John Carlos Baez

I'm a mathematical physicist who likes explaining stuff. I'm the Maxwell Fellow of Public Engagement at the School of Mathematics and the School of Physics and Astronomy at the University of Edinburgh.

Check out my blog Azimuth! 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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Last Notes

2026-04-28 16:06:28 GMT

BIOMIMETIC TECHNOLOGIES
How can we learn from nature? One of the most obvious ways is to look at natural systems and design technologies based on them. These are called biomimetic technologies. A single example can illustrate some of the issues that arise.
Termites maintain nearly constant internal temperatures in their mounds through a system of channels. They don’t need fans that require power. For a time, it was believed that they used a simple convective cooling system, where hot air rises through the central chimney, drawing in cool air at the base. In 1996, a large office and retail building was built based on this idea: the Eastgate Centre in Harare, Zimbabwe, designed by the architect Mick Pearce [TS]. It has chimneys and ventilation channels that draw cool night air through the building’s thermal mass. It uses roughly 90% less energy for climate control than a conventional building of comparable size! That translates directly into far lower carbon emissions from heating and cooling.
This success inspired emulation. Pearce himself used similar termite-chimney-inspired designs in a Melbourne office building [HB]. More recently the Startup Lions Campus in Kenya, designed by Kéré Architecture on the banks of Lake Turkana, features three tall terracotta-colored ventilation towers modeled after local termite mounds.
(1/n)
[TS] Turner, J.S. & Soar, R.C. (2008). Beyond biomimicry: What termites can tell us about realizing the living building, Proc. I3CON, p. 18.
[HB] Hes, D. & Bayudi, R. (2005). Council House 2 (CH2), Melbourne CBD: a green building showcase in the making. Proceedings of Conference on Sustainable Building South East Asia, pp. 231-241.
https://media.mathstodon.xyz/media_attachments/files/116/483/175/250/666/344/original/7e790bce5fea0853.png

2026-03-27 04:22:31 GMT

As confirmed by a 2020 Supreme Court decision, 15% of Oklahoma is under jurisdiction of the Choctaw Nation. Now the Choctaw have used their power to prevent ICE from getting a big detention center!
https://www.projectsaltbox.com/p/choctaw-nation-buys-former-big-lots

2026-03-21 02:18:23 GMT

The proton, 1836 times heavier than the electron, is made of two up quarks and a down, with two of their spins aligned and one pointing the other way.
The same quarks with all spins aligned give a new particle, the Δ⁺, that's 2411 times heavier than the electron!
But the Δ⁺ is just the first of many 'excited states' of the proton: particles made of two up quarks and a down, but arranged in different ways, with higher energy and thus more mass. They quickly decay, often turning back into a proton.
There are two main kinds:
• If two quarks have spin pointing the same way and one points the other way, you get a particle with total spin
1/2 + 1/2 - 1/2 = 1/2
It could be a proton, but there are lots of others. Any particle of this kind is called an N*⁺.
• If all three quarks have their spins aligned, you get a particle with spin
1/2 + 1/2 + 1/2 = 3/2
Any particle of this kind is called a Δ⁺.
When we want to be precise, the Δ⁺ I mentioned before is called Δ(1232)⁺, because its energy at rest is 1232 MeV. That corresponds to its mass being 2411 electron masses. But then come a family of increasingly overweight relatives: the Δ(1600)⁺, Δ(1620)⁺, Δ(1700)⁺, Δ(1750)⁺, and so on, all of spin 3/2.
Similarly the proton can be called N(939)⁺, though it'd be like calling water dihydrogen monoxide. Then come the N(1440)⁺, N(1520)⁺, N(1535)⁺, N(1650)⁺, N(1675)⁺, N(1680)⁺, and so on - a seemingly endless series of increasingly heavy relatives, this time all of spin 1/2.
Physicists started studying these excited states, or 'resonances', in 1952. By the late 1960s, people were cranking them out. How to understand them???
(1/n)
https://media.mathstodon.xyz/media_attachments/files/116/264/701/322/929/059/original/ef37ec6500bcc4ba.png

2026-02-25 01:21:03 GMT

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.
https://media.mathstodon.xyz/media_attachments/files/116/128/639/831/911/927/original/1acd7709b64aa2cc.webp

2026-02-18 00:18:12 GMT

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Even if we go up to 1000 ≤ x ≤ 1100, we see the simple function
1/(1 - 2⁻ˢ)(1 - 3⁻ˢ)(1 - 5⁻ˢ) (where s = ½ + ix )
making a noble attempt to mimic the biggest peaks in the Riemann zeta function! By now there are places where it doesn't quite manage the job. But still, they look a bit alike.
(2/n)
https://media.mathstodon.xyz/media_attachments/files/116/088/757/223/861/800/original/05325bff144db563.png

2026-02-18 00:16:43 GMT

It's a bit surprising that even on the so-called 'critical line' where s = ½ + ix for x real, the Riemann zeta function ζ(𝑠) looks a lot like the much simpler function
1/(1 - 2⁻ˢ)(1 - 3⁻ˢ)(1 - 5⁻ˢ)
This is just the start of an infinite product over all primes, called the 'Euler product'. That product converges to the Riemann zeta function when Re(s) > 1, but not on the critical line.
(1/n)
https://media.mathstodon.xyz/media_attachments/files/116/088/748/075/657/005/original/45be345408d806d0.png

2026-02-17 07:27:25 GMT

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It's a lot easier to do the Fourier transform of the zeta function without absolute values around it, since we can take
1/(1 - 3⁻ˢ)(1 - 5⁻ˢ)
and use the geometric series to write it as
$$ g(x) = \sum_{j=0}^{\infty} \sum_{k=0}^{\infty} 3^{-j/2} 5^{-k/2} e^{-ix(j\ln 3 + k\ln 5)}$$
(Sorry, I had to use LaTeX there, which only Mathstodon users will see rendered here.)
But enough for now. Good night!
(10/n, n = 10)

2026-02-17 06:42:42 GMT

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So far I've been trying to understand the complicated waves in the function
|1/(1−3^{-(1/2) - ix})(1−5^{-(1/2) - ix})|
@nprofile…920r suggested that to do this I should compute the Fourier transform this function.
This was indeed very revealing. Check out the graph below!
There are the expected big peaks at
ln(3)/2π
ln(5)/2π
ln(15)/2π
which we expect from part 3. But there are many more - and many with musical significance! Let me list them - but instead of writing each frequencies ω, which are always of the form ln(a)/2π for rational numbers a, I'll just write the numbers a. Some have fairly simple musical names:
0.0122 27/25 large diatonic semitone
0.0813 5/3 major sixth
0.0935 9/5 minor seventh
0.1626 25/9 two major thirds
0.1748 3 perfect twelfth
0.2561 5 major third + two octaves
0.2684 27/5
0.3375 25/3
0.3497 9 two twelfths
0.4188 125/9
0.4310 15
0.4432 81/5
The musical names are probably less informative than the patterns here.
Some of these peaks are barely visible. There are probably more too small to see - infinitely many of them.
(9/n)
https://media.mathstodon.xyz/media_attachments/files/116/084/515/538/035/299/original/7d78fa68fab8bc77.png

2026-02-17 01:53:12 GMT

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For starters, neither
15 × [2π/ln(3)] ≈ 85.8
nor
13 × [2π/ln(3)] ≈ 74.3
are especially close to the true period of the long wave crests in the absolute value of the zeta function. One drifts ahead, while the other drifts behind.
But both do a pretty good job of landing on sharp spikes!
I suspect that I'm just seeing the beauty of continued fraction expansions playing itself out on this playing field. There's an infinite wealth of structure and substructure, just like in the rings of Saturn - which are also caused by resonance phenomena, governed in part by continued fractions.
But this particular function is a lot simpler than the rings of Saturn!
(8/n)
https://media.mathstodon.xyz/media_attachments/files/116/083/454/154/889/152/original/5cd1f826a1cdb75f.png

2026-02-17 01:31:43 GMT

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Aha! 💡
Now I see what the number 13/19 being close to ln(3)/ln(5) does for us.
I said it predicts peaks in our zeta function spaced at a distance of roughly 74.3. And indeed, there are big peaks at x = 0 and x ≈ 74.3!
There's a bigger peak at x ≈ 85.84, since the number 15/22 is even closer to ln(3)/ln(5).
But I was wrong in suggesting that the period of the really big waves is 85.84! In fact the multiples of 85.84 drift away from crests of those waves.
But you'll notice they do lie on sharp spikes.
And this is only possible because
85.84 - 74.3 = 11.54
which is very close to the distance between the sharp spikes!!! Remember, I computed distance in part 4, that using the fact that 2/3 is another rational approximation to ln(3)/ln(5).
But where does the above equation come from? Is it a coincidence?
No, earlier in this thread we approximately got the numbers 85.84, 74.3 and 11.54 in two different ways. If we use one of these ways, 85.84 - 74.3 = 11.54 is telling us
15 × [2π/ln(3)] - 13 × [2π/ln(3)] = 2 × [2π/ln(3)]
If we use the other, it's telling us
22 × [2π/ln(5)] - 19 × [2π/ln(5)] = 3 × [2π/ln(5)]
Both of these are of course true.
So, something very nice is going on here. But I'm still confused about what.
(7/n)
https://media.mathstodon.xyz/media_attachments/files/116/083/327/896/010/609/original/85c4a086ad93e119.png

2026-02-17 00:59:19 GMT

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@nprofile…uv8h - Nice! Thanks! So far I'm entranced by its values on the critical line, and haven't dared venture off that line - since I don't know how it would help me.

2026-02-17 00:36:01 GMT

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Now I think I see why the distance between crests of the big long waves in this graph is about 85.84!
We saw
ln(3)/ln(5) ≈ 2/3
gives spikes in this graph spaced at a distance of about
2 × [2π/ln(3)] ≈ 3 × [2π/ln(5)]
These numbers are 11.44 and 11.71, respectively. The actual spike spacing is about halfway between these two.
Similarly, the better approximation
ln(3)/ln(5) ≈ 13/19
should give peaks spaced at a distance of about
13 × [2π/ln(3)] ≈ 19 × [2π/ln(5)]
These numbers are 74.35 and 74.17, respectively.
Alas, that's not explaining our number 85.84. 😢
But there's an even better approximation
ln(3)/ln(5) ≈ 15/22
which should give peaks spaced at a distance of about
15 × [2π/ln(3)] ≈ 22 × [2π/ln(5)]
and these numbers are 85.79 and 85.89. The number 85.84 is about halfway between these two!!! 🎉
Of course this is still a bit mysterious. Why does 15/22 do something for us, but apparently not 13/19?
Actually I believe 13/19 *does* do something for us. I'm just not sure what.
I also haven'st studied what even better rational approximations to ln(3)/ln(5) do for us. But I imagine they create subtler waves in the zeta function, with even longer periods.
(6/n)
https://media.mathstodon.xyz/media_attachments/files/116/083/076/496/793/756/original/578d1c03af47a7d1.png

2026-02-16 22:41:51 GMT

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More interestingly, the fact that 2/3 is a good rational approximation to ln(3)/ln(5) says that
3 ln(3) ≈ 2 ln(5)
or exponentiating both sides,
3³ ≈ 2⁵
i.e.
27 ≈ 25.
The ratio 27/25 is so important in music that it has a name! It's called the 'large diatonic semitone'. It's one of four semitones that naturally show up in just intonation.
Just intonation is ruled by the primes 2, 3, and 5, but today I'm just looking at the primes 3 and 5. That's why I'm looking at the zeta function of the commutative ring ℤ/3 × ℤ/5, and that's why the number 27/25 showed up!
(5/n)
https://media.mathstodon.xyz/media_attachments/files/116/082/712/140/724/441/original/d11a95c69a961edf.jpg

2026-02-16 22:22:02 GMT

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The slow 'beats' in the absolute value of the zeta function
1/(1 - 3⁻ˢ)(1 - 5⁻ˢ)
arise because it's the product of two functions: 1/1 - 3⁻ˢ with period
2π/ln(3) ≈ 5.719
and 1/1 - 3⁻ˢ with period
2π/ln(5) ≈ 3.904
These functions both become big simultaneously when there are integers m,n with
m 2π/ln(3) ≈ n 2π/ln(5)
or in other words
m/n ≈ ln(3)/ln(5)
So finding the tallest peaks in the absolute value of the zeta function amounts to looking for good rational approximations of this number:
ln(3)/ln(5) ≈ 0.682606194
Here are the first few:
2/3 ≈ 0.6667
13/19 ≈ 0.6842
15/22 ≈ 0.6818
The first says we expect tall peaks spaced apart by roughly
2 × [2π/ln(3)] ≈ 3 × [2π/ln(5)]
These numbers are close but not equal! Look at them:
4π/ln(3) ≈ 11.44
6π/ln(5) ≈ 11.71
This explains why I empirically found that the very tall spikes in the graph below are separated by a distance of about 11.58. However, my guess that this distance is really
5 × [2π/ln(15)] ≈ 11.6009
may be completely wrong.
(4/n)
https://media.mathstodon.xyz/media_attachments/files/116/082/638/110/150/050/original/156231969416dcca.png

2026-02-16 20:31:43 GMT

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In the graph here, the sharp peaks seem to be spaced by a distance of
11.58 ≈ 5 × [2π/ln(15)]
The slow beating has a period of roughly
62.69 ≈ 27 × [2π/ln(15)]
I don't know if these approximate formulas are good - based on some deeper math - or just coincidences. With luck I can figure this out pretty soon, but I thought I'd throw it out here for y'all to play around with.
We can take the reciprocal of the zeta function and notice that
(1 - 3⁻ˢ)(1 - 5⁻ˢ) = 1 - √3 e^(ix ln 3) - √5 e^(ix ln 5) + √15 e^(ix ln 15)
So for *this* function we expect oscillations with periods
2π/ln(3) ≈ 5.719
2π/ln(5) ≈ 3.904
2π/ln(15) ≈ 2.320
but this does not instantly explain the longer periods that stand out so dramatically in the graph here.
(3/n)
https://media.mathstodon.xyz/media_attachments/files/116/082/180/348/029/195/original/1497f0dc712c8afb.png

2026-02-16 20:09:14 GMT

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To get a better picture of the *slow* oscillations in
|1/(1 - 3⁻ˢ)(1 - 5⁻ˢ)|
where s = ½ + ix, let's plot it from x = 0 to x = 300.
https://media.mathstodon.xyz/media_attachments/files/116/082/100/436/960/223/original/1875827997fb8e9d.png

2026-02-16 20:00:51 GMT

There are fascinating connections between the Riemann zeta function and music theory. I'll probably write a paper about this, but I can't resist talking about a little piece of the story. I will *not* explain what this has to do with music, since I want to tell that exciting story later on, and do a really good job of it.
Any commutative ring has a zeta function! The Riemann zeta function is the zeta function of ℤ, but the zeta function of ℤ/3 × ℤ/5 is simpler: it's just
1/(1 - 3⁻ˢ)(1 - 5⁻ˢ)
Let's graph this along the 'critical line' where the famous zeros of the Riemann zeta function live. So, let's take
s = ½ + ix
and plot
|1/(1 - 3⁻ˢ)(1 - 5⁻ˢ)|
as a function of x from x = 0 to x = 100. We get this picture here:
(1/n)
https://media.mathstodon.xyz/media_attachments/files/116/082/076/373/675/815/original/798686c7fe217f56.png

2026-02-13 07:51:01 GMT

This fairly dumb article in a magazine of the Institute of Electrical and Electronics Engineers claims there's a "crisis" at Wikipedia because the editors rejected a push for AI summaries of Wikipedia articles.
A couple of reasons why the article is dumb:
• Where the article says "Research has shown that many readers today greatly value quick overviews of any article," the link leads to something completely different: an article titled "In the AI era, Wikipedia has never been more valuable", containing no such research.
• The article says "But the volunteer base is aging. A 2010 study found the average Wikipedia contributor was in their mid-twenties; today, many of those same editors are now in their forties or fifties."
So volunteers at Wikipedia are aging faster than other people, with some of *the same people* moving from their mid-twenties to their fifties in just 16 years?!? Maybe it just feels that way. 😆

2026-02-12 03:49:17 GMT

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@nprofile…nuf5 - yes, in physics (and elsewhere) it's important to have, for each fact, not just a bit saying whether you believe it or not, but a little file recording the evidence for or against it.

2026-02-11 02:11:57 GMT

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@nprofile…3mqd - are you trying to tell me something?

2026-02-10 19:24:08 GMT

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@nprofile…xdkd - we might have a lot to talk about.

2026-02-10 17:54:26 GMT

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Gro-Tsen continues:
This C = 584 283 or “GMT” correlation value places the “Long Count epoch” 0.0.0.0.0 on August 11, 3114BCE in the proleptic Gregorian calendar (the day with Julian Date 584 283), although IIUC it's not clear if this precise date held any particular importance to the Olmecs (or later Mayans).
Maybe it was just arbitrary like the start of our own Julian Date (because, no, Julius Scalier didn't think the world started on November 24, 4714BCE proleptic Gregorian).
One Mayan inscription suggest that the Long Count was the truncation to the last 5 “digits” of an even longer count, and that a Long Count value such as 9.15.13.6.9 was in fact 13.13.13.13.13.13.13.13.9.15.13.6.9 in this Even Longer Count (why 13 everywhere? I don't know!). But this may be one particular astronomer's weird ideas, I guess we'll never know.
But back to the Mayan correlation constant C.
Wikipedia suggests that this “GMT” value C = 584 283 for the Mayan correlation is now settled and firmly established. But between 1905 and now there was some going back and forth with various authors (including the three Goodman, Martínez and Thompson after which it is named) adding or removing a day or two (I think Goodman first proposed 584 283, then changed his mind to 584 280, but nobody really cared, Hernández resurrected the proposal in 1926 but altered it to 584 284, then Thompson to 584 285 in 1927, and then Thompson later said Goodman's initial value of 584 283 had been right all long, and while this is now accepted, the confusion of ±3 days might still linger).
(5/n)

2026-02-10 17:51:49 GMT

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Gro-Tsen writes:
I did the math. 🙋
👉 It's Sept. 3, 32BCE (reminder: “32BCE” actually means “−31” 😒) in the proleptic Julian calendar = Sept. 1 prol. Gregorian.
The Western equivalent of the Mesoamerican Long Count is the “Julian Date” (NB: “Julian” here refers not to Julius Cæsar as in “Julian Calendar” but to the 16th century scholar Julius Scaliger). The Julian Date simply counts the number of days from an arbitrary remote reference point (Nov. 24, 4714BCE proleptic Gregorian). More practically, on 2000-01-01 it equaled 2 451 545 (at 12:00 UTC if we want to use fractional Julian dates).
For example, today as I write is Julian Date 2 461 082 (well, 2 461 081.9 because it's not yet noon UTC). And the date of Sept. 1, 32BCE [prol. Greg.] we're talking about corresponds to Julian Date 1 709 981. More convenient than all this dealing with complicated calendar conventions.
So to convert a Long Count date to the Western calendar, we first convert the Long Count to an integer (trivial: it's already just an integer written in base 20-except-18-in-the-penultimate-digit), we add a constant (C) to get a Julian Date, and we convert to our messy calendars.
BUT! What is this constant C? This is known as the “Mayan correlation”. For a long time in the 20th century there was a debate about its value: scholars could relate any two Mayan dates, but not situate them exactly w.r.t. our own calendar. Various values were proposed, ranging from the (frankly rather ludicrous) 394 483 to 774 078, an interval of about 1000 years! (😅)
https://bsky.app/profile/gro-tsen.bsky.social/post/3meiqswj7b22a
(4/n)

2026-02-10 17:31:52 GMT

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30 years later a farmer found the other half of the rock and confirmed Marion Stirlling's guess: yes, the date was September 3, 32 BC!
That's a wonderful story of delayed gratification. But here's the absolutely chilling part: the Mesoamerican Long Count calendar was so damn good that we can look at that date and know it meant September 3, 32 BC... to within a few days.
I'll explain how, quoting my friend Gro-Tsen (who alas chose Bluesky rather than Mastodon because it's easier to move your posts somewhere else).
(2/n)
https://media.mathstodon.xyz/media_attachments/files/116/047/488/961/743/570/original/4218f0bcc8606fc7.jpg

2026-02-10 17:20:34 GMT

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People used to think the Olmecs, who made those colossal stone heads, were not very old.
But in 1939, an archaeologist couple, Marion and Matthew Stirling, found the bottom half of this Olmec rock, which had part of a date carved on it.
They guessed the date was 7.16.6.16.18. In the Meso-American Long Count calendar this corresponds to September 3, 32 BC. That meant the Olmecs were extremely old!
But the first digit was missing - Marion just *guessed* it was a 7 - so few believed them.
(2/n)
https://media.mathstodon.xyz/media_attachments/files/116/047/470/216/672/514/original/85bbe97ef822b850.jpg

2026-02-10 17:13:39 GMT

The problem with archeologists is that the successful ones get a big head.
(1/n)
https://media.mathstodon.xyz/media_attachments/files/116/047/449/437/146/590/original/bb1076bf7a7d5c31.webp

2026-02-09 18:20:12 GMT

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@nprofile…8qh7 @nprofile…xdkd - I used to read Gene Ward Smith on sci.math and such, but I never paid much attention to his tuning theory stuff. I feel terribly sorry that I didn't, since he died of COVID in 2021:
https://en.xen.wiki/w/Gene_Ward_Smith
His work seems to be scattered in various newsgroups and chat rooms, articles on the Xenharmonic Wiki, etc. Besides the material on the Riemann zeta function another exciting thing is his study of "Don Page commas":
https://en.xen.wiki/w/Don_Page_comma

2026-02-09 17:45:03 GMT

My god! What an amazing blog article! For the last few weeks I've been studying the work of Gene Ward Smith, who discovered a connection between muic theory and the Riemann zeta function. But it turns out @nprofile…xdkd has been thinking abou this for years... and what she has discovered is much richer and more beautiful than I had imagined. This changes everything!

2026-02-09 05:32:21 GMT

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@nprofile…uk82 - scarlet macaws in New Mexico? Or cacao?

2026-02-08 22:42:15 GMT

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For more on parrots and scarlet macaws at Chaco Canyon:
https://archaeologymag.com/2025/12/ancient-puebloans-macaws-ceremonial-use/
"Chaco Canyon saw occupation from the mid-9th to the mid-12th centuries, which also coincided with the growth of what eventually became monumental masonry pueblos, known as Great Houses. While macaw and parrot remains have intrigued researchers for decades, the last analysis of them was published more than half a century ago. This study reexamines that old material using modern zooarchaeological methods and contextual reconstruction.
The reanalysis identified the remains of 45 birds from five different sites within the canyon. Most of them were scarlet macaws, with a small number of thick-billed parrots, a species that is not native to the region, and provide evidence of long-distance acquisition. Most of the birds were found in the Great Houses, particularly Pueblo Bonito, the largest and most studied Chacoan building. There, archaeologists found dozens of macaws in large plastered rooms, which often included thermal features, indicating a deliberate effort was put into keeping the birds warm in a harsh environment.
Many of the rooms showed clear signs that live birds had been held inside for long periods. Researchers observed thick layers of droppings, food debris, and what looked like perches, which provides proof that macaws lived in these spaces rather than just being put there for a short time or processed. Individuals ranged widely in age from juveniles to those over the age of twenty, which points to long-term care rather than short-lived use."
(3/3)

2026-02-08 22:40:50 GMT

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For more on chocolate at Chaco Canyon:
https://www.nps.gov/chcu/learn/historyculture/pre-columbian-chocolate-discovered-at-chaco.htm
"From 2004-2007 a University of New Mexico (UNM) research project re-excavated the trenches first dug in Pueblo Bonito’s middens under Neil Judd in the 1920s. Of the hundreds of thousands of pot sherds that were recovered, archaeologist Patricia Crown selected five for her research. She is a ceramics specialist at UNM’s Department of Anthropology. She designed the project, and W. Jeffrey Hurst from The Hershey Center for Health and Nutrition performed the research. They chose five pot sherds for organic residue analysis, three of which were likely from cylinder jars. The pieces date to between 1000 and 1125 AD based on their decorative styles.
Only the three sherds most likely from cylinder jars exhibited trace theobromine, a conclusive indicator of cacao or chocolate. The implications of this find are extraordinary. The cacao plant grows only in certain tropical climates, and the nearest possibility for Chaco is Central Mexico. We already know the Chacoan people traded with Mesoamerican cultures for exotics like copper bells and scarlet macaws, but cacao suggests a more ritual connection than other Mesoamerican goods. In some Maya ceremonies a cacao beverage was frothed by pouring the liquid from one vessel to another. Likewise, the cacao found at Chaco was probably in liquid form because the residue had absorbed into the clay itself. Further, the limited distribution of the cylinder jars could be evidence that only an elite or small segment of the population consumed the beverage."
(2/3)

2026-02-08 22:38:47 GMT

Chaco Canyon should be called Choco Canyon, because researchers have found traces of *chocolate* in cups found at this site dating back to 1000 - 1125 AD.
This is amazing: Chaco Canyon is in a dry part of New Mexico, 1900 kilometers north of where cacao grows. But the cups look like those that Mayans used for chocolate-drinking rituals! And archeologists have also found remains of parrots and macaws in Chaco Canyon. This suggests enormous trading routes.
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2026-02-05 17:34:08 GMT

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@nprofile…xuq7 - nice! It looks like a rocket-fuel-propelled approach to algebraic geometry where he talks about zeta functions and cohomology theories on page 3 and then gets serious. 😆

2026-02-05 17:29:46 GMT

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@nprofile…xuq7 - Oh great! I haven't ever looked at a book by Raskin, in fact I don't even know the name Raskin. But I've been thinking there should be an approach to algebraic geometry that goes like this. I only heard about this approach long after I suffered through the Hartshorne approach (listening to people talk about it, not actually studying it very hard).

2026-02-05 17:15:33 GMT

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@nprofile…xuq7 - Neat! I was just joking: I actually skipped commutative algebra entirely myself, too - I just zoned out whenever anyone tried to explain this stuff to me.
I just posted a comment sketching in a very sketchy way the "modern" approach to defining schemes, via the Zariski site. Maybe this is what you're studying now. Hartshorne takes a more traditional approach - or maybe we should say the traditional approach is to follow Hartshorne.

2026-02-05 16:48:20 GMT

If like me you slept through part of your algebra class and spent years later trying to catch up on algebraic geometry, this is the thread for you!

2026-02-03 18:22:52 GMT

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@nprofile…5ahq - I'd never say radically improved democracy is impossible. We need not only an appealing new system, but a way to get there from here. I hope you've written, or will write, a detailed tactical manual on how to get there from here.

2026-02-03 17:46:15 GMT

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@nprofile…5ahq - I don't read Medium, but try these:
https://www.youtube.com/watch?v=d-4OPJaqkP4
https://www.alternet.org/capitalism-trump/
https://www.hks.harvard.edu/faculty-research/policycast/oligarchy-open-what-happens-now-us-forced-confront-its-plutocracy
and many more.

2026-02-03 17:33:18 GMT

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@nprofile…5ahq - Lots of people are discussing those questions. But it's also important to avoid falling off the cliff right now.

2026-02-03 16:28:38 GMT

Trump now says the quiet part out loud:
"The Republicans should say: 'We want to take over. We should take over the voting in at least 15 places.' The Republicans ought to nationalise the voting."
For folks outside the US:
Elections in the US are run, not nationally, but by the individual states - until the final stage. Thus, nationalizing the elections would be a way to end fair elections in the US. We can guess that the "15 places" include states where ICE thugs currently roam in large numbers: states with Democrat-dominated cities such as Minneapolis, Portland, Chicago, New York City, Los Angeles and Atlanta. Trump can use ICE, the National Guard and - if he invokes the Insurrection Act - the military to suppress demonstrations against this takeover.
Funding DHS, the department ICE belongs to, is a very bad idea.
https://www.bbc.com/news/articles/c0mke841zj0o

2026-02-02 18:58:14 GMT

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Ecologist Alison Anastasio visited a former US Steel South Works site in Chicago. She expected to find “all crap plants” — common invasive weeds. To her surprise she spotted little bluestem and three species of native milkweed. She had already decided she didn't want a career as an academic scientist. But she came up with the idea of forming a group to study this ecosystem: “a dream team of people I wanted to work with.”
She knew Laura Merwin from the University of Chicago, and later she met Lauren Umek, a project manager for the Chicago Park District. She invited them to brunch to pitch her idea to research plants growing on slag. Not for any obvious career goal. Just from sheer curiosity.
Merwin and Umek were excited to join her project - a “reverse side hustle,” since it involved a lot of work, but cost money.
Their first paper, “Urban post-industrial landscapes have unrealized ecological potential,” was published in Restoration Ecology in 2022. It argues that slag fields don't need to be fixed. They have ecological value in and of themselves. And land managers should forget whatever ecosystem was there before. Instead, they should look to more exotic ecosystems as a guide, like the dolomite prairies of Illinois, where magnesium-rich rock near the surface makes it hard for ordinary plants to thrive. Slag too is rich in magnesium.
The Slag Queens are continuing their revolutionary work even now! For more, start here:
• Carrie Gous, The beauty of slag, https://mag.uchicago.edu/science-medicine/beauty-slag
Some of what I just wrote is a paraphrase of this article.
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2026-02-02 18:34:35 GMT

Here's a tale of how nature triumphs in the end. Steel mills dumped molten slag in parts of Chicago and nearby areas. The slag hardened in layers up to 15 feet deep. These places became barren wastelands. Some were also dumping grounds for hot ash and cinders.
Eventually the steel mills closed. The deep layers of hard, toxic material were not friendly to plants. Cottonwoods are usually 30 meters tall or more. In the slag fields, stunted cottonwoods grow to just 2 meters.
But rare species that could handle these conditions began to thrive. The lakeside daisy, a federally threatened species lost to Illinois for decades, turns out to grow taller on slag than on topsoil! The capitate spike-rush, last recorded in Illinois in 1894 and considered locally extinct, was rediscovered growing on slag.
And more! Native prairie grasses like little bluestem. Native milkweeds. Even tiny white orchids called sphinx ladies' tresses.
A team of women ecologists began studying these unusual landscapes. They call themselves the Slag Queens.
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2026-02-01 21:55:57 GMT

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@nprofile…t6k2 - My wife is reading Katabasis. She says it's too long (450 pages), and could have used editing to remove repetition. But she likes some of the ideas, e.g. how people in the Department of Analytic Magick at Cambridge need to compare Western and Chinese accounts of the geography of Hell to get a good understanding of the place.

2026-02-01 03:47:44 GMT

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@nprofile…ldyp @nprofile…s7d2 @nprofile…t6k2 @nprofile…z2n3 - when you folks figure out what's going on here, I hope you explain it to me in simple terms.

2026-01-31 17:45:45 GMT

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@nprofile…s7d2 - You're making me think I was mixed up. (I wrote that stuff a long time ago and haven't thought about it in years.) The proof-theoretic ordinal of PRA is $\omega^\omega
$ but it sounds like PRA is only supposed to be able to define primitive recursive functions, hence not the Ackermann function.
@nprofile…t6k2 @nprofile…z2n3

2026-01-31 16:05:45 GMT

I thought they wouldn't release the Epstein files until hell freezes over... but in fact it only took frost quakes. Yes, FROST QUAKES, also called cryoseisms! It got so cold so fast in the US that freezing soil created earthquakes. They even cracked the wall in someone's house.
https://www.npr.org/2026/01/30/nx-s1-5693186/winter-storm-causes-weather-phenomenon-known-as-frost-quakes-in-parts-of-the-south

2026-01-31 05:11:19 GMT

The movement is starting in Minneapolis, and the movement needs songs.
youtu.be/hyPFxh_XXwY

2026-01-30 16:18:30 GMT

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@nprofile…t6k2 @nprofile…z2n3 @nprofile…s7d2 - just to be clear, where @nprofile…z2n3 quoted me as writing ω2 and ωω, I actually wrote ω^2 (ω squared) and ω^ω (ω to the ω). I was trying to say that PRA is powerful enough to prove that the Ackermann function is total because it can handle recurious up to ω^ω, but proving the Ackermann function is total only requires recursion up to ω^2.

2026-01-30 02:44:44 GMT

Wow! Once the Rhine, the Thames and the Seine were all tributaries of the same big river that flowed out where the English Channel is now!
It's called the Channel River:
https://en.wikipedia.org/wiki/Channel_River
This river system began forming after catastrophic megafloods breached the Dover Strait between roughly 450,000 and 180,000 years ago
Whenever there have been glacial periods since then, the sea level drops, Britain becomes connected to Europe, and the Channel River forms. Its most recent appearance was during the Last Glacial Maximum 20,000 years ago, when sea levels were about 130 meters lower than today.
(hat-tip to
@julesh.mathstodon.xyz.ap.brid.gy
)
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2026-01-28 22:01:14 GMT

Fractions built from the numbers 2, 3 and 5 play a big role in music theory! Here I list those as close to 1 as possible for a given complexity. First we get a bunch of famous ones like the major third:
5/4 = 1.25
the minor third:
6/5 = 1.2
the diatonic semitone:
16/15 = 1.06666...
and the syntonic comma:
81/80 = 1.0125
Eventually we get absurdly small ones like the quark of Baez:
2⁻⁵⁷³ 3²³⁷ 5⁸⁵ = 1.0000005104....
Yes, it's bad to name things after yourself. In fact that's item 25 on the crackpot index! So if you do it, you should do it for something silly.
But the math here is not silly. I think it's really cool how musicians systematically explored fractions close to 1.

2026-01-28 00:42:52 GMT

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@nprofile…g0zc @nprofile…gu9p @nprofile…ck2y - If Gödel really did get angry and threw out Chaitin, this could be one of the first examples of scientists getting disgusted by people trying to connect Gödel's theorem and Heisenberg's uncertainty principle. By now it's a routine occurrence.
Speaking of "adding axioms", Grothendieck added an extra axiom to ZFC to carry out his work on algebraic geometry: the axiom of universes. This is a trick to get around the problem with "the set of all sets". The idea is that for any cardinal, there's a larger cardinal so big that the collection of sets with cardinality smaller than that is almost indistinguishable from the set of all sets.
Since Grothendieck's work relying on the axiom of universe was later used to prove Fermat's Last Theorem, this raises the question of whether we only know FLT conditional on the axiom of universes!
My friend Colin McLarty has been trying to sort this out. He believes that in all cases the axiom of universes can be sidestepped - at the expense of making various arguments more technical.
Here's a surprisingly readable introduction to this business:
https://www.jstor.org/stable/20749620

2026-01-27 03:05:49 GMT

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@nprofile…zyzh - Interesting. For some kinds of math questions there's a right answer, e.g. true-or-false questions (which can be very powerful in mathematics). For these it can be really helpful for beginners to know what's the right answer. For other questions, more open-ended, there's nothing at all like a unique right answer.
I personally never even paid much attention to this check mark business.
Personally I think the competitiveness comes in when we see charts like this:
https://mathoverflow.net/users/2893/john-c-baez
I'm the 100th best person on MathOverflow this month, etc.

2026-01-27 02:20:42 GMT

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@nprofile…zyzh - nothing in MathOverflow says a question has a single answer. Many questions there have lots.

2026-01-27 01:03:49 GMT

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@nprofile…fh5j - I find MathOverflow extremely useful myself, because it attracts lots of experts who can help me with problems I'm stuck on. However, I suspect you're right that "gamifying" the process of asking and answering questions by giving people points and publicly displaying their "reputation" scores could be off-putting to lots of women (and also certain kinds of men).
When I suggested that women might not enjoy this competitive approach, someone reacted by pointing out the existence of highly competitive women tennis players, and then the conversation digressed to the meta-question of whether this analogy was useful - and the meta-meta-question of whether analogies of this sort were ever useful. I found it quite unsatisfying to see my point shunted off in this way.

2026-01-26 23:57:43 GMT

Not very urgent compared to some things, but okay:
MathOverflow is a website for asking and answering math questions. Very few women participate. After hearing some women discuss the reasons why, I asked what has been done to make MathOverflow more welcoming to women:
https://meta.mathoverflow.net/questions/6367/what-is-being-done-to-make-mathoverflow-more-welcoming-to-women
Answer: something was done once, but nothing is being done now.
But some preferred to argue against the very question. And now I see that someone has posted a followup, proposing that
"questions that invite general discussion about how to address the gender imbalance on MO are off-topic and should be closed/locked/deleted"
https://meta.mathoverflow.net/questions/6390/proposal-free-floating-discussions-of-things-like-women-and-mo-should-be-bann
The reason, supposedly, is that my question started a conversation that "did not reflect well on MathOverflow".
Personally I think the solution is not to shut people up, but to start doing things to make MathOverflow more welcoming to women. Some women say this is hopeless - they'll just ignore the place and/or hope it dies. I hope people who think that start up a new site. But I hope people who like MathOverflow try to improve it.
(For nonexperts I should clarify that I asked my question on MathOverflow Meta. This is precisely the place to talk, not about math, but MathOverflow policies. So it's not as if I were interrupting math conversations with "off-topic" discussion of policy. The objection was solely that I started a conversation that became unpleasant in some ways. By now the moderators have deleted the most obnoxious comments.)

2026-01-24 00:46:13 GMT

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@nprofile…g0zc - opposite of feckless?

2026-01-23 19:12:35 GMT

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@nprofile…g0zc - "This shepherd's pie is f***ing good!"

2026-01-22 17:46:35 GMT

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@nprofile…3nhw - I don't really know any references that do what you want. Wikipedia lists a bunch of references for the classical case:
https://en.wikipedia.org/wiki/Schur%E2%80%93Weyl_duality
but not the q-deformed case.

2026-01-21 06:35:02 GMT

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@nprofile…jrcx - @nprofile…t6k2 would have opinions on this, though he may be tired of talking about it.

2026-01-21 03:19:18 GMT

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@nprofile…a3ey - I guess this is early enough that he's not saying it with the somewhat exasperated tone of a modern mathematician?

2026-01-21 02:22:07 GMT

Mood.

2026-01-20 21:33:15 GMT

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@nprofile…npxu - I was only commenting on the headline and the first explanatory paragraph:
"Microsoft chief executive Satya Nadella has warned that artificial intelligence (AI) risks becoming a speculative bubble unless its use spreads beyond big tech companies and wealthy economies."
So the massive investment in AI risks becoming a speculative bubble if not enough people use AI. That seems close enough to a tautology to be funny.
You're saying something more interesting: "the massive success and wide adoption of life-changing technology does not, in itself, mean there is no bubble." That's the converse, which is a lot less obvious.
And then you're saying a lot of other interesting things....

2026-01-20 17:18:31 GMT

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@nprofile…j4wh - really??? Can't AI solve that problem?

2026-01-20 17:14:12 GMT

"There's a danger that our technology could fail if it fails!"
Maybe this man's impressively large dome makes such remarks sound profound.
https://www.irishtimes.com/business/2026/01/20/ai-boom-could-falter-without-wider-adoption-microsoft-chief-satya-nadella-warns/

2026-01-20 07:30:45 GMT

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@nprofile…a3ey - noble of you!

2026-01-20 06:42:04 GMT

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@nprofile…a3ey - I too promise to no longer publish more than 7 papers per year. 😆

2026-01-20 06:39:25 GMT

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@nprofile…2sx2 - they've got lots of nice posts here.

2026-01-20 06:28:57 GMT

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@nprofile…2sx2 - there are some very nice experts on constructive mathematics here on Mathstodon, e.g. @nprofile…t6k2 and @nprofile…ldyp.

2026-01-20 05:56:43 GMT

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@nprofile…2sx2 - there are mathematicians who question all the things you're questioning, and they write papers about that. However, it pays to learn the standard rules. This doesn't mean "believing" in them, indeed one great thing about math is that checking the validity of a proof never requires belief. It pays to learn the standard rules because if you don't, you're cut off from an enormous body of centuries of work.

2026-01-20 03:03:13 GMT

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@nprofile…hs9s - thanks! The connection of music and math is a wonderful thing. I've got more on this:
https://math.ucr.edu/home/baez/tuning_book/
Probably more than you want!

2026-01-20 01:32:53 GMT

Life in Nunavut - the largest and northernmost territory of Canada.

2026-01-20 01:29:29 GMT

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@nprofile…hs9s - Here is a talk Dmitri gave in Edinburgh:
https://johncarlosbaez.wordpress.com/2025/11/22/dmitri-tymoczko/

2026-01-20 01:29:05 GMT

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@nprofile…hs9s - Here is a talk Dmitri gave in Edinburgh:
https://johncarlosbaez.wordpress.com/2025/11/22/dmitri-tymoczko/

2026-01-20 01:28:35 GMT

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@nprofile…hs9s - Here is a talk Dmitri gave in Edinburgh:
https://johncarlosbaez.wordpress.com/2025/11/22/dmitri-tymoczko/

2026-01-19 17:20:05 GMT

He lives in northern Norway, inside the Arctic circle, in an old house full of animals - rabbits and 2 macaw parrots. The sun just rose for the first time this year, and it's BEAUTIFUL.
And thanks to the Fediverse, we get to share that feeling.

2026-01-18 17:14:48 GMT

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@nprofile…e2fl - now I get to check out #SolarPunkSunday!

2026-01-18 17:10:44 GMT

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@nprofile…avcw - I'd never thought much about Felis lybica before, thanks! This Wikipedia page has a chart showing that nuclear DNA says both F. sylvestris and F. catus emerged from the same common ancestor as F. lybica, while mitochondrial DNA tells the story you learned:
https://en.wikipedia.org/wiki/African_wildcat
I'm not sure what that means.

2026-01-18 08:33:54 GMT

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@nprofile…6fwm - what a great story! I could enjoy a whole novel about beavers on the San Pedro, written in a similar style, with maps. But I'll be frequenting your blog now, for other tales.

2026-01-18 06:48:52 GMT

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@nprofile…m5tp @nprofile…6txz
🦫

2026-01-18 02:51:53 GMT

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@nprofile…l7em @nprofile…yj83 - really cool! If you haven't already read it it, check out my post on the Cat Sith:
https://mathstodon.xyz/@johncarlosbaez/115043274642634616

2026-01-18 02:32:56 GMT

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@nprofile…9xae - I'm glad you enjoy my stuff. Don't worry about missing anything. If you ever feel bored, you can see many years of such posts here:
https://math.ucr.edu/home/baez/diary/july_2006.html

2026-01-18 01:27:55 GMT

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@nprofile…9xae - I tend to talk less about personal things here, more about math and physics. But I like to link to the occasional good news, often with a green slant. There are green shoots poking up here and there.
https://mathstodon.xyz/@johncarlosbaez/115559260956688871

2026-01-17 22:37:53 GMT

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@nprofile…nekg - that would definitely be disturbing if I were out camping.

2026-01-17 22:22:17 GMT

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@nprofile…80ng - will do!

2026-01-17 20:49:38 GMT

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@nprofile…ccq6 - indeed, that's how I often feel these days.

2026-01-17 20:47:45 GMT

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@nprofile…x8fk - we see animals called bobcats in our back yard in Southern California, but these are actually lynxes... so I'm a big fan of the lynx. Hurrah for the Iberian lynx!

2026-01-17 20:45:10 GMT

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@nprofile…9xae - that's a great approach to thinking about the Fediverse and what social media *should* be, or at least should include. There's a lot of reportage about new political outrages, and I could certainly broadcast those incipiencies, but I've decided to focus mainly on others, since we're oversaturated with those.
An animal called the "fisher" is also coming back - in the US:
https://mathstodon.xyz/@johncarlosbaez/115734476846026424

2026-01-17 18:12:05 GMT

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@nprofile…pklk -
"Wildcat hybrids have shown up mingling undetected among housecats around farms in Hungary.
In an animal shelter in Germany, a seemingly abandoned, unusually fierce kitten turned out to be a wildcat, according to media reports."
https://www.upi.com/Odd_News/2024/10/25/germany-kitten-European-wild-cat-animal-shelter/9601729888048/

2026-01-17 18:06:10 GMT

Some good news in a time of darkness: the European wildcat, Felis sylvestris, is making a comeback! This thoughtful-looking example was photographed in a mountainous region of the Czech republic.
The European wildcat's extreme elusiveness may have helped it avoid hunters in places where a larger native cat, the lynx, has been killed off. There may be about 140,000 European wildcats spread across more than two dozen countries. But they are very hard to find!
Wildlife photographer Andrea Giovanni, who made a video of one, writes:
"I'd never even thought of taking photos of wildcats, for a simple reason: I thought it was impossible, or at least, extremely difficult. It's considered 'the ghost of the forests' because it's very, very elusive, and it's hard to predict where it can be spotted. Other animals tend to follow the same trails through the forest. The wildcat goes wherever she wants to."
One reason the European wildcat is coming back is increased legal protections. But another is that villages in Italy and other regions are becoming depopulated! Some are very worried about declining human populations. But it does make room for other species.
I got this picture, taken by Vladimír Čech Jr in the Doupov mountains, from a very nice article on the European wildcat:
https://www.bbc.com/future/article/20260112-rare-images-of-europes-ghost-cat
For more on this species:
https://en.wikipedia.org/wiki/European_wildcat
https://media.mathstodon.xyz/media_attachments/files/115/911/651/790/519/381/original/c65d70c11d981b0c.jpg

2026-01-17 17:29:28 GMT

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@nprofile…659g - is that on the California coast south of Big Sur? That's where I've seen such rocks.

2026-01-16 20:58:54 GMT

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@nprofile…r0xq - I'm afraid so!

2026-01-16 16:55:13 GMT

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@nprofile…r0xq - I hope they had a big party at the Department of Humanities at MIT. Yes, they have just one department for all "humanities".
https://catalog.mit.edu/schools/humanities-arts-social-sciences/humanities/

2026-01-16 16:41:51 GMT

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@nprofile…r0xq - Okay, I can't figure out how to see after the 4th on my laptop. Let's just say it's not optimally designed.
But it turns out they really *do* claim MIT is the top university in the world for arts and humanities, and that's a complete joke. I did my PhD there, it's a great place, but it has very few people working on arts and humanities.

2026-01-16 01:44:18 GMT

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@nprofile…fzye - thanks very much for that quote!
@nprofile…t6k2

2026-01-16 01:43:13 GMT

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@nprofile…qlgd - okay. I think there was a grad student with that name at UCR, and then he left.

2026-01-16 01:27:33 GMT

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@nprofile…qlgd - Was Lior Pachter a grad student at UCR for a while?

2026-01-16 01:19:45 GMT

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@nprofile…t6k2 - I don't even know who is to blame for the website, the Times Higher Education folks or Elsevier. It claims to list 750 schools in order but I only see 4.

2026-01-16 00:35:05 GMT

https://media.mathstodon.xyz/media_attachments/files/115/901/959/123/405/845/original/c511796ccfb00580.jpg

2026-01-16 00:30:31 GMT

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@nprofile…eytx - yes, it's progress.
Nobody has ever shown there was a string theory setup that gives a metastable positive cosmological constant in 4d spacetime. The KKLT mechanism was a popular strategy, and the hype fooled many nonexperts into thinking there were zillions of stable vacua with a positive cosmological constant, but that was never really show. This recent paper searched through > 300,000 candidates and found 5 that *might* work:
https://arxiv.org/html/2505.00149

2026-01-15 22:28:32 GMT

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@nprofile…llgf - I did my PhD at MIT, where I took a course on continental philosophy in the math department.