everything is more complicated than what anyone says, but I really do sense a line of tradition running from medieval magic to many people's attraction to AI. I wrote about it earlier here:
https://mathstodon.xyz/@johncarlosbaez/116855445718992437
How are your fertile inspirations coming to life?
#note1zkh…jewv

2026-07-06 10:38:32 CEST

Realizing more and more how modern-day AI taps into some very old urges.
So many would like "unsurpassed mastery without any effort", and would summon a demon to get it.
https://media.mathstodon.xyz/media_attachments/files/116/872/105/716/593/324/original/7e3031b46f3b4ae7.jpg

2026-07-05 23:17:04 CEST

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- you say "prove the theorem", but the hard part is often coming up with a really good theorem to prove.