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• Post #4496817

  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/

• Post #4470326

  I hope you're enjoying the weather this summer. I expect it will be one of the coolest summers in the next two decades!

• Post #4446771

  𝐇𝐀𝐑𝐃𝐂𝐎𝐑𝐄 𝐏𝐇𝐘𝐒𝐈𝐂𝐒 𝐏𝐎𝐒𝐓 It's cool how the SU(5) grand unified theory uses a Higgs to break down the symmetry to the Standard Model gauge group. The Standard Model gauge group is really S(U(3) x U(2)): that is, the subgroup of SU(5) that preserves a 3d subspace of ℂ⁵, or equivalently the 2d subspace orthogonal to that. So the desired symmetry breaking amounts to a choice of 2d subspace in ℂ⁵: that is, a point in a certain Grassmannian called Gr(2,5). So we're looking for a field, a &#...

• Post #4445248

  I just learned that John Lennon made a half-hearted pass at my cousin Joan while the Beatles were holed up in Hollywood during their 1965 tour. Later John and George Harrison dropped acid with Ringo but Paul refused. Peter Fonda was there. He tried to calm down George, who thought he might be dying, by talking about when he shot himself as a kid, saying "I know what it is like to be dead". But this just freaked out everyone. Lennon urged him to drop the subject, saying "Who...

• Post #4423255

  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 noticeably on its...

• Post #4382491

  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

• Post #4360749

  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...

• Post #4277362

  An article at NikkeiAsia claims that hidden debt at Alphabet, Microsoft, Amazon, Meta and Oracle has shot up to about $1.65 trillion this year, due to huge AI investments. 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 mention it in annotations to their quarterly financial statements. So you have to dig a bit to find it. This includes money that tech companies b...

• Post #4234878

  Brand new old stuff! Tiny ivory figurines of birds made around 40,000 BC, recently found 1.5 meters apart in a cave called Hohle Fels in Germany. This art is typical of the Early Aurignacian culture - Cro-Magnons known for their fine flint blades, ivory carvings, and some of the earliest cave paintings. The glaciers had receded in Germany by this time: it was a steppe-tundra with regions of permafrost that supported mammoth, reindeer, horse, and bison. It was a time of rapid climate...

• Post #4142729

  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...

• Post #4097712

  I would go into biology if I could write a paper with a title like "Dynamic instability of Asgard archaeal tubulins". So intense! 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 toge...

• Post #4060072

  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* interested in the battle for the soul of ma...

• Post #4050997

  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 [pⱼ,qₖ] = −iℏδⱼₖ, all other commutators zero. Concretely, it consists of polynomial differential operators on ℝ³ - roughly...

• Post #3913862

  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)

• Post #3803022

  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!

• Post #3736714

  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 surfac...

• Post #3668246

  RE: https://mathstodon.xyz/@foldworks/116879349561576734 The mathematical skill of those old tilers was astounding.

• Post #3657517

  I don't like algebra anymore. (1/2)

• Post #3624877

  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.

• Post #3555357

  Ireland's oldest pub, established in 1198, is called The Brazen Head. And I just learned it's named after a very early dream of AI. In stories going back to the Middle Ages, a "brazen head" was a brass head that could correctly answer any question put to it - or maybe just yes/no questions. Often the ownership of this head was attributed to a famous scholar like Roger Bacon, or Albertus Magnus, or Robert Grosseteste. Apparently these early proto-scientsts seemed like wizar...

• Post #3547552

  If you write an 878-page treatise revolutionizing all aspects of natural science, from grammar and logic to mathematics, physics, and philosophy, you get a grubby little lane in Oxford.

• Post #3525899

  The new CEO of the Wikimedia Foundation worked at J.P. Morgan and Lehman Brothers. The Foundation has now fired a longtime lead developer and disbanded the team whose job was to listen to volunteers. Most of the people they fired were union organizers. Wikipedia’s editors are now threatening to strike. To stand in solidarity with them, sign the petition: https://en.wikipedia.org/wiki/Wikipedia:Wiki_Workers_United_solidarity For more, read on! (1/2)

• Post #3447536

  The publisher Springer claims it can only discuss why it retracted some of Planck's papers with... Planck! But until they hold a seance, they are happy to sell you completely redacted versions: blank pages for $39.95 or more each. https://www.science.org/content/article/why-have-papers-one-history-s-most-famous-physicists-been-retracted?utm_campaign=News

• Post #3431352

  While France roasted and England sweltered, Scotland had some fine sunny days - a rarety. Yesterday, however, a cold fog rolled into Edinburgh.

• Post #3398352

  "Derived stacks" are a powerful modern framework for geometry. But what's the idea behind them? Proceeding in steps: A variety or scheme X has a tangent space at each point, and these assemble to form the tangent sheaf TX. If X is smooth is locally free: a vector bundle. But at singularities, the dimension of the tangent space jumps up! For example, the tangent space of this curve y² = x³ has dimension 1 except at the origin, where it jumps up to 2. (1/n)

• Post #3352895

  Back in 2011 when I worked in Singapore at the Centre for Quantum Technologies, I visited Angkor Wat and the surrounding temples in Cambodia. On the outskirts, overgrown by the jungle, is Ta Prohm, made famous by that Tomb Raider movie. I thought I might as well share my photos. I generally try to minimize travel these days, and I try not to glamorize travel either. But it also seems a bit silly to sit on these photos forever and die with them on my laptop. So here they are. Not grea...

• Post #3301722

  The JUNO detector, a huge sphere full of 20,000 tonnes of transparent organic goop, is detecting antineutrinos produced by two nuclear reactors each located the same distance away. After just 2 months, it measured some things about neutrino masses better than ever before! A paper was just published on this. We're trying to understand the masses of the 3 neutrinos. A big problem is that most experiments can't detect the actual masses, only fancier things that are easier to measure....

• Post #3071332

  More geometry in music: the 'cube dance'. This shows all the chords called major, minor and augmented triads. Two of these chords are connected by an edge if they differ by just one note. We get 4 cubes connected at their corners! Augmented triads are choke points: if you want to get from one cube to another, you have to go through an augmented triad. (1/n)

• Post #2854440

  Bad at lab work? Don't give up hope. The man who discovered the electron through some of the most delicate vacuum-tube experiments ever performed was a complete disaster with his hands.

• Post #2742490

  My friend's son was a math prodigy, and now he has one-upped OpenAI, which recently wowed the world by disproving a conjecture due to Erdos. OpenAI found a way to put n points on the plane with more pairs at distance 1 from each other than anyone thought possible. Will Sawin quickly found a way to get even more pairs at distance 1. https://arxiv.org/abs/2605.20579