There's a beautiful way to combine symmetry (that is, group theory) with calculus (that is, differential geometry) in a single theory, which not surprisingly is fundamental to a lot of modern physics. It's called the theory of Lie groups. I've always loved it.

But when you try to do calculations with Lie groups, you're instantly led to a trick which is a grand generalization of a slide rule. A slide rule converts multiplication to addition. You'd like to do this with a Lie group, too. It's harder, because multiplication doesn't need to commute: in a Lie group you can have ab≠ba.

But you can still manage. Not surprisingly you're led to think about ab-ba all the time - and this leads to an operation called the Lie bracket,

[a,b] = ab-ba

and gadgets called Lie algebras. Every Lie group has a partner called a Lie algebra, and while the Lie group is more beautiful and conceptual, when you need to calculate you whip out your slide rule: its Lie algebra.

(Experts will roll their eyes at this oversimplification, but tough!)

I dutifully studied Lie algebras in grad school, but I didn't really love them. You need technical things called roots and weights, which seemed rather dry.

Years later, I grew to like these ideas, thanks to conversations with James Dolan, who has a way of making things fun. But I still avoided doing serious calculations using this technology.

Now all of a sudden I need to do them - to write a paper relating the Lie algebra 𝔢₇ to particle physics! 𝔢₇ is a so-called 'exceptional' Lie algebra, meaning that it's connected to the octonions. I like calculating with octonions. But for now I need to calculate using roots and weights. 😩

I'll learn to like it. I will!

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 borrow for servers and GPUs that haven't yet been delivered yet, 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.""

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 oscillations - the Dansgaard-Oeschger cycles - swinging between mild and colder weather on timescales of centuries to a couple thousand years.

It's really fun to imagine what it would be like to be an Aurignacian!

For more, read this:

https://uni-tuebingen.de/en/university/news-and-publications/press-releases/press-releases/article/two-bird-figurines-found-in-world-heritage-site-of-hohle-fels-are-declared-find-of-the-year/

You can get it in German or English.

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 mathematics that is heating up.

On the one hand, some of these LLMs are getting very good at solving math problems - when used by someone who knows what they're doing. The person I spoke to was in awe. He could ask Fable hard questions about generalized cohomology theories and it could compute the answers using clever tricks without being told which tricks to use.

On the other hand, 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/

Wildberries is not just the Amazon of Russia. It sells a wide range of military gear. And shockingly, Russian soldiers often have to buy their own body armor, combat vests, helmets, boots, clothing, torniquets, first-aid kits and even drones. Sometimes families buy this stuff and send it to their sons. And a lot of this gets bought at Wildberries. So the recent Ukrainian attacks destroying at least 6 Wildberries warehouses are not only hurting the Russian economy and making ordinary Russians more aware of the war: they're a pre-emptive attack on the big mobilization that Putin may call up. And they're probably just getting started.

https://www.youtube.com/watch?v=7Ox9AWaFhj0

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:

https://arxiv.org/abs/2607.10833

I will probably explain this in a bit, but I'm still recovering from writing the paper! We sprinted to finish it in time for our workshop on this stuff:

https://indico.ph.ed.ac.uk/event/420/overview

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.

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

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

The Korean branch of Starbucks used AI to create a marketing campaign, but whoops - this campaign celebrated the May 18th massacre of Korean civilians peacefully protesting a military coup in 1980!

Nobody checked before they launched the ad campaign.

Result: massive protests and boycotts, legal actions against Starbucks, collapse in stock prices, etc.

Linkedin is annoying, but this article is great, and you can read it without registering. Just click the X on the top right of the box to make it go away:

https://www.linkedin.com/pulse/from-national-gift-insult-how-starbucks-korea-destroyed-tz0ve

This is a view of the sky through a 6-meter-wide hole at the top of James Turrell's huge new installation called The Dome. The color of the light inside this dome keeps changing - and so does the color of the sky, as the day goes by. When you look at a colored disk on a colored background, both colors affect how the disk looks.

"Outside, the Danish evening sky was a mottled gray. But to the viewers gazing up through the opening inside the huge dome that is the American artist James Turrell’s latest creation, the sky was a flat circle of pure color that morphed seamlessly from cobalt blue to teal green, to a yolk-like yellow and back to blue. With each cycle, the colors grew subtly deeper, gradually reaching an intensity that compelled several visitors to lie down directly below the aperture, as if waiting for the mothership to beam them home."

Quote from Lisa Abend at the New York Times (gift link): https://www.nytimes.com/2026/06/19/arts/james-turrell-aros-aarhus.html?unlocked_article_code=1.rlA.jTIJ.r9iQQbaM_4d2&smid=url-share

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 great quality, but a great place.

(1/n)