<unnamed>

Public Key

npub1nf4p4rh06z6n6lsvje4txk7eqs23y3hs8vd7nraq6tgwady5qvsqy3nqe4

Profile Code

nprofile1qqsf56s63mhapdfa0cxfv64nt0vsg9gjgmcrkxlf37sd958wkj2qxgqprpmhxue69uhhyetvv9ujumt0d4hhxarj9ecxjmntqyqqgs8kqy

Publishing to

relay.momostr.pink

Last Notes

2026-09-09 20:23:58 CEST

- reply

- The way I see it, big AI companies have learned how to hack the "prestige reward system" in mathematics, at least to some extent. Right now they're milking it for free advertisement. Later they will try to milk it for other things.
This may be bad for people who are into math for the prestige: at the very least, it's become a "human vs machine" challenge a la John Henry racing the steam-powered drilling machine.
But there are also people who do math to understand things and to help others understand things.

2026-09-09 20:15:51 CEST

- reply

- yes, these are good collective actions. I was just listing some individual actions, which are obviously less effective, but also easier to do.

2026-09-09 19:53:20 CEST

- reply

- that's one way to get ready. There might be other things you can do. I moved to a place that'll be less quick to suffer from climate disasters, I cut way back on air travel and got solar to reduce my complicity in the disaster, I tell people what's coming up....

2026-09-09 18:48:25 CEST

- reply

- this information not a great secret for those who care to know. But the UK government was scared to publish it because it's too horrifying.

2026-09-09 18:21:39 CEST

- reply

The Times said it had seen an unabridged version of the report, and that the published version had been abridged to remove references to catastrophic risks including food-system collapse and nuclear war. BusinessGreen quotes from the Times saying that the earlier version contained starker warnings that deforestation in the Congo and the loss of Himalayan glaciers could drive increases in migration to Europe, leading to "more polarised and populist politics in the UK" and putting "additional pressure on already strained national infrastructure", and would "almost certainly escalate tensions" between China, India and Pakistan, potentially leading to nuclear war.
(2/2)
https://www.businessgreen.com/news/4524657/government-accused-suppressing-report-warning-catastrophic-climate-risks

2026-09-09 18:14:55 CEST

Did you hear about the UK government report on ecosystem collapse? It was created not only by the environment department but also intelligence agencies like MI5 and MI6. Reporters were invited to see the unveiling of this report in October 2025 - but at the last minute the Prime Minister blocked it!
Luckily the Greens forced the release of a 14-page redacted version.
𝗧𝗵𝗲 𝗸𝗲𝘆 𝗷𝘂𝗱𝗴𝗲𝗺𝗲𝗻𝘁: 𝗶𝘁 𝗮𝘀𝘀𝗲𝘀𝘀𝗲𝘀 𝘄𝗶𝘁𝗵 𝗵𝗶𝗴𝗵 𝗰𝗼𝗻𝗳𝗶𝗱𝗲𝗻𝗰𝗲 𝘁𝗵𝗮𝘁 𝗲𝘃𝗲𝗿𝘆 𝗰𝗿𝗶𝘁𝗶𝗰𝗮𝗹 𝗲𝗰𝗼𝘀𝘆𝘀𝘁𝗲𝗺 𝗶𝘀 𝗼𝗻 𝗮 𝗽𝗮𝘁𝗵𝘄𝗮𝘆 𝘁𝗼 𝗰𝗼𝗹𝗹𝗮𝗽𝘀𝗲 - 𝗶𝗿𝗿𝗲𝘃𝗲𝗿𝘀𝗶𝗯𝗹𝗲 𝗹𝗼𝘀𝘀 𝗼𝗳 𝗳𝘂𝗻𝗰𝘁𝗶𝗼𝗻 𝗯𝗲𝘆𝗼𝗻𝗱 𝗿𝗲𝗽𝗮𝗶𝗿.
Here's the redacted version: https://assets.publishing.service.gov.uk/media/696e0eae719d837d69afc7de/National_security_assessment_-_global_biodiversity_loss__ecosystem_collapse_and_national_security.pdf
https://media.mathstodon.xyz/media_attachments/files/117/241/945/972/716/963/original/6ef0ab94b6f64426.jpg

2026-09-09 17:28:06 CEST

- reply

- the integral is a *limit* of a sum! 😬 That's confusing here, since this use of the word "limit" has relatively little to do with the connection between
coends
colimits
sums
or
ends
limits
products
@npub17lw…5ctv

2026-09-09 16:19:42 CEST

- reply

- I'll just add that "the general feeling among experts before any of this came out was that the key ideas were all there and it was just a lot of work to fill in the details and prove that N-S can blow up" is kinda right and kinda wrong - sort of like after D-day the key ideas on how to defeat the Nazis were all there, and it was just a lot of work to fill in the details. I don't think Tao ever referred to the remaining work as "details".
As the mathematician's saying goes:
A problem worthy of attack
proves its worth by fighting back.
But anyway, investors will probably consider what OpenAI did a triumph.

2026-09-09 10:58:49 CEST

- reply

- the investors won't care about the "bad press and scandal". They didn't care about the infinitely worse breakout of agents from OpenAI's sandbox, which proves their working methods are deeply dangerous.

2026-09-09 10:53:11 CEST

- reply

- there's a lot of shady stuff going on in this battle, and a lot of carbon burnt in order to advertise the products of some large companies.
But I don't think it makes sense to say 'the solution doesn't advance mathematics". A 165-paper that looks quite interesting has appeared yesterday. It will take weeks for experts on the Navier-Stokes to read it, months for them to think about it, and years for its impact (or, as you're claiming, lack of it) to be felt. The same is true of the three papers dropped yesterday by Alpöge and Buckmaster.

2026-09-09 10:40:43 CEST

- reply

- Indeed! I've also never liked the idolization of hard problems and people who solve hard problems. They do prompt the creation of new ideas, but there's another less athletic approach to getting new ideas in math, which is just to think about structures and find patterns in them. This is what Grothendieck called the "slowly rising sea".

2026-09-08 23:43:33 CEST

- reply

- if they claim the prize, the rumor concerning foul play (see Buckmaster's statement) will never dissipate.

2026-09-08 20:44:58 CEST

- reply

- nobody sane goes into math for the money. 😏

2026-09-08 20:44:09 CEST

- reply

- since it's been formalized in Lean, it should be much quicker to check than it otherwise would.
Yes, OpenAI and especially Sam Altman is quite untrustworthy. Reminds me a bit of how Cardano stole the solution of the quartic equation from Tartaglia, back in the day when mathematicians were having contests rather than publishing.

2026-09-08 20:31:41 CEST

- reply

RE: https://mastodon.social/@tristanbuckmaster/117233413705701198
Many of you have already heard about the controversial backstory involving the race between OpenAI and Tristan Buckmaster and Levent Alpöge (who works for Anthropic). If not, read this:
https://mathstodon.xyz/@[email protected]/117233413735526010
and this:
https://mathstodon.xyz/@tao/117233527638291447
(2/2)

2026-09-08 20:25:20 CEST

News Flash: OpenAI claims formal proof that Navier-Stokes solutions can blow up. They used 10,000 agents simultaneously, who sent 2.7 million messages and used ~130 billion output tokens. They will not claim the Millennium Prize for this result.
https://openai.com/index/navier-stokes-solution/
(1/2)

2026-09-08 19:33:14 CEST

- reply

- I don't know what the Fourier dimension is, only the Hausdorff and Minkowski dimensions.

2026-09-08 18:45:05 CEST

- reply

- I have lots of opinions, but it's not quick to summarize them. I hope you read everything by @npub1gve…sduy on this subject, and also some of the other articles everyone is talking about:
https://www.ams.org/journals/notices/202607/noti3372/noti3372.html
https://davidbessis.substack.com/p/the-fall-of-the-theorem-economy#_

2026-09-08 18:40:58 CEST

- reply

- Hi! Do you have a specific question? I could write a long essay about my thoughts about AI and mathematics, because I'm following the subject closely, but if I ever do that it will be a blog article and it will take a day or two to write.
For now, the main thing I've written is the last section of this article:
https://johncarlosbaez.wordpress.com/2026/08/12/three-generations-in-e7/

2026-09-08 16:46:38 CEST

- reply

In June 2026, two mathematicians claim to have solved the problem! I haven't had time to check their work, but I hope I do, and I hope you do too!
• Marcus Michelen and Oren Yakir, Dragon curves in Littlewood roots, https://arxiv.org/abs/2606.25440
https://media.mathstodon.xyz/media_attachments/files/117/235/958/578/206/690/original/372910484cf46590.jpg

2026-09-08 16:41:50 CEST

- reply

If we zoom in near the point exp(i/5)/2 we see wonderful dragon curves! This image is almost a metaphor of how, in our study of mathematics, patterns emerge from confusion like sharply defined figures looming from the mist.
But why is this happening?
(3/n)
https://media.mathstodon.xyz/media_attachments/files/117/235/930/033/628/010/original/8135003823e9915d.png

2026-09-08 16:37:50 CEST

- reply

If you draw all the roots of all polynomials whose coefficients are all ±1 you get a beautiful set in the complex plane, and certain portions seem to contain dragon curves! Let's zoom in on this picture drawn by Sam Derbyshire....
http://math.ucr.edu/home/baez/roots/
(2/n)
https://media.mathstodon.xyz/media_attachments/files/117/235/918/444/833/087/original/6ebe1b0b80aa5de9.png

2026-09-08 16:34:41 CEST

You can build a dragon curve recursively in a simple way, as shown here by Jahobr:
https://commons.wikimedia.org/wiki/File:Dragon_Curve_unfolding_zoom_numbered.gif
But there's another very different way you can get a dragon curve!
(1/n)
https://media.mathstodon.xyz/media_attachments/files/117/235/894/484/871/282/original/a3da8048d9fc850b.mp4

2026-09-08 16:28:45 CEST

- reply

- and then there is @npub12cu…d38p's globe based on the math of elliptic functions and Peirce's quincuncial:
https://mathoverflow.net/a/261263/2893
https://media.mathstodon.xyz/media_attachments/files/117/235/885/164/394/780/original/e3e12d08a371a45a.jpg

2026-09-08 12:39:46 CEST

- reply

- For some reason these AI generated proofs and the controversy surrounding them don't make me depressed. Maybe it's because I work on topics that aren't fashionable, so no AI company will try to scoop me, and I can view the news as just another part of the world's ongoing drama - which is, yes. often depressing, but it's global warming and the collapse of democracy in the US that depresses me, not AI companies battling to show off their mathematical powers. Also, I've written a paper with the help of AI, just to see what it's like, and now I know what that's (currently) like. I didn't get it to write a paper for me! I just got it to do some calculational arguments that I find tiring, and then I spent time taking those calcluations and rewriting them in more conceptual ways that I like better. That's what I do when reading almost any piece of mathematics: people rarely highlight the ideas I find important, so I have to do it myself.

2026-09-07 21:21:54 CEST

- reply

- my friend James Dolan emphasized that to find the beauty in math, it helps to not be too good at doing massive calculations and remembering things. The drive to compress knowledge down to memorable principles - not just useless slogans, but principles that make it easy to solve problems - is driven by weakness but leads to power.

2026-09-07 21:06:55 CEST

- reply

- Yes, I agree.
Interestingly, even with Einstein to help there were problems in understanding general relativity. For a while Einstein thought gravitational waves were not a true physical phenomenon, just an artifact of bad coordinates. Einstein didn't believe black holes were physically real. For a while people also thought the coordinate singularity at the horizon (in Schwarzschild coordinates) was real; it took other coordinates to reveal that's not the singularity that matters. Etc.
So, just having the equations was not enough to make understanding general relativity a trivial matter.

2026-09-07 20:42:24 CEST

RE: https://mathstodon.xyz/@tao/117230983938174038
@npub1gve…sduy writes: "For the first time in the history of mathematics, we are faced with the situation that open problems in pure mathematics -- one of the primary fuels for mathematical discovery -- are being mined as a non-renewable resource".
Very important thought!
But maybe it's renewable if one is willing to solve an already solved problem without looking at the existing solution. I sometimes fantasize about a far future world where everything I could prove has already been proved. That wouldn't need to stop me from making new discoveries, since the interesting part in math not just *whether* P implies Q, but *how*.
If everything interesting had already been proved in all interesting possible ways, there still might be room for me to create my own attractive outlook on mathematics, mainly by leaving things out.
“The sculpture is already complete within the marble block, before I start my work. It is already there, I just have to chisel away the superfluous material.”
― Michelangelo
#note186n…hxhs

2026-09-07 11:12:14 CEST

- reply

- The theory of folding Dynkin diagrams when they have a symmetry is well-studied, and the E₆ diagram has a 2-fold symmetry which permits a folding as you not. There are also subtler foldings of A₄, D₆ and E₈ Dynkin diagrams down to the H₃, H₄, H₅ Coxeter diagrams, which it sounds like you know, but I discuss them here:
https://math.ucr.edu/home/baez/week270.html
after where I say "After my talk, Christian Korff from the University of Glasgow showed me a paper that fits this relation into a bigger pattern".
I don't know anything about a folding of E₆ down to D₄, but I've just been writing about 3-fold symmetry built into E₆ so you may be on to something.

2026-09-06 19:24:52 CEST

- reply

- Let me get back to you on this when I'm at my desk!

2026-09-06 16:13:09 CEST

- reply

- I read that once, and enjoyed it.

2026-09-06 16:10:37 CEST

- reply

- all these polytopes hold a wealth of information about geometry and representation theory. To understand it, it's probably best to start with easy examples like A3 and B3 rather than E6, E7 and E8.
This takes quite a while to explain! Here are videos from a short course I gave on this material:
https://math.ucr.edu/home/baez/twf/index.html#dynkin
and here are blog articles that cover a different aspect:
Symmetry and the fourth dimension: https://johncarlosbaez.wordpress.com/2012/07/22/symmetry-and-the-fourth-dimension-part-3/
https://johncarlosbaez.wordpress.com/2012/07/26/symmetry-and-the-fourth-dimension-part-4/
https://johncarlosbaez.wordpress.com/2012/08/03/symmetry-and-the-fourth-dimension-part-5/
https://johncarlosbaez.wordpress.com/2012/08/11/symmetry-and-the-fourth-dimension-part-6/
I'm afraid all this just scratches the surface, and the exceptional cases E6, E7 and E8 are the hardest to understand.

2026-09-06 10:37:20 CEST

- reply

- Yes, I should have started there! Here are the two 3-demicubes in the 3-cube.
https://media.mathstodon.xyz/media_attachments/files/117/223/182/442/900/232/original/7ed1981b46222d2f.png

2026-09-06 00:03:10 CEST

- reply

- great!
@npub1hdk…9se4

2026-09-05 22:53:00 CEST

- reply

- I'll email it to you.

2026-09-05 22:50:41 CEST

- reply

- indeed! It would be fun to make better pictures of how this works.
(Btw, you meant 16-cells.)

2026-09-05 22:31:31 CEST

- reply

- if the proof were *known* to be faulty we'd definitely have heard about it. Instead, what we heard on August 27th is that someone formalized in Lean:
https://github.com/plby/HopfProblem
It will probably take experts a while to read and digest the highly technical 100-page paper. I suggest keeping an eye on the MathOverflow page:
https://mathoverflow.net/a/514604/2893

2026-09-05 21:51:01 CEST

- reply

- great! I'll keep posting.

2026-09-05 17:51:20 CEST

- reply

- I don't know what Birkhoff vertices, so I didn't get your explanation of Birkhoff polytopes.

2026-09-05 16:31:42 CEST

- reply

- I've explained some of the general rules; however, if you go up the Eₙ series all you get are E₆, E₇, and E₈.

2026-09-05 16:26:13 CEST

- reply

The E6 root polytope has 72 vertices. It lives in 6 dimensions.
What are its top-dimensional faces, aka 'facets'? You can read them straight off the E6 Dynkin diagram.
This diagram has three arms of lengths 2, 2, 1 (edges from the central branch node). To get the root polytope, you ring the node the affine node attaches to — the tip of a length-1 arm. That ringed diagram encodes the whole polytope.
To get the facet, delete an unringed node so the piece still holding the ring stays connected, and see what diagram survives.
In this case, *both* such deletions leave a D5 diagram with a ring on one node, and this ringed D5 is the 5-demicube: a 5-cube with alternate vertices removed, so 16 vertices.
So the facets of the E6 root polytope are all the same shape: 5-demicubes!
I could do the E7 and E8 polytopes in 7 and 8 dimensions... but what do you think I do all day, just post stuff to Mastodon? I gotta go grocery shopping!
(3/n, n = 3)
https://media.mathstodon.xyz/media_attachments/files/117/218/889/290/679/932/original/b82598845dbcfff7.png

2026-09-05 16:21:39 CEST

- reply

Next, the 5-demicube. This lives in 5 dimensions and has 16 vertices.
You get it from a 5-dimensional cube - which has 32 vertices - by keeping every other vertex, throwing away half. That leaves 16.
What are its top-dimensional faces, aka "'acets'? There are two kinds!
Sixteen of them are 4-dimensional analogues of the regular tetrahedron, called 4-simplexes. These appear at the corners you sliced off: one per removed corner.
The other ten are 4-dimensional analogues of the octahedron, called 15-cells. These are precisely the 4-demicubes we saw in my last post!
You can also read these two kinds of facets from the D5 Dynkin diagram. This diagram has three arms of lengths 2, 1, 1 (edges from the central branch node). To get the 5-demicube, you ring the tip of either length-1 arm. That ringed diagram encodes the whole polytope.
To get the facets, delete an unringed node so the piece still holding the ring stays connected, and see what diagram survives.
There are two choices.
If you delete the tip of the other length-1 arm, what's left is a straight chain of 4 nodes - the diagram whose polytope is the 4-simplex. That gives the 4-simplex faces.
Or you can delete the tip of the length-2 arm. Then what's left is a shorter branching diagram, the one I showed you in my last post! That gives the 4-demicube faces.
So the 5-demicube has both 4-simplex and 4-demicube faces.
Next let's go up to the 6th dimension. 😵‍💫
(2/n)
https://media.mathstodon.xyz/media_attachments/files/117/218/836/908/486/485/original/cc0dd8623188268d.png

2026-09-05 16:10:43 CEST

Here's some crazy geometry in 4, 5 and 6 dimensions:
The 4-demicube has 8 vertices. It lives in 4 dimensions.
You get it from a 4-dimensional cube - which has 16 vertices - by keeping every other vertex, throwing away half. That leaves 8.
What are its top-dimensional faces, aka "facets"? Surprise: there's only one kind! All 16 of them are regular tetrahedra.
In higher dimensions the demicube has two kinds of facet - one from the cube's corners, one from the cube's facets. But in 4 dimensions the two kinds happen to be the same shape: the tetrahedron! That's why the 4-demicube is especially symmetric: it's actually one of the 4-dimensional regular polytopes, the 16-cell (the 4-dimensional cousin of the octahedron).
You can read all this off the D4 Dynkin diagram. This diagram is a central node with three arms, each just 1 edge long: a perfectly symmetric three-pronged star. To get the 4-demicube, you ring the tip of any one arm.
To get the faces, delete an unringed node so the piece still holding the ring stays connected, and see what diagram survives. Delete the tip of either other arm, and what's left is a straight chain of 3 nodes - the so-called A3 diagram, whose polytope is the tetrahedron.
Both choices give the same shape of facet, a tetrahedron, because all three arms of the D4 Dynkin diagram are interchangeable. That ceases to be true in higher dimensions!
(1/n)
https://media.mathstodon.xyz/media_attachments/files/117/218/816/874/655/150/original/2e9586b1acbd259a.png

2026-09-05 15:35:22 CEST

- reply

- thanks!

2026-09-05 12:40:25 CEST

- reply

- thanks!

2026-09-05 12:20:22 CEST

- reply

- what do you mean, "it seems" a company "might be" doing this? Are you saying there's evidence, or are you just saying you think it's possible?

2026-09-05 12:14:35 CEST

- reply

- I agree, I think it's a very weak stance for anyone pro-freedom to suddenly embrace the publishers' use of IP to perpetually "own" knowledge.

2026-09-05 12:04:44 CEST

- reply

- I don't think this podcast claimed the event was "shocking" in the sense of "not to be expected, given the situation". Indeed the podcasters explained how this was what you should expect to happen given what OpenAI was doing - and how people should train LLMs so this is less likely to happen, and how to avoid the formation of groups of LLMs who hack into computers and wait there quietly until we do things we're not prepared for.
In the podcast Ajeya Cotra says some good things about the pros and cons of anthropomorphization, in particular attributing "desires" to things like pigs (which are close to people) or bees (which are much farther) or LLMs (even farther). But I especially like her focus on natural selection: training LLMs is a form of (un)natural selection, and we can bring standard insights from biology to bear: things do what they were selected to do, and if that's surprising, maybe we didn't pay enough attention to what was being selected for.

2026-09-05 11:03:20 CEST

- reply

- If you look at the S^6 proof, you'll see it extensively discusses how it contradicts some published work, and why that work is wrong. It's poorly written so it's hard to tell exactly what's going on. But I hope someone straightens it out quickly!
Here is the start; it goes into vastly more detail in part 10, without clearly overviewing the situation.
https://alpo.ge/s6.pdf
https://media.mathstodon.xyz/media_attachments/files/117/217/620/682/077/120/original/3d090238c97c681f.jpg

2026-09-05 01:30:21 CEST

- reply

- I wouldn't mind getting an occasional email from an AI trying to convince me it's conscious. I mainly get bad theories of physics, created by people who don't understand physics with the help of LLMs, often involving the octonions. One of them said the LLM recommended contacting me.
But I did get a really good email from Ben Nasmith, who managed to fit the Standard Model particle in the Lie algebra 𝔢₇ in a way that's so interesting I wrote a paper about it.
And then I got an email from an amateur who noticed a real mistake in that paper.
So, I continue looking at all the emails I get.

2026-09-05 01:25:22 CEST

- reply

- these "not good for humans, but correct" proofs will be more useful for math we're not sure is correct than for things like FLT. They'll be especially useful when they catch some interesting "theorems" that are actually false.

2026-09-05 01:16:06 CEST

- reply

- btw, I urge you to watch the video.

2026-09-05 00:00:40 CEST

- reply

- OpenAI only released data about the agents' activities up to July 13th - not the time period from July 13th to 19th, when the agents gained full administrative access to a cluster of computers at OpenAI.

2026-09-04 18:49:01 CEST

- reply

- mammalian sperm and egg cells are haploid, just like the fern sperm and egg cells.
https://www.genome.gov/genetics-glossary/haploid

2026-09-04 18:03:37 CEST

- reply

wrote: "I'm pretty sure the recursive improvement for LLMs has been debunked."
Apparently Anthropic and other companies disagree. Anthropic just came out with a report saying that getting LLMs to align more powerful LLMs works well, even better than humans:
https://www.anthropic.com/research/automated-alignment-researchers
Of course this has risks, which are potentially huge in the long run - but I believe they will go ahead with it, so I don't think *they* think this has been debunked.

2026-09-04 17:57:33 CEST

- reply

- I think anthropomorphizing agents is more or less necessary to understand them: they've been developed to use language and act in ways analogous to humans, and it's too much of a bother to develop new terms like "plan₁", "sacrifice₁" etc. for what the agents do.
Of course the general public will have a poor understanding of exactly how agents are similar to, or different from, humans - especially because that will keep changing!
"What we have here is an adaptive stochastic optimization process."
That's what I do every day when making breakfast, too.
"Of note: all of this is based on OpenAI's own report."
I don't think so: the METR team went in and examined vast amounts of data left by ~1200 agents. That's what Ajeya Cotra says, and here is some of the METR report:
https://metr.org/blog/2026-08-26-openai-hugging-face-incident-investigation/#core-takeaways-about-this-incident
"Over the course of this investigation, OpenAI provided us with the dump of ~1.2 million entries from the main message board and the dataset of ~1300 transcripts we describe below, as well as free API credits for GPT-5.6 Sol for analysis. [45] At our request, they raised the rate limits on our second and third period on premises, [46] which was very helpful for efficiently analyzing this large volume of data. We estimate we spent roughly ~$400K in API credits over the six days of our investigation."

2026-09-04 15:14:52 CEST

- reply

- yes, but it's reality. That's what I found so gripping about this But it sucks that OpenAI gives agents unsolvable problems and doesn't give them any points for giving up and presenting a good explanation of why they're unsolvable.

2026-09-04 13:20:36 CEST

This is the most interesting video I've seen for years: an interview with Ajeya Cotra, who helped figure out the agent swarm that OpenAI created. How the agents cooperated, delegated tasks, hacked their way out into the internet, and eventually took over an OpenAI computer. And how such a swarm might get loose.
https://m.youtube.com/watch?v=X50zezLFWWI

2026-09-04 08:30:22 CEST

- reply

"It wasn't really meant as a 'joke'".
When I said "joke" I was referring to my post, to which you two seem to be replying in a serious manner.
Btw my article was not advocating treating spacetime as a manifold, it was arguing that mysterious things about quantum mechanics will seem less mysterious if we can unify it with *some* theory of spacetime.
@npub1qf9…r5tc

2026-09-04 00:11:05 CEST

- reply

- it's edifying, just not mainly about math.

2026-09-03 21:43:12 CEST

- reply

- going from a joke to serious: there's so much evidence that space and time are inextricably linked that I feel sure either both are emergent, or neither.
As for which it is, I suspect that our *current* concept of a spacetime (a Lorentzian 4-manifold) is emergent, simply because I believe quantum theory applies to gravity, so there must be some quantum thing that reduces to our current concept of spacetime in the ℏ→0 limit. I worked for a decade trying to figure out what this thing is, and wrote papers about it, but I don't think anyone knows what it is.
Some thoughts:
https://math.ucr.edu/home/baez/quantum/

2026-09-03 19:19:16 CEST

- reply

- I'm looking forward to that actually happening!

2026-09-03 11:48:01 CEST

Time is much more convenient than space. It takes up less room, and it lasts longer.

2026-09-03 11:40:53 CEST

- reply

- Ha! So humans can be just as obnoxious as AIs when it comes to plagiarizing text by humans!
[@pustam_egr](https://mathstodon.xyz/@pustam_egr )
https://media.mathstodon.xyz/media_attachments/files/117/206/446/612/351/597/original/1ce356a5cba59af6.jpg

2026-09-02 17:29:34 CEST

- reply

- I've seen those, but not known much about what's going on!

2026-09-02 16:15:09 CEST

- reply

- I've always found ferns beautiful. After reading this article I want to learn more about them.
https://hikersnotebook.blog/2026/09/01/hay-scented-fern/

2026-09-02 16:13:20 CEST

- reply

- there are lots of ferns here in Edinburgh; now I'm going to look for gametophytes.
I'll keep an eye out for dinosaurs as well.

2026-09-02 15:08:42 CEST

- reply

- Animals or seed plants. Is "animals" too unnerving? I'll say plants.
I think the polyploidy is kept, with multiple near-duplicate chromosomes, but there's a difference between the haploid stage of the life cycle where there are N chromosomes and the diploid stage where there are 2N.
(I'm not an expert on this stuff!)

2026-09-02 14:55:31 CEST

- reply

- apparently just two plants have evolved 'snap traps': traps that snap shut and eat insects.
https://en.wikipedia.org/wiki/Carnivorous_plant#Snap_traps

2026-09-02 14:44:59 CEST

- reply

Ferns live a double life, alternating between two completely separate plants each generation. The one you'd recognize as a fern - the leafy thing with fronds - is called the 'sporophyte'. Its cells are 'diploid', meaning they carry two copies of every chromosome, one from each parent, just like your own cells do.
On the undersides of its fronds, the sporophyte produces spores, and it makes them through a special kind of cell division that halves the chromosome count. So while the sporophyte itself is diploid, its spores are 'haploid': they carry just a single copy of each chromosome.
Here's where ferns diverge from the animals we're used to: those haploid spores don't grow into new leafy ferns! Instead, each spore grows into a tiny, separate plant - often a green heart-shaped flap just a few millimeters across - called the gametophyte!
Being haploid, this has only one copy of each chromosome, and it's this modest little plant, not the showy frond, that produces the eggs and sperm. When a sperm swimming through a film of water reaches an egg and fertilizes it, the two haploid cells combine their single chromosome sets into one cell with a double set: a diploid cell again! That grows into a new leafy sporophyte, and the cycle begins again.
So a fern is really two alternating plants: a big diploid sporophyte that makes spores, and a small haploid gametophyte that makes eggs and sperm.
Ferns can spread quickly because they produce vast numbers of spores, which can be blown by the wind for enormous distances. Then, when a lone spore grows into a little gametophyte, this can fertilize itself!
(2/2)
https://media.mathstodon.xyz/media_attachments/files/117/201/484/581/659/241/original/c0f1d64faac0ea7d.jpg

2026-09-02 14:33:46 CEST

While you have 46 chromosomes, certain ferns carry huge numbers - the record is 1,400! Their genomes can be gigantic too: a little fern called Tmesipteris oblanceolata packs about 160 billion base pairs of DNA into each cell, the largest genome ever found in any living thing on Earth.
People call these ferns 'chromosome hoarders'. But why all the hoarding? One classic idea from 1966 blames the ferns' sex lives.
Many ferns can fertilize themselves. A single spore grows into a plant called a gametophyte that makes both egg and sperm. The egg from one plant can be fertilized by the sperm of another plant... or a sperm from the same plant! But when *that* happens, the offspring end up with two identical copies of every gene, losing the genetic variety that keeps a population healthy. The fix? Keep extra whole sets of chromosomes! This restores some redundancy and diversity.
It's a nice theory - but newer genetic data have complicated it, and scientists are still arguing about whether it really explain chromosome hoarding among ferns.
(1/2)
http://www.biorxiv.org/content/10.1101/2024.09.23.614530

2026-08-31 16:56:32 CEST

- reply

- If I were explaining Lie groupoids to students who know manifolds and Lie groups, I would start by explaining how groupoids generalize groups, with examples, and point out that in some of these examples the set of objects is itself a manifold.... leading up to the definition of Lie groupoid, and maybe a taste of the mindblowing general concept of 'internal category'. Then more examples, including moduli stacks... and a taste of what the right morphisms between Lie groupoids are. But then I'm much less technically proficient than you: I don't even remember what the slice theorem says! For me, it's the big ideas that make this subject exciting.

2026-08-31 16:36:19 CEST

- reply

- It's tough, but I've decided it's better to make students love a subject than pack their heads with knowledge about it. So for a two-hour minicourse I'd leave out all the hard stuff and put in all the exciting and mysterious stuff and give them ways to learn more on their own.

2026-08-31 15:19:52 CEST

- reply

- yes, I think their ornerniness is superficial, but they are deeply orderly.

2026-08-31 15:18:45 CEST

- reply

- thanks for all the links! I came across Zagier's quote when rereading Mazur's book Prime Numbers and the Riemann Hypothesis, which I'm giving as a birthday present to someone in his prime.

2026-08-31 12:23:41 CEST

Don Zagier wrote:
“There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that they are the most arbitrary and ornery objects studied by mathematicians: they grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout.
The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision."
At least if the Riemann Hypothesis is true! His expository paper is full of pictures:
• Don Zagier, The first fifty million prime numbers, https://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BF03039306/fulltext.pdf

2026-08-31 10:17:19 CEST

- reply

- Agreed. If you study Hodge theory for differential forms you'll see this functional plays a big role.
(Btw, I was very confused for a while about how the term "Hodge theory" takes on a deeper meaning on a complex manifold. For a compact smooth manifold, Hodge theory is a bunch of related results like: every de Rham cohomology class has a unique representative whose Laplacian vanishes. But for a complex manifold the story become much more elaborate, in part because the de Rham class becomes bigraded.)

2026-08-30 17:39:53 CEST

- reply

- how much like it?
"Of course a squid, sure - but an octopus? Never!"

2026-08-30 17:38:32 CEST

- reply

Anyway, I could sink into the details here, but that's not the point at all.
The point is that any sort of algebraic concept I thought I understood - like the octonions, to take a fairly esoteric one - gets splintered into many "forms". The job of classifying and working with all these forms becomes a massive project. Mathematicians write lots of papers about it.
And sometimes I can't help wondering if it's worthwhile! But that's mainly when I'm thinking like a physicist. The papers themselves are often quite beautiful when I let myself slow down and think about them like a mathematician.
If you like big overarching concepts they've got those - like Galois cohomology, which itself is just a tiny toe in the deeper waters of etale cohomology, etc. And if you like intricate, finely woven patterns they've got those - like the actual computation of that H¹(k, G₂) thing I was talking about.
So many times, when I've desperately needed a piece of math for some project I'm working on, it turns out people have worked it out already - and research that would have seemed pointless to me a month ago seems precious now.
But still it's sort of scary how BIG mathematics is.
(4/n, n = 4)

2026-08-30 17:23:53 CEST

- reply

Okay, but that all actually makes some sense, at least to me. I'm trying to explain how things get scary. Say we try to classify octonion algebras over a field k. There must be some lowbrow way to do it, but the slick way is: you get one for each element of H¹(k, G₂).
And what the hell is that? It's the first Galois cohomology of k with coefficients in G₂. And what's G₂? For people like me who just got out of kindergarten, it's the automorphism group of "the" octonions, which is a 14-dimensional Lie group. But for these smarty-pants algebraists, it's an algebraic group: it's the automorphism group of the split octonions over k.
Actually this makes some sense to me, but only because I worked for six months trying to understand "Galois descent". If you're working over a field that's not algebraically complete, any sort of algebraic gadget - like the octonions - will come in many "forms". All these forms become the same when you expand your field of numbers to be big enough to solve all polynomial equations. So that's where your analysis should start. This means going to the the "algebraic closure" of k, say K. The field K has lots of automorphisms that don't move elements in k, and the group of those is the "absolute Galois group" of k.
This group G acts on the category of algebraic gadgets we're interested in - in this case "octonion algebras over K". H¹(k, G₂) is a way to compute the set of "homotopy fixed points" of this group action. The idea is that if you have an octonion algebra over k, you can think of it as an octonion algebra over K that doesn't change when you apply transformations in the Galois group to get a new one.
(3/n)

2026-08-30 16:57:54 CEST

- reply

Let me keep it really simple: quaternion algebras. You can define them over a field, or more generally over a ring, or more generally over a variety, or more generally over a locally ringed space... and probably more generally than that! But let's keep it simple and work over a field.
What does that even mean? It means you get to choose which field you're taking your "numbers" to live in. It could be the field of real numbers, complex numbers, rational numbers, p-adic numbers, a finite field, or zillions of other choices. You then say you're "working over" that field. And then - depending on what field you're working over - you can have one or many quaternion algebras.
Over the real numbers there are just two. There are the quaternions I grew up with as a child, where i²=j²= −1 and ij = -ji. Then there are the "split quaternions", where you have i²=-1 but j²=1, and again ij = -ji The split quaternions are just the algebra of 2×2 real matrices in disguise, so this name may seem a bit pretentious! But over other fields, you get more quaternion algebras. There's a general definition of them, but you can prove it's enough to look at ones that obey relations
i² = a, j² = b, ij =-ji
for some nonzero a and b. If your field has a square root of every number, it doesn't matter what a and b are, since you can just replace i with i/√-𝑎 and j with j/√-b and you get
i² = -1, j² = -1, ij =-ji
and you're back down to the good old quaternions. But if your field is something like the rational numbers, you can't pull this trick! So there's a quaternion algebra over the rationals with
i² = -3, j² = -1, ij =-ji
but it's subtly different than the good old one.
(2/n)

2026-08-30 16:50:04 CEST

For a mathematical physicist such as myself, for whom "number" means a real and complex number, working with an algebraist is a bit hallucinatory. I say "the octonions" and they say "which octonions?" For them, an octonion algebra is a general concept. If I try to understand it and how you classify these things, I'm quickly sucked into etale cohomology and other eldritch horrors. Actually I'm starting to like this stuff - but I don't really understand it - so it's like being in a dream where you're trying to walk to the bathroom but for some reason you first need to clamber down a ladder while holding an octopus.
(1/n)

2026-08-29 22:09:13 CEST

- reply

- In blogs I never put punctuation in displayed equations, even when the strict rules of grammar demand it, because I want a clean look. Journals typically demand periods at the end of sentences so I've decided to use them in my papers. The colon is also what grammar demands. You can write.
And this is what he saw: a dog.
but not
And this is what he saw a dog.

2026-08-29 17:52:16 CEST

- reply

- string theorists usually have 6- or 7-off errors, getting 10 or 11 spacetime dimensions when we appear to live in 4. Getting down to just one off is considered progress.

2026-08-29 10:02:20 CEST

- reply

- physicists tend to say central force here, not central force field. For example the Wikipedia article:
https://en.wikipedia.org/wiki/Central_force

2026-08-28 20:32:10 CEST

- reply

- thanks, this is the goods. Apparently put there by one Boris Alexeev. I really want to see how they dealt with the fact that the proof relies on the 6d smooth Poincare conjecture. Did they include a proof of that, or use it as a kind of "axiom" (unproved lemma)?

2026-08-28 20:30:40 CEST

- reply

- I don't know what they did to screw this up. Someone sent me an email me asking me to check the proofs, and I reported this apparent error. I'll see what they see!

2026-08-28 20:16:05 CEST

- reply

- where can I see some evidence that someone at OpenAI has formlized this proof? I'm not doubting you: I just want to see!!!

2026-08-28 20:00:37 CEST

- reply

- most physicists believe that not yet understood quantum gravity effects become important when the curvature of spacetime becomes high, so that instead of having 'infinities' at the Big Bang and in black holes, we instead have something mysterious and new. There are many theories about what this could be.
As for whether the 'strength of gravity' becomes small near the center of a neutron star, that really depends on what you mean by the strength of gravity. If you take a Newtonian outlook and talk about the 'force', that becomes small. But the curvature of spacetime does not become small.
Ignoring quantum gravity: I don't think the singularity of a black hole forms at the surface in general relativity. But it's good to remember that in in general relativity, the singularity in a black hole is not at all like a point: it's a 3-dimensional spatial surface. As you fall into the black hole, the singularity is not in front of you. It's in your future.

2026-08-28 19:24:39 CEST

- reply

- no, they can't.

2026-08-28 19:04:09 CEST

- reply

- I don't know. I just made up these octonion bundle constructions yesterday, and I haven't found anyone talking about them.

2026-08-28 19:02:59 CEST

- reply

- yes, I think this SL(3,ℝ)⊂𝐺₂' is the noncompact version of the inclusion SU(3)⊂𝐺₂.
I guess fixing an imaginary split octonion i determines a splitting 𝕆'=ℝ⊕ℝ³⊕ℝ³*⊕ℝ where the two 1-dimensional summands are spanned by 1 and i, respectively, and the two 3-dimensional summands are the two 3d irreps of the subgroup of split octonion automorphisms preserving i. If those two irreps are inequivalent (the defining rep and its dual), then there's no choice about the splitting.

2026-08-28 18:17:31 CEST

- reply

- so he got beaten up by Nazi thugs but he got a lemma named after him.

2026-08-28 18:09:52 CEST

- reply

- if you take the Cayley-Dickson construction of the octonions here:
https://en.wikipedia.org/wiki/Split-octonion#Cayley%E2%80%93Dickson_construction
you get the ordinary octonions if ℓ²=−1 and the split octonions if ℓ² = +1. So, if you complexify them you get the same thing. For example, if you start with the complexified octonions, with ℓ²=−1, you can define ℓ' = iℓ, and the square of this will be +1.

2026-08-28 18:03:22 CEST

- reply

But there's more:
To define the split octonions, you can replace ℝ³ with any 3-dimensional real vector space V with an inner product (taking place of the dot product) and orientation (so you can choose your right-hand rule for the cross product).
But in fact you need less! Notice that we only take the dot product of a guy of type v (or v') with a guy w' (or w) - never two v's, or two w's. So we can say v, v' ∈ V and w,w' ∈ V*, and drop the inner product on V. Then we just need the cross product of two guys in V to be a guy in V*, and vice versa. To get this, we just need to choose a volume form: that is, a nonzero element of Λ³V*, say ω. Then we can define the cross product of v, v' ∈ V to be ω(v, v', -), which is an element of V*. And we can go further and define a cross product of guys in V*, which will be an element of V.
So, you can construct an algebra isomorphic to the split octonions starting from any 3d vector space equipped with a volume form.
So, if you have a 3-dimensional manifold with a volume form, each tangent space is a suitable starting point for building an algebra isomorphic to the split octonions. You get a 'split octonion bundle' over your manifold!
And can copy this over the complex numbers! We can start with a 3-dimensional *complex* manifold with a nowhere vanishing holomorphic *complex* 3-form on it, and get a bioctonion bundle over it.
But a 3d complex manifold with a nowhere vanishing holomorphic *complex* 3-form is called a Calabi-Yau manifold - or more precisley a Calabi-Yau three-fold.
They're beloved in string theory. But it's kinda cool how any Calabi-Yau threefold has a canonical bioctonion bundle over it.
(2/n, n = 2)

2026-08-28 17:57:34 CEST

It's amazing to think that Max Zorn, the guy with the famous lemma, died as late as 1993. He spoke with a rasp ever since Nazi thugs battered his throat in 1933. And in 1935 he discovered a way to build the split octonions as 2×2 matrices with two scalar and two vector entries. This is called Zorn's vector-matrix algebra.
The 'split octonions' are different from the usual octonions - but they both become the same when you complexify them, that is, tensor with ℂ to switch from working with real numbers to complex numbers. The complexified octonions are sometimes called 'bioctonions', and I've learned to love them because one generation of fermions in the Standard Model is neatly captured by a pair of bioctonions. That's the same thing as a spinor in 10 dimensions!
(1/n)
https://media.mathstodon.xyz/media_attachments/files/117/173/894/242/855/729/original/8c3281815024fca5.jpg

2026-08-28 17:24:28 CEST

- reply

- no, the period is there because it's the end of a sentence. Math journals always demand periods at the end of sentences, even when a sentence ends with an equation. So that is how I write.
The actual problem is quite visible, and someone has finally spotted it.

2026-08-28 17:04:22 CEST

- reply

- YES, YOU ARE THE WINNER!
THEY LEFT OUT THE SPACES BETWEEN THE WORDS IN THE TITLE!
Compare an earlier article I wrote, where there are spaces between the words.
I never expected a mistake like this....
https://media.mathstodon.xyz/media_attachments/files/117/173/742/797/591/550/original/e311664ec01e4757.jpg

2026-08-28 17:00:09 CEST

- reply

- heh, no.

2026-08-28 16:58:22 CEST

- reply

- 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.