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  I've been thinking about life after the ecological collapse. Many of the species we love will be extinct. But probably not liverworts. Liverworts were among the first plants to bounce back after the world-wide rain of molten quartz set all the forests on fire when an asteroid hit the Earth 65 million years ago. They were among the first to bounce back after the much worse end-Permian extinction. They've been around for 420 million years, but with bursts of new species beginnin...

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  I hope folks make lots of copies of this map of 300,000 Flock cameras before Flock gets the "official" map taken down. https://theintercept.com/2026/09/24/how-many-flock-devices-in-united-states-300000/

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  I went to Glasgow to give a talk on the 4d rotational symmetry of the hydrogen atom, but what sticks in my mind is the beauty of ferns, tree ferns, mosses, liverworts and other "primitive" plants on display at the Glasgow botanical garden! (1/2)

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  RE: https://mathstodon.xyz/@highergeometer/117323453342622319 Wow, more progress on the Riemann zeta function! We know in our bones that for all positive integers n, ζ(n)=1⁻ⁿ + 2⁻ⁿ + 3⁻ⁿ + ⋯ is irrational. But it's only been proved for n even, n = 3, .... and now, it seems, n = 5. This may seem small, but it's huge.

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  As the view from Moscow becomes increasingly embarrassing, it seems increasingly likely Putin will start a new more desperate phase of his war against Ukraine. Experts expect him to start a quiet mobilization of troops shortly after tomorrow's parliamentary elections. He seems to want 300,000 - 500,000 new soldiers. Despite increasing payments, recruitment has been failing to keep pace with the rising casualties. We can also expect Putin to step up attacks against Europe. More drones...

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  @rtn@chaos.social - in modern math, an 'associative algebra' is a structure where you can add, subtract, and multiply, obeying certain rules - for example, that multiplication is associative. Often we call this simply an 'algebra'. But a 'Lie algebra' is a different kind of thing, where you can add, subtract, and take the 'bracket' of two elements, obeying some other rules. And then there are other things, like 'nonassociative algebras'. All these...

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  A vector looks like an arrow; anyone studying math or physics in college learns about these. Bivectors are important too - though less widely taught. You can create a bivector by taking the 'wedge product' u ∧ v of two vectors u and v, and you can visualize this as the parallelogram shown at left here. But in fact, an ellipse or any other flat surface with the same area lying in the same plane is just as good a picture of this bivector. Further, you can add bivectors, so there are...

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  I'm fascinated by the laurel forests of La Gomera, one the Canary Islands. They're full of different species of trees with shiny, waxy leaves. Mosses and lichens grow on the branches, and the ground is covered with a green mantle of dripping ferns and liverworts thriving in the permanent damp. The place has a mysterious, prehistoric air. The lower altitudes of this island are deserts, but the upper parts catch fog, known locally as "horizontal rain" (lluvia horizontal)....

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  The Republican Party's midterm convention used the traditional red, white and blue decorations - but added Trump's favorite color: gold. I recommend that they use a new US flag. The traditional colors were red for blood, white for purity and blue for justice. The new version features red for blood, gold for wealth and purple for monarchy.

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  Mathilde Marcolli writes: "What is it about all of these famous and less famous, invariably hard, Big Problems™? There is an undeniable sense in which these questions are interesting. Typically they are because, in the process of thinking about a particularly hard question, a lot of new mathematics gets developed that advances the field as a whole. The Riemann hypothesis is currently unsolved (at the time of this writing), but many people, in the process of thinking about the problem, have...

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  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...

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  In the northern hemisphere, the sun moves clockwise in the sky. This is why clocks, which were based on sundials, have hands that move clockwise. In 2014 the Bolivians finally decided to break free of this colonial legacy. They're in the southern hemisphere, after all! So the clock on their parliament now looks like this. I like it. But it must make a tempting target for counter-revolutionaries.

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  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/

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

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  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 &#...

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  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...

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  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...

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  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

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  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...

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  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...

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  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...

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  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...

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  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...

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  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...

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  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...

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  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)

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  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!

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  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...

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

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