John Carlos Baez

I'm a mathematical physicist who likes explaining stuff. Sometimes I work at the Topos Institute. Check out my blog! I'm also a member of the n-Category Café, a group blog on math with an emphasis on category theory. I also have a YouTube channel, full of talks about math, physics and the future.

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nprofile1qqs0wdrwk2peq2k6n5suzzdf86p39rvlslv0elns4kqekwlj4kduu9spz4mhxue69uhhyetvv9ujumt0wd68ytnsw43q4sswgn

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Last Notes

2024-11-17 17:29:53

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@nprofile…wlc0 - actually I think people would accept other systems that made it easy for them to find interesting people. It's quite hard here, and I dislike that, but I stick around because I'm tired of social media run by corporations.
@nprofile…ccu8

2024-11-17 17:24:59

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@nprofile…wlc0 - You asked "Why do people need followpacks?" and I answered. Perhaps your intent was to persuade people to be different (by insulting them, calling them "sheep").
@nprofile…ccu8

2024-10-18 04:34:48

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@npub1305…k70l - super-cool!
This statue of a Buddha was found in a Viking's grave in Sweden. It dates from about the 6th century AD and was probably made in Kashmir.
http://irisharchaeology.ie/2013/12/the-helgo-treasure-a-viking-age-buddha/
https://media.mathstodon.xyz/media_attachments/files/113/326/560/541/148/109/original/62d67fa9e7f6d296.webp

2024-10-15 16:10:45

The Europa Clipper finally took off! It'll reach this icy, water-filled moon of Jupiter in 2030. But it won't orbit that moon. Instead, it will fly by Europa 44 times, tracing out these wonderful tracks.
Why not orbit Europa? There's too much radiation that close to Jupiter, and the Clipper can collect data much faster than it can transmit it to Earth. If it orbited Europa it would fry before it could tell us very much!
Thus, the Clipper will spend most of its time farther from Jupiter, transmitting data to Earth. Using gravity assists from 2 other moons, it will occasionally swoop down, zip past Europa, and make more measurements.
In its 3.5 year life, it will get 2.8 megarads of radiation from Jupiter. This is about 280 times the amount that would kill you! Jupiter is pretty, but don't try to live nearby.
(1/n)
https://media.mathstodon.xyz/media_attachments/files/113/312/253/329/856/740/original/3031a2f4b7e17066.jpg

2024-10-15 15:36:10

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@npub1ulf…sjfm - there is one, but they've never gotten around to publishing it.

2024-10-13 16:13:27

Hey, they did it! People have known for a long time that in 4 dimensions you can pack equal-sized balls so that each touches 24 others. In 2008 someone showed that's the best you can do. But in April this year, three mathematicians showed there's essentially just 𝑜𝑛𝑒 𝑤𝑎𝑦 you can accomplish this!
This way is easy to describe. Take the points whose coordinates are all integers, together with those whose coordinates are all integers plus 1/2. Use each of these points as the center of a ball of radius 1/2. Then each ball touches 24 others. This pattern is called the 𝗗𝟰 𝗹𝗮𝘁𝘁𝗶𝗰𝗲.
The new result shows that if you pack equal-sized balls in 4 dimensions so that each touches 24 others, you're 𝑓𝑜𝑟𝑐𝑒𝑑 to create a version of the D4 lattice - possibly rotated, translated and/or rescaled.
This sort of thing is hard to show: it took a lot of fancy math and two weeks of calculation on a standard desktop computer. I believe only other dimensions where we know a result like this are 1, 2, 8 and 24. In dimension 3 it's just false: you can pack equal-sized balls so that each touches 12 others, and this is the best you can do, but there are uncountably many different ways to do it!
https://arxiv.org/abs/2404.18794

2024-10-11 04:54:17

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@npub1g7v…scf3 - a bunch of people were wondering about such things on the Zulip.

2024-10-09 15:34:09

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@npub1jzg…hc3a - a lot of people don't like AI and they think machine learning researchers have essentially managed to steal the physics Nobel.

2024-10-09 01:06:31

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@npub10l8…ath2 - I don't really care much who wins Nobel prizes, but yes a lot of my favorite physicists never won Nobels. Heck, Einstein didn't win one for special relativity, and he didn't win one for general relativity - just one for understanding photons, which is fine as far as it goes.
Btw, actually the point of LIGO was not to test our theory of gravitational waves: that would be a lot of money to check something we all believed in. The real point was to revolutionize astronomy by creating a new kind of telescope: we've seen a lot of unexpected things, like lots of black holes of masses 30-60 times the Sun, and we don't know where they came from.

2024-10-08 23:36:43

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@npub10l8…ath2 - You mean like LIGO's detection of gravitational waves? The physics was worked out long before, and we all knew gravitational waves existed - Hulse and Taylor had already won the Nobel for that - so one could argue that all LIGO did was an (magnificent, awesome) application of physics. But apparently the Nobel committee thought that was fine.
Or how about the 1912 physics prize, to Gustaf Dalén, for "for his invention of automatic valves designed to be used in combination with gas accumulators in lighthouses and buoys"? 😆

2024-10-08 22:52:57

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@npub10l8…ath2 - but I didn't say "neural networks research is physics". I say "it all started with physics", and gave evidence for how Hopfield and Hinton were using ideas from statistical mechanics. Yes, neural network research has grown beyond this now - it's now its own subject. I don't think anyone is claiming otherwise.
Wikipedia on Hopfield networks:
"The Sherrington–Kirkpatrick model of spin glass, published in 1975, is the Hopfield network with random initialization. Sherrington and Kirkpatrick found that it is highly likely for the energy function of the SK model to have many local minima. In the 1982 paper, Hopfield applied this recently developed theory to study the Hopfield network with binary activation functions. In a 1984 paper he extended this to continuous activation functions. It became a standard model for the study of neural networks through statistical mechanics."
Wikipedia on Boltzmann machines:
"They are named after the Boltzmann distribution in statistical mechanics, which is used in their sampling function. They were heavily popularized and promoted by Geoffrey Hinton, Terry Sejnowski and Yann LeCun in cognitive sciences communities, particularly in machine learning, as part of "energy-based models" (EBM), because Hamiltonians of spin glasses as energy are used as a starting point to define the learning task."

2024-10-08 16:51:32

Yay! John Hopfield and Geoffrey Hinton won the physics Nobel for their work on neural networks... work that ultimately led to modern-day machine learning.
Some of you are wondering why this counts as physics.
In the 1980s, Hopfield invented the 'Hopfield network' based on how atomic spins interact in a chunk of solid matter. Each atom's spin makes it into a tiny magnet. Depending on the material, these spins may tend to line up, or point opposite to their neighbors, or interact in even more complicated ways. No matter what they do, at very low temperatures they tend to minimize energy.
A Hopfield network is a simulation of such a system that's been cleverly set up so that the spins store data, like images. When the Hopfield network is fed a distorted or incomplete image, it keeps updating the spins so that the energy decreases... and works its way toward the saved image that's most like the imperfect one it was fed with.
Later, Geoffrey Hinton and others generalized Hopfield's ideas and developed 'Boltzmann machines'. These exploit a discovery by the famous physicist Boltzmann!
Boltzmann realized that the probability that a chunk of matter in equilibrium has energy E is proportional to
exp(-E/kT)
where E is its energy, T is its temperature and k is a number now called Boltzmann's constant. When the temperature is low, this makes it overwhelmingly probable that the stuff will have low energy. But at higher temperatures this is less true. By exploiting this formula and cleverly adjusting the temperature, we can make neural networks do very useful things.
There's been a vast amount of work since then, refining these ideas. But it started with physics.
https://arstechnica.com/ai/2024/10/in-stunning-nobel-win-ai-researchers-hopfield-and-hinton-take-2024-physics-prize/

2024-10-05 16:18:12

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@npub1t76…4l63 - what's the idea of 'tidal squeeze' and 'tidal stretch'? I guess these are forces that appear when we have a central gravitational field and we're working in a rotating coordinate system, maybe with its origin at a point that's rotating in a circular orbit around the center?

2024-10-04 04:46:41

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@npub12n4…5dmn - interesting puzzle! To me the obvious approach is to start with some low 𝑛.
Let's compare SO(3) to S² × SO(2). Since π₁(SO(3)) ≅ ℤ/2 while π₁(S² × SO(2)) ≅ ℤ, they're not homeomorphic. By the way, SO(3) ≅ ℝP² since it's SU(2)/±1 and SU(2) is a 3-sphere.
Next let's compare SO(4) to S³ × SO(3). Since π₁(SO(n)) ≅ ℤ/2 for all n ≥ 3 now both these spaces have π₁ ≅ ℤ/2 so the above fundamental group argument no longer works, and we see it will never work to distinguish SO(n+1) from Sⁿ × SO(n) for n ≥ 3: they both have π₁ ≅ ℤ/2.
So we need a deeper idea.
(1/n)

2024-10-03 17:17:44

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"“The Neptunian Ridge stands tall above the Desert and Savanna. It provides us with a key to understanding the physical mechanisms shaping the Desert,” points out Vincent Bourrier. Most Neptunes may be distributed over the Savanna and Desert early in their life by migrating within the disk in which they formed. The existence of the Ridge suggests that some Neptune-size planets are brought to this region by a type of migration called high-eccentricity migration, which occurs later in their life and allow them to survive erosion from the stars.
These migration processes, coupled with photoevaporation, likely shape the distinct features observed in the Neptunian landscape. The similarities between the Neptunian Ridge and another feature in the exoplanet distribution, the hot Jupiter pileup, suggests that similar evolutionary processes may influence both groups of planets."
(2/2)
https://astrobiology.com/2024/09/hidden-exoplanets-between-the-neptunian-desert-and-savanna.html

2024-10-03 17:14:10

𝗧𝗛𝗘 𝗡𝗘𝗣𝗧𝗨𝗡𝗜𝗔𝗡 𝗗𝗘𝗦𝗘𝗥𝗧
Once we knew just one solar system. Now we can see 𝘩𝘶𝘯𝘥𝘳𝘦𝘥𝘴 of solar systems and learn how they work on average! We are learning amazing things, like this:
Here's a chart showing planets whose year is 25 Earth days long or less. These planets are very close to their suns. There's a big blob of planets with mass 10-20 times the Earth's mass, whose year is just 1-10 days long! These are called 𝗵𝗼𝘁 𝗡𝗲𝗽𝘁𝘂𝗻𝗲𝘀.
But there's an interesting shortage of slightly less massive planets that are quite close to their sun. This is called the 𝗡𝗲𝗽𝘁𝘂𝗻𝗶𝗮𝗻 𝗱𝗲𝘀𝗲𝗿𝘁. And there's another less dramatic shortage of planets in the 𝗡𝗲𝗽𝘁𝘂𝗻𝗶𝗮𝗻 𝘀𝗮𝘃𝗮𝗻𝗻𝗮. Both these need to be explained.
Now we have a new clue. This September astronomers discovered that the desert and savanna are separated by a region with more planets, which they call the 𝗡𝗲𝗽𝘁𝘂𝗻𝗶𝗮𝗻 𝗿𝗶𝗱𝗴𝗲.
What does it all mean? There are lots of theories, but it will take a while to figure it out.
Here's the new paper:
• A. Castro-González, V. Bourrier, J. Lillo-Box, J.-B. Delisle, D. J. Armstrong, D. Barrado and A. C. M. Correia, Mapping the exo-Neptunian landscape: a ridge between the desert and savanna, https://arxiv.org/abs/2409.10517.
With all this work we need to think about 'selection bias': some planets are easier to see than others, so what we see is not the same as the real situation! So, a lot of this paper is about compensating for selection bias.
For a link to a gentler, less technical article, read on.
(1/2)
https://media.mathstodon.xyz/media_attachments/files/113/244/496/342/306/359/original/8a986be8fc9391fa.jpg

2024-10-01 05:32:37

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@npub1rf7…zwfx - this is cool, but the subtitle "This is very unexpected" is an exaggeration: the issue has been around for a long time:
https://en.wikipedia.org/wiki/Cuspy_halo_problem

2024-09-30 16:34:21

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@npub1z4e…aasw - yes, I hope you read the image description.

2024-09-29 16:02:08

𝗤𝗨𝗜𝗖𝗞! 𝗙𝗘𝗘𝗗 𝗧𝗛𝗘 𝗕𝗘𝗔𝗦𝗧!
Academic publisher Taylor & Francis recently sold many of its authors’ works to Microsoft for $10 million, without asking or paying the authors — to train Microsoft’s large language models!
Taylor & Francis asked their journal "Learning, Media and Technology" to cut peer review time to 15 days — absurdly little time — to crank out more content.
And Taylor & Francis's subsidiary Routledge told staff that it was “extra important” to meet publishing targets for 2024. It moved some book deadlines from 2025 to 2024. Why? To meet its deadline with Microsoft.
Another academic publisher, Wiley, made a $44 million deal to feed academic books to LLMs — with no way for authors to opt out. They say “it is in the public interest for these emerging technologies to be trained on high-quality, reliable information.”
When you publish with one of the big academic publishers, they try to make you sign a contract saying they can do whatever they want with your work. That means anything.
Hat-tip to @npub1yt3…wmv6 for pointing this out.
These articles have links to original sources:
https://pivot-to-ai.com/2024/08/04/more-academic-publishers-are-doing-ai-deals/
https://pivot-to-ai.com/2024/09/28/routledge-nags-academics-to-finish-books-asap-to-feed-microsofts-ai/
https://media.mathstodon.xyz/media_attachments/files/113/221/626/482/836/881/original/b380b5ccede93fcd.jpg

2024-09-26 21:06:31

Professor emeritus blues:
You know you've been retired a while when the department staff accidentally puts some new lecturer's photo up with your name on it, because they don't remember who you are.
https://media.mathstodon.xyz/media_attachments/files/113/205/883/520/123/142/original/0547f399b930ed86.jpg

2024-09-20 06:12:47

What would happen if the Moon broke into seven pieces? It would not be good. Check out this simulation, including a video.
https://jasmcole.com/2017/09/20/the-moon-blew-up-without-warning-and-for-no-apparent-reason/

2024-09-18 00:09:11

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@npub1d8z…zgls - "no" to the first question, in the math department at U. C. Riverside.

2024-09-17 20:23:56

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@npub12nw…uegw - I used the "explosion" feature in Irfanview, setting it to different levels in different patches of the picture.

2024-09-17 16:46:23

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@npub18vx…zvkl - if it's easy, why do you have to steel youself for it? 😈

2024-09-16 15:19:49

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@npub1sqt…hmh2 - beautiful!

2024-09-13 23:38:25

This feature in Hudson Bay is probably 𝑛𝑜𝑡 a large meteor crater. It's called the Nastapoka Arc. While the dark red part of the circle really is the coastline, the rest of the circle is not anything real.
Details:
https://en.wikipedia.org/wiki/Nastapoka_arc
https://media.mathstodon.xyz/media_attachments/files/113/132/859/294/301/181/original/abfcc15cc37dda23.png

2024-09-13 16:08:36

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@npub17q0…je6u - if you block the email address and/or journal/conference name for each one, does it help at all? Or are they mostly new instances?

2024-09-13 01:43:08

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@npub1cd3…paq7 - you meant to say rig.

2024-09-10 21:17:08

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@npub1ez3…xnxl - I was in a debate society for a little while, and I've never seen an official debate where the debaters get to ask each other questions. Maybe you're thinking of a 'discussion'?

2024-09-10 14:20:09

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@npub1crf…qu6y - presumably you are thinking, talking and pacing around a lot?

2024-09-09 15:12:21

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@npub1a0a…kv03 - yikes!

2024-09-08 19:29:49

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@npub1syx…d6lz - I haven't seen one in my 2007 Honda. Let me just get in the trunk and see....

2024-09-08 19:11:09

I rented a car, but they didn't have the sensible compact I wanted so they gave me Ford Mustang. What the hell does this pull chain in the trunk do? "Pull to let kidnapped victim scamper to freedom?" Is this something Mustang owners regularly need? And why the improbably high arc on the parabola? "Pull to let gremlin jump out and run off gibbering manically?"
https://media.mathstodon.xyz/media_attachments/files/113/103/490/851/026/152/original/d9b55de735b5579b.jpg

2024-09-08 14:52:35

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@npub16sc…hgj4 - yes, indeed it's a bummer.

2024-08-30 14:03:36

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@npub1a0a…kv03 - I'm keeping it real.

2024-08-28 06:17:31

No, there are no "new, exceptionally bright photons". The energy of a photon is still related to its frequency by
E = h ν
Please go back to what you were doing.
https://www.livescience.com/technology/communications/unbreakable-quantum-communication-closer-to-reality-thanks-to-new-exceptionally-bright-photons

2024-08-25 09:34:45

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@npub1kc7…tf4u - give in! Whenever I see a Bayesian I give them a swift kick in the posterior.

2024-08-25 09:33:36

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@npub15v3…luuc - I'd never heard it's name until today, when the New York Times acted like we all knew it.

2024-08-25 08:04:44

What's a "Bayesian luxury yacht" and why does this New York Times article assume we all know what it is?
Is it a yacht whose probability of sinking is computed by Bayesian methods?
Full article free here:
https://archive.is/tjAvY
https://media.mathstodon.xyz/media_attachments/files/113/021/612/342/309/376/original/5032eefa6aee11fa.jpg

2024-08-24 23:50:29

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@npub10gg…a0hf
https://en.wikipedia.org/wiki/Mouse_(set_theory)

2024-08-24 13:04:37

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@npub1lap…elxz - there are lots of theories to test, but so far most of them look pretty implausible. So yes, we should take a break on big particle colliders. There's also this thing called climate change....

2024-08-24 10:54:26

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@npub10l8…ath2 - yeah, I almost stopped reading the thesis in disgust when I saw this:
2.2 Hypercharge Quantisation and Fermat’s Last Theorem

2024-08-24 09:17:09

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@npub1683…qclh - excellent caption!

2024-08-20 09:55:13

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@npub16hx…u6hc - did he seem to be following it?

2024-08-18 11:12:29

Have you ever tried learning physics from a teacher who does calculations without explaining them well? Setting constants equal to 1, then putting them back later using dimensional analysis. Setting terms equal to zero when they're negligible - which you're supposed to know by instinct. And so on. You're just supposed to watch it and pick it up.
It's like trying to learn gymnastics by watching Simona Biles.
https://media.mathstodon.xyz/media_attachments/files/112/982/718/965/079/950/original/527ba85e06bdcf32.mp4

2024-08-17 13:22:46

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@npub1swy…snpj - thanks, your praise is worth more than most.

2024-08-12 08:26:03

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@npub12hx…78rc - it's called Gargannock House, near Stirling in Scotland.

2024-08-11 14:19:24

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@npub1c0s…pcx4 - I haven't found a cellar! But lots of staircases.
https://media.mathstodon.xyz/media_attachments/files/112/943/822/339/915/378/original/c34756ba2d1fc5e9.jpg

2024-08-10 14:07:06

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I explored it today. It led to another stairway, and then....
https://media.mathstodon.xyz/media_attachments/files/112/938/089/098/925/171/original/8538bf42714d8280.jpg

2024-08-09 13:25:15

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@npub10p4…jn3m the "expected" or average momentum of a standing wave is zero, because it's a sum of two or more waves going in different directions, whose momenta cancel out.

2024-08-07 13:32:19

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@npub19g8…9445 - I'm surprised it would take that long. I'm willing to believe Wikipedia, but I should look up estimates of the wave speed and distance.

2024-08-07 12:32:54

Don't call these "protests", the word is "riots".
https://www.bbc.co.uk/news/live/cy8497l7dx8t

2024-08-06 11:12:26

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@npub1r7e…0vz0 - welcome! Follow me if you like fun pop explanations of science and music theory (and, sorry, occasional hardcore math).

2024-08-06 07:30:03

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@npub178l…zhp6 - which lies is Google spreading that benefit my side - the side that's against installing right-wing extremists across the globe? I'm against lies even for a good cause.

2024-08-05 11:33:54

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@npub1nad…2adz - it's not published yet! That's why I said "is coming out with" instead of "has come out with".

2024-08-05 11:08:05

The AMS Notices is coming out with a collection of papers on how to publish and present math. I'm glad I persuaded them to not put my paper first, like they did here.
It would be amusing, but not really a good idea. It looks like a subtitle.
https://media.mathstodon.xyz/media_attachments/files/112/909/083/481/946/456/original/aea3c5911d487a85.jpg

2024-08-04 21:53:57

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@npub19g8…9445 - wow, cool. I hadn't known they'd gotten evidence that the asteroid hit in the spring.

2024-08-04 16:16:18

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@npub1c9m…9xjd - I thought you pretending to be a physicist. Physicists love quoting theorems, but they forget the hypotheses.

2024-08-04 15:55:22

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@npub17hm…7lag - punctured disk.

2024-08-04 15:51:20

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@npub1c9m…9xjd - if two unproved conjecture are true!

2024-08-03 14:59:02

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@npub1kvv…y5yh - are you talking about the origin of life? LUCA came quite a while after that, so it's almost a separate question.

2024-08-03 11:48:47

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@npub1ky2…ghvc - thanks, I hadn't spotted that!
But was there spam?

2024-08-02 13:15:03

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If you don't anything about Æon Flux, here's a show about it - the animations that Peter Chung made, not the later movie with Charlize Theron.
(2/3)
https://www.youtube.com/watch?v=BfOT_-VQfi8

2024-07-31 13:30:15

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@npub1t76…4l63 - but that's all about how *easy* it is to solve the sine-Gordon equation!

2024-07-29 19:36:55

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@npub160h…xddr - groovy! 🎃

2024-07-29 16:12:48

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@npub1jsy…tp7j - Thanks. I'm glad to hear it's sort of addictive, at least for some people!

2024-07-25 14:41:25

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@npub1teu…w5lp - actually not weird, right? They wilt under female scorn.

2024-07-24 13:11:45

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@npub1lau…uqs9 - maybe I will wait 5 or so years. But someday I want to write expository math papers with moving images!

2024-07-24 08:29:52

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@npub1ky2…ghvc - Thanks! That's a heinous error in bright red.

2024-07-23 15:25:07

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Inversions and other variants of the Petrushka chord have a lot of different intriguing sounds!
(2/2)
https://www.youtube.com/watch?v=noJLcu7gHJ0

2024-07-23 13:07:32

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@npub1ky2…ghvc - thanks! Got it.

2024-07-23 08:11:38

https://www.youtube.com/watch?v=r-SUdkc5XI4

2024-07-22 22:10:10

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@npub1drg…7yxw - Kamala Harris has raised $81 million in new funds since Biden stepped down on Sunday.
https://www.bbc.co.uk/news/articles/cxx2d25l634o

2024-07-22 15:55:48

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@npub1tve…ejsx - great! For a lot more on information theory I highly recommend Cover and Thomas' book - it's in the list of recommendations on page 113.

2024-07-22 15:42:02

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@npub1ma3…q8k6 - I bet you'll enjoy the concepts. By the way, I liked your paper about networks and biology... umm, sorry, I've already forgotten the technical terms.

2024-07-22 12:46:21

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@npub19g8…9445 - have fun - and please report typos and places where it's hard to understand!

2024-07-22 12:28:53

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@npub1ntc…9r9g - by the way, that conclusion in red is slightly wrong! I noticed only because you made me look at it again. I've fixed it in the new version available at the same place.

2024-07-22 11:34:29

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@npub1zly…j8gf - those sailors and their quaint superstitions, which people who never sail don't understand.

2024-07-22 06:51:54

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@npub1m2w…yhad - see other comments.

2024-07-21 14:13:03

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@npub1ne0…p3ud - extrapolation.

2024-07-20 16:24:58

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@npub14em…wg0c - yes, dimensional analysis is very powerful!