Nice garden shed, now! But an amazing structure.

1/1² - 1/2² + 1/4² - 1/5² + 1/7² - 1/8² + ⋯ Announcement that Dimitrov and Tang proved this number is irrational... and on April 18th they won the $100,000 2026 New Horizons in Mathematics prize!

1/1² - 1/2² + 1/4² - 1/5² + 1/7² - 1/8² + ⋯

Together with Calegari, Dimitrov and Tang proved this number is irrational... and they just won a $100,000 prize!

It's the biggest advance in irrationality since Apéry showed

ζ(3) = 1/1³ + 1/2³ + 1/3³ + ⋯

is irrational back in 1978.

(1/n)

Ringwoodite: (Mg,Fe)_2 SiO_4 in a crystal made of stacked octahedra with tetrahedral gaps. SiO_4 tetrahedra are shown in blue with MO_6 octahedra in brown, Mg and Fe atoms in orange (at the octahedra centers), and O atoms in red. F From https://home.hiroshima-u.ac.jp/kawazoe/html/Kawazoe03-Crystal-EN.html

Indeed, I've decided to tackle my shocking ignorance of rocks. Why study neutron stars when there's a phase transition from wadsleyite to ringwoodite as we go down 520 kilometers beneath my feet? And it gets very mathematical down there!

Yes! I'm a crusty old guy, but even I have to admit that perioditite rocks!

Why Are Palantir and OpenAI Scared of Alex Bores? | The Ezra Klein Show YouTube video by The Ezra Klein Show

Alex Bores is running for Congress in New York. Ads are coming out attacking him because he worked for Palantir.

These ads are funded by... Palantir.

Why? Because Bores wants to regulate AI.

youtu.be/znKb71kLG5c

Serpentine Barrens This is the Soldiers Delight Natural Environmental Area, a nature reserve in Maryland. The early colonial records of Maryland describe the area as a hunting ground for Native Americans. In 1…

Usually in its guise as serpentine!

Speaking of which....

johncarlosbaez.wordpress.com/2021/02/19/s...

Got it. I replied.

Peridotite is a dense, coarse-grained igneous rock consisting mostly of the silicate minerals olivine and pyroxene. Most of the Earth's upper mantle is peridotite. https://en.wikipedia.org/wiki/Peridotite

People are talking about a lot of things here - but sadly, not about how most of the Earth is made of peridotite.

You will ace it. But if you have any questions about Royden's book, you know who you can ask.

You will start by getting to know the real numbers better than you probably ever did....

Trade it for a better world? Sure, if I could!

PLACES TO INTERVENE IN A SYSTEM (in increasing order of effectiveness) 12. Constants, parameters, numbers (such as subsidies, taxes, standards). 11. The sizes of buffers and other stabilizing stocks, relative to their flows. 10. The structure of material stocks and flows (such as transport networks, population age structures). 9. The lengths of delays, relative to the rate of system change. 8. The strength of negative feedback loops, relative to the impacts they are trying to correct against. 7. The gain around driving positive feedback loops. 6. The structure of information flows (who does and does not have access to information). 5. The rules of the system (such as incentives, punishments, constraints). 4. The power to add, change, evolve, or self-organize system structure. 3. The goals of the system. 2. The mindset or paradigm out of which the system — its goals, structure, rules, delays, parameters — arises. 1. The power to transcend paradigms.

Try Donella Meadows' short essay "Leverage points: Places to intervene in a system", at donellameadows.org/wp-content/u...

"Scientists from NASA, from Japan, from many institutes worldwide who have discovered her work have visited her to learn more, yet there is relatively little followup given its scope and importance except for the remarkable installations for which she has been more or less directly responsible. "Why?" I asked her, as we sat in the huge old mansion that served as headquarters for the Limnology Group, in the midst of a nature preserve near Krefeld in Germany, where she now lives alone until her immanent eviction, due to sale of the property by the Max Planck Institute. Her answer was telling. "Men always reach for technology, for development. They insist it will bring us to higher levels of progress. They haven't the patience to work with slow-growing plants, nor do they understand natural cycles as women do. They see my work as farming, not engineering, so they go away and return to their machinery." "Wait till we run out of water," I countered. "They'll all come running to you then." "It will be too late then," she said matter-of-factly in her rich, low register voice. "Neither these cabinets, nor the research reports in them, nor I, will be around." [Sadly, yet perhaps predictably, Dr. Seidel passed away a few months after her eviction.]" From https://www.ratical.org/LifeWeb/Articles/rushes.html

Käthe Seidel was a pioneer in ecological engineering, using plants like reeds and rushes to clean waste water. This is standard now, but mocked when she started doing it in the 1950s.

Elisabet Sahtouris interviewed her in the 1990s - before her work caught on:

Persistence hunting - Wikipedia

You say our design is wretched but "Humans are some of the best long distance runners in the animal kingdom", able to tire out and kill our prey.

en.wikipedia.org/wiki/Persist...

Nice! The only other place I've seen natural arches is in - no surprise - Arches National Monument in Utah. They say the hot morning sun shining on ice-coated east-facing cliffs makes them crack and flake off. So it's a systematic thing, yet with sometimes insane results.

No, the actual Window Rock is even cooler! Not my photograph:

That is one of my favorite albums ever. Baia is so far ahead of its time, once it gets cooking.

Note the word "quipped".

I've never thought about the Kähler-Dirac equation, just the ordinary Dirac equation and its chiral version, the Weyl equation. The DeWitts were at loggerheads because on a nonorientable manifold we need to use pinor rather than spinor bundles, and the choice really matters.

Today busy working on a paper on how we can learn from nature, for the 2026 UN Human Development Report (which may or may not accept it). Last weekend was more fun, at the zoo in Window Rock, capital of the Navaho Nation. Who wants a selfie when there's a beautiful rock instead?

In good legal system, yes.

In a famous paper, some economists estimated the annual value of the whole biosphere's ecosystem services at $33 trillion per year.

Michael Toman quipped that this is "a serious underestimate of infinity".

www.nature.com/articles/387...

I can imagine other ingredient labels committing the same sin. Maybe wheat and wheat germ, or something. Maybe few logicians have jobs making those labels, though I bet whoever does it has lots of rules to follow.

What's "this"?

We beat Google’s zero-knowledge proof of quantum cryptanalysis Trail of Bits discovered and exploited memory safety and logic vulnerabilities in Google’s Rust zero-knowledge proof code to forge a proof claiming better quantum circuit performance metrics than Goog...

This goes over my head, but it seems to prove Google's original zero-knowledge proof was not a prank:

blog.trailofbits.com/2026/04/17/w...

The topic "quantum mechanics on a Klein bottle" reminds me of a story DeWitt's wife Cecile DeWitt-Morette once told in a talk. She and her husband both did QFT calculations on a nonorientable spacetime and got different answers - because they were using different pinor bundles!

If you're going to bother with supersymmetric theories, you might as well use supermanifolds. For a first pass you can think of one as being a manifold equipped with a bundle of exterior algebras. That lets us neatly handle how fermionic fields anticommute, even classically.

For what it's worth, GPT was prompted not to search the internet while constructing this proof. It could have "cheated" - but anyway, the proof it came up with was quite novel... and far from ideal.

Ohio and Indiana had the greatest mileage of interurbans in the United States, at 2,798 miles (4,503 km) and 1,825 miles (2,937 km), respectively. This map shows the network across the Midwestern United States in 1918. From here: https://commons.wikimedia.org/wiki/File:Electric_Railways_of_the_Central_States_1918.jpg

The good old days, when the midwestern US had a large network of electric railways... wait, what?

Yes: this map shows what it looked like! By 1920 there were 25,000 kilometers of electric passenger trains, called 'interurbans'.

en.wikipedia.org/wiki/Interur...

From https://www.erdosproblems.com/forum/thread/1196#post-5561 : That log was somewhat helpful, but still inconclusive at the most critical components of the problem solving process, which remain frustratingly opaque. My tentative theory is that these models are still quite weak at developing strategy and constructing novel coherent narratives (as opposed to explaining existing, human-generated, narratives, for which they are now rather good at); this may also be related to the tendency of AI-generated proofs (such as this one) to dwell at length on rather routine components of an argument, while not stressing the most original and important aspects of a proof. For me, there are two points in the thought process that would be particularly illuminating to locate. The first, obviously, is where the idea of using the von Mangoldt process first emerged. But dual to this is why this idea was not later abandoned, like so many other ideas generated in the chain of thought. Problem solving is not just about coming up with the one right idea amongst a sea of bad ideas; it is also about the filtering that isolates that one viable idea from all the unviable ideas. This filtering seems to be largely absent in the published chain of thought; and yet the AI was somehow able to perform the remaining technical steps needed to convert the viable idea into an actual proof. Perhaps one has to compensate for the effect of "survivor bias": amongst the Erdos problems alone, thousands of instances of these models are being thrown at these problems, but it is pretty much only the successful (or partially successful) instances which are being reported here, and one possible explanation for the lack of coherent strategic narrative in this instance is that there simply isn't one: it could instead be more of a numbers game of "throwing things repeatedly at the wall and seeing what sticks". Perhaps one experiment which would be helpful would be to [....]

Mathematicians are still scratching their heads about precisely how GPT-5.4 Pro came up with its proof. Terence Tao read a log of its "thinking", and wrote this analysis:

www.erdosproblems.com/forum/thread...

(3/n, n = 3)

The proof involved Markov processes. Then people used Math Inc.'s AI tool called Gauss to formalize the proof, to make sure it was legit. Yesterday, Tao simplified the proof immensely:

www.
erdosproblems.com/forum/thread/1196#post-5521

Markov processes don't show up in his new proof!

(2/n)