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

2026-07-24 09:32:43 UTC

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/

2026-05-27 12:13:14 UTC

The new CEO of Wikipedia worked at J.P. Morgan and Lehman Brothers. The Wikimedia Foundation has now fired the a longtime lead developer and disbanded the team whose job was to listen to volunteers. Most of the people they fired were union organizers. Wikipedia’s editors are now threatening to strike in solidarity. To stand in solidarity with them, sign the petition:
https://en.wikipedia.org/wiki/Wikipedia:Wiki_Workers_United_solidarity
For more, read on!
(1/2)

2026-04-28 16:06:28 UTC

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

2026-03-29 06:03:51 UTC

Kings Day shows yet again that the least powerful Americans are the most courageous.
https://media.mathstodon.xyz/media_attachments/files/116/310/935/159/404/387/original/067696ef1be2cc14.webp

2026-03-27 05:45:07 UTC

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@nprofile…j9f5 - the Choctaw don't own most of the land over which they have jurisdiction, and I can't quickly find out how much land they do own. So far they seem to be doing solar on a smaller scale:
The Choctaw Nation Solar Farm in Durant, Oklahoma sits on 35 acres of land and contains more than 15,300 solar panels. It's a partnership with Oklahoma Gas & Electric (OG&E), which constructed and operates the facility.
The farm first came online in August 2020 producing 5 megawatts, and expanded it to 10 megawatts by the end of 2021. Many Choctaw Nation facilities — including tribal government, culture, and health centers — receive a portion of their power from the farm, with each of the 58 connected facilities using solar for up to half of their total power.
The Nation saved $69,000 on energy costs in the first 90 days after the farm went online. The tribe also avoided price spikes during the devastating February 2021 winter storm because half their power consumption came from solar.
The utility authority director has mentioned the longer-term possibility of the Choctaw Nation moving into energy production itself, though that was described as still far off.

2026-03-27 04:22:31 UTC

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 UTC

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

2026-03-13 21:18:40 UTC

The arXiv is separating from Cornell University, and is hiring a CEO, who will be paid roughly $300,000/year:
https://jobs.chronicle.com/job/37961678/chief-executive-officer
They say:
"After decades of productive partnership with Cornell University, and with support from the Simons Foundation, arXiv is establishing itself as an independent nonprofit organization, marking the next stage in its 35-year history as a pioneer of open-access science."
The arXiv’s current annual budget is approximately $6 million and they employ ~27 staff members, most of whom work remotely, primarily in the U.S. The new chief executive officer (CEO) will be responsible for all aspects of arXiv, including strategic planning, financial management, technical infrastructure, personnel oversight and stakeholder engagement. They will work closely with board member representatives of Cornell University and the Simons Foundation to establish the organization’s independence.
A firm called Spencer Stuart is recruiting the CEO. For confidential nominations and expressions of interest, you can contact them at [email protected]. The salary is expected to be around $300,000, though the actual salary offered may differ.

2026-03-12 16:25:27 UTC

Beato isn't really saying AI will fail: despite the denialists who claim it's never good for anything, it's too useful for too many things to simply "fail". He's claiming something more interesting: for many purposes, big AI can be replaced by LLMs that you can run on your laptop. They already exist and you can download them for free. So the massive investment in data centers, expecting huge profits when we all start paying monthly fees to run LLMs, may soon be undercut by something cheaper.
(1/2)
https://www.youtube.com/watch?v=YTLnnoZPALI

2026-03-07 16:59:14 UTC

Manet's famous painting Un Bar aux Folies-Bergère never appealed to me. But now I realize its genius, and my spine tingles every time I see it.
The perspective looks all wrong. You're staring straight at this barmaid, but her reflection in the mirror is way off to right. Even worse, her reflection is facing a guy who doesn't appear in the main view!
But in 2000, a researcher showed this perspective is actually possible!!! To prove it, he did a photographic reconstruction of this scene. Check it out in my next post.
This blows my mind.
(1/3)
https://media.mathstodon.xyz/media_attachments/files/116/188/884/486/749/574/original/54d69d2cfe515f37.jpg

2026-03-02 23:32:22 UTC

Another place I want to visit on my trip to New Mexico is called Aztec Ruins - named that before white folks realized the ancestral Puebloans were a very different civilization.
It's huge, with ~450 rooms, some 3 stories high! People lived here from about 1050 to 1200 CE.
Aztec Ruins National Monument: https://www.nps.gov/azru/index.htm
(1/n)
https://media.mathstodon.xyz/media_attachments/files/116/162/185/973/630/938/original/f0f8b280a6618c34.jpg

2026-02-26 18:16:50 UTC

Have you read There Is No Antimemetics Division? Would I like it?
A review in The Guardian says:
"Memes are ideas that easily spread; antimemes are literally unthinkable, “self-keeping secrets”, impossible to record or to remember. Some feed on memories and pose an existential threat. But how is it possible to win a war when there’s no identifiable enemy, and every attack is immediately forgotten? Against these odds, the Antimemetics Division somehow exists, part of a secret organisation with bases deep underground in the English countryside, as related in this unforgettable, mind-bendingly brilliant novel."
https://en.wikipedia.org/wiki/There_Is_No_Antimemetics_Division

2026-02-25 01:21:03 UTC

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 07:03:44 UTC

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@nprofile…uv8h - Yes, leaving out extra terms in the Euler product formula for the Riemann zeta function tends to smooth it, because those extra terms induce more rapid oscillations.

2026-02-18 00:48:45 UTC

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@nprofile…fh5j - yes, it sounds like that, and I hope it's something like that! But I haven't seen it proved, and I don't even know exactly what's the statement to prove. Since the Riemann Hypothesis is a big deal, someone might have done some of the work already.

2026-02-17 22:03:31 UTC

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@nprofile…fd79 - thanks, all this is very interesting! I agree wholeheartedly that the Fourier transform of
|1/(1−3^{-(1/2) - ix})(1−5^{-(1/2) - ix})|²
is nicer to study than the Fourier transform of
|1/(1−3^{-(1/2) - ix})(1−5^{-(1/2) - ix})|
The Fourier transform of |f| is a somewhat cursed thing to study; there are no nice formulas for it. I just sort of backed into thinking about it.
Your pictures of envelopes are very nice.
27/25 is important for the following reason - I explained it in my posts. The functions above are not periodic but only "quasi-periodic" in the technical sense. Suppose P is a number for which there are integers m, n with
P ≈ m 2π/ln(3) ≈ n 2π/ln(5)
Then
f(x) = 1/(1−3^{-(1/2) - ix})
has
f(x+P) ≈ f(x)
and
g(x) = 1/(1−5^{-(1/2) - ix})
has
g(x+P) ≈ g(x)
so their product is almost periodic with period P - we say P is a "quasi-period".
When we look for m,n such that
m 2π/ln(3) ≈ n 2π/ln(5)
we are looking for rational numbers m,n with
m/n ≈ ln(3)/ln(5) ≈ 0.682606194...
The first few good approximations are
2/3 ≈ 0.6667
13/19 ≈ 0.6842
15/22 ≈ 0.6818
The very first one being a good approximation to ln(3)/ln(5) implies that
3 ln(3) ≈ 2 ln(5)
or exponentiating both sides,
3³ ≈ 5²
i.e.
27 ≈ 25
So 27/25 being close to 1 makes the function
1/(1−3^{-(1/2) - ix})(1−5^{-(1/2) - ix})
quasiperiodic with quasiperiod
P ≈ 2 × 2π/ln(3) ≈ 3 × 2π/ln(5)
But of course these two numbers aren't exactly equal:
2 × 2π/ln(3) ≈ 11.44
3 × 2π/ln(5) ≈ 11.71
So our function is approximately periodic with a quasiperiod around 11.5, but its two factors f and g drift out of phase... and to understand that we need better rational approximations of
ln(3)/ln(5)

2026-02-17 20:40:49 UTC

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@nprofile…920r
https://mathstodon.xyz/@johncarlosbaez/116041915039490427

2026-02-17 08:41:41 UTC

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@nprofile…w9hq - If I didn't have good theoretical/intuitive reasons to expect these beats I would be more suspicious of what I'm seeing. I am suspicious of the height and especially the *width* of the spikes in the Fourier transform of the absolute value of this zeta function - they could easily be delta functions. But I'm not surprised to be getting spikes at frequencies of the form ln(3ʲ5ᵏ)/2π), because if we omit the absolute value, the Fourier transform of the zeta function itself is a weighted sum of delta functions at all possible numbers of the form ln(3ʲ5ᵏ)/2π).
Still, your point is a good one: if I ever want to publish anything about this I should check it more carefully.

2026-02-17 07:27:25 UTC

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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 07:05:15 UTC

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@nprofile…gvxt - I don't know what "sort the partials out to see what it looks like without them" means.
I'm trying to understand all the rational numbers a for which the Fourier transform spikes at ln(a)/2π.

2026-02-17 06:52:44 UTC

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@nprofile…gvxt @nprofile…920r - yes, you can call some of them overtones. But I'm not sure that's the full explanation of the pattern we're seeing here! It should be related to the continued fraction expansion of ln(3)/ln(5) - or in other words, which powers of 3 are close to powers of 5. I showed that's part of the story. But I don't know the full story.

2026-02-17 06:42:42 UTC

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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 06:13:10 UTC

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@nprofile…920r - yes, that was a great idea.

2026-02-17 04:44:26 UTC

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@nprofile…920r - yes, now that I''ve learned a lot by guessing things that might be good. It would have terrible earlier.

2026-02-17 01:53:12 UTC

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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 UTC

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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 UTC

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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 UTC

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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 23:46:48 UTC

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@nprofile…3dur - I saw those too, and I'm trying to understand them, but I hadn't noticed that there were 3 interlocking waves. So thanks for pointing that out: that's important.
I had said that the distance between successive wave crests is about 62.69, which actually looks wrong to me now. Anyway, the right thing to look at might be the distance between successive wave crests of the same color. Or maybe not!

2026-02-16 23:43:56 UTC

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@nprofile…zkp2 - you're right - thanks!

2026-02-16 22:41:51 UTC

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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 UTC

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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 21:44:50 UTC

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@nprofile…3n8d - umm, no, because no commtuative ring where 1+1 = 0 embeds in any commutative ring without that property, and none without that property embeds in one that does have that property. This is why commutative algebraists (dually known as algebraic geometers) divide the world into different 'characteristics'. A ring of characteristic n has 1 + 1 + ... + 1 = 0 where you sum n ones. They especially like fields, and for fields each field either has characteristic p for some prime p or 'characteristic zero', meaning no sum of 1's can equal zero.
I am studying ℤ/3 × ℤ/5, which is a product of a field of characteristic 3 and one of characteristic 5. For many purposes these should be studied separately, but I enjoy seeing these 'beats' which appear when you combine different primes.

2026-02-16 21:18:28 UTC

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@nprofile…3n8d - if the rings are "too infinite", like the ring of continuous real-valued functions on the real interval, the sum in the definition of the Hasse-Weil zeta function doesn't converge and we should probably leave it alone.
For finite boolean rings it's not so bad! For the 2-element ring with 'and' as times and 'exor' as plus, the zeta function is
1/(1 - 2ˢ).
(Whoops - just noticed a pile of minus sign typos I need to fix. Thanks!)

2026-02-16 20:45:30 UTC

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@nprofile…7k0h - No, that's not how it works - the actual definition is here:
https://en.wikipedia.org/wiki/Hasse%E2%80%93Weil_zeta_function#Definition
But for rings that are products of fields with a prime number of elements, like
ℤ/3 × ℤ/11
or
ℤ/2 × ℤ/7 × ℤ/19 × ℤ/23
the zeta function takes a simple form, namely
1/(1 - 3⁻ˢ)(1 - 11⁻ˢ)
or
1/(1 - 2⁻ˢ)(1 - 7⁻ˢ)(1 - 19⁻ˢ)(1 - 23⁻ˢ)

2026-02-16 20:31:43 UTC

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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 UTC

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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 UTC

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

2026-02-16 06:23:06 UTC

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@nprofile…3mqd - got it. Fixed! Thanks!

2026-02-14 06:36:04 UTC

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@nprofile…6fjx - I remember: that's a weird one.

2026-02-13 23:06:48 UTC

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@nprofile…lvpm - "it feels so trivial sharing science communication or other fun things". Yes, and I've also decided I don't add a lot of value warning people about all the dangerous things going on now, since there are so many other doing this. So one thing I can do that seems useful and not trivial is sharing ecological good news, like this:
https://mathstodon.xyz/@johncarlosbaez/115911764922880272

2026-02-13 08:14:42 UTC

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@nprofile…84yf - I believe the only problem with "g-adics" for non-prime g is that they don't form a field. They still form a commutative ring.

2026-02-13 07:51:01 UTC

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 21:01:35 UTC

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@nprofile…qh5j - thanks, I am very interested in this 13 stuff. You'll note it reappears in the Tsol'kin calendar:
https://mathstodon.xyz/@johncarlosbaez/116058959548787702

2026-02-12 20:58:18 UTC

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@nprofile…8qh7 - I fixed the "360 years" typo.
"Julius Scalier" was Gro-Tsen's typo for "Julius Scaliger" - he thought that the "Julian day" concept was named after the 16th-century scholar Julius Scaliger. In fact that "Julian day" concept is based on the Julian Period proposed by Joseph Scaliger, a classical scholar, in 1583. Joseph's father was Julius Scaliger! Yet apparently the Julian day was named after the Julian calendar, which was named after Julius Caesar.
So I will just delete that passage from my toot - the whole subject is too confusing!
https://en.wikipedia.org/wiki/Julian_day

2026-02-12 20:00:59 UTC

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@nprofile…84yf - it would be cool if some far-out Mayan priest dreamt up the 20-adics!

2026-02-12 17:59:42 UTC

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The Meso-American Long Count identifies any day by counting how many days passed since the world was created. This count is more or less in base 20, except that the second “digit” is in base 18, since they liked a year that was 18 × 20 = 360 years long. So,
7.16.6.16.18
would mean
7 × 144,000 + 16 × 7,200 + 6 × 360 + 16 × 20 + 18 = 1,125,698 days
after the world was created. But if the first digit was 8 instead of 7, the date would be 144,000 days later.
The Olmec's other calendar, the Tzolkʼin, uses a 260-day cycle. Each day gets its own number and name: there are 13 numbers and 20 names. And the rock the Stirlings found had inscribed not only the last four digits of the Mesoamerican Long Count digits, but also the Tzolkʼin day: 6 Etz’nab.
Here’s why 7 was the only possible choice of the missing digit. Because the last four Long Count digits (16.6.16.18) are fixed, the total day count must be
B × 144,000 + 117,698.
for some B. But
144,000 = 0 mod 20,
and there are 20 different Tzolkʼin day names, so changing B by one doesn't change Tzolkʼin day name. But there are 13 different Tzolkʼin day numbers, so changing B by one adds
144,000 ≡ –1 (mod 13) days to the Tzolkʼin day number.
This means that after the day
7.16.6.16.18 and 6 Etz’nab
the next day of the form
N.16.6.16.18 and 6 Etz’nab
happens when N = 7+13. But this is 13 × 144,000 days later: that is, roughly 5,094 years after 32 BC! Far in the future!
So, while 32 BC seemed awfully early for the Olmecs to carve this stone, there’s no way they could have done it later. (Or earlier, for that matter.)
(2/2)

2026-02-12 17:45:56 UTC

Math is everywhere. Here's a tale where a bit of number theory let a couple of archeologists, Marion and Matthew Stirling, figure out that the Olmec civilization was incredibly old.
The Stirlings found the stone here in Mexico, and guessed that the date written on it was 7.16.6.16.18. In the calendar used by the Olmecs and other Central American civilizations, this corresponds to September 3, 32 BC.
But the first digit was missing from this part of the stone! All the Stirlings actually saw was 16.6.16.18. And the first digit was the most significant one! If it were 8 instead of 7, the date of the stone would be much later: roughly 362 AD, when the Mayans were in full swing.
The Stirlings guessed that the first digit must be 7 using a clever indirect argument.
The Olmecs and Mayans used two calendars! In addition to the calendar I just mentioned, called the Mesoamerican Long Count, they also used one called the Tzolkʼin. This uses a 260-day cycle, where each day gets its own number and name: there are 13 numbers and 20 names. And inscribed on this stone are not only the last four digits of the Mesoamerican Long Count digits, but also the Tzolkʼin day: 6 Etz’nab.
This is what made the reconstruction possible!
(1/n)
https://media.mathstodon.xyz/media_attachments/files/116/058/840/198/220/342/original/c637fb5a58de01af.jpg

2026-02-12 17:18:26 UTC

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@nprofile…vxcd - complicated mixed feelings about this... thanks.

2026-02-12 17:16:25 UTC

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@nprofile…lvpm - it's interesting to hear your story. My height of solidarity and fun on the internet came in the early 90's on usenet. Then came the era of blogs, and then I put a lot of energy into Twitter. The day Musk bought that, I quit. I soon moved to Mastodon. It started out fun; then some of my best friends moved to Bluesky and things got complicated.... and then bad political news overshadowed everything, making it embarassing to post anything lighthearted or even serious math and physics. After a while I decided to continue posting mostly things like that and also good news about ecology (the bad news is too easy to find). I find it necessary for my own equilibrium.

2026-02-12 03:49:17 UTC

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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 08:50:43 UTC

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@nprofile…chx4 - it's not me, it's Gro-Tsen you have your quarrel with. You may well be right. It's an interesting issue. But right now I'm obsessed with how Marion Stirling figured out the missing digit on that broken stone.

2026-02-11 08:48:57 UTC

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@nprofile…3mqd - In that case you would have succeeded.

2026-02-11 02:11:57 UTC

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

2026-02-10 19:24:08 UTC

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

2026-02-10 19:18:30 UTC

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@nprofile…kep0 @nprofile…8qh7 - An archeologist may get a big head yet still date mummies.

2026-02-10 19:10:15 UTC

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@nprofile…2q3z - that's a great question, and there should be a paper that answers. I doubt Marion Stirling simply guessed that initial 7 based on no evidence at all! She (or her husband) would have given some explanation.
Time for some research!

2026-02-10 17:54:26 UTC

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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 UTC

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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 UTC

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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 UTC

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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 UTC

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

2026-02-10 17:10:56 UTC

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@nprofile…um4f - I hadn't known either until very recently when my research on the math of music led me to his work. Very sad, and like you say, part of the sadness is the dissolution of the old internet.
@nprofile…8qh7 @nprofile…xdkd

2026-02-10 17:09:49 UTC

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@nprofile…8qh7 - Jesus effing Christ! I've been having lunch with Dave Benson after the category theory seminar for months last year and never knew he had written this! (We started out talking a bit about exceptional groups but he knows too much about sporadic finite simple groups and I know too much about physics for this to be entirely comfortable.)

2026-02-10 06:29:08 UTC

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@nprofile…6re0 - that's fake. Here's the truth:
https://wsbt.com/news/nation-world/reps-khanna-and-massie-review-unredacted-epstein-files-at-doj-amid-transparency-push

2026-02-10 02:41:52 UTC

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@nprofile…h6a3 - thanks, I'll check out this show in preparation for my own pilgrimage!

2026-02-09 18:20:12 UTC

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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 UTC

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 17:17:07 UTC

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@nprofile…uk82 - oh! I'm starting to read about this, and so far it seems like there were extensive trading networks in the Americas. I haven't read about anything particularly linking Puebloans to Mayans genetically. But I'll keep reading.
The Aztecs started up around 1300, while the Chaco Canyon civilization ended around 1126, probably due to a drought.

2026-02-09 05:33:34 UTC

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@nprofile…ygne - was it easy to find everything when you got there?

2026-02-09 05:32:21 UTC

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

2026-02-09 01:07:42 UTC

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@nprofile…f5pz - oh, interesting! I didn't know that about the thick-billed parrots.

2026-02-08 23:41:01 UTC

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@nprofile…ju9m - this is amazing! Or rather: many of us don't learn enough about this in school, so we tend to underestimate these people.

2026-02-08 23:10:04 UTC

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@nprofile…a638 - sounds cool. I went to White Sands once, but have no clear picture of what's between that and Chaco Canyon.

2026-02-08 23:08:28 UTC

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@nprofile…8qh7 @nprofile…kep0 - indeed the active chemical of chocolate, found on the cups at Chaco Canyon, is called 'theobromine'. And the 'theo-' is no accident!
The word theobromine has nothing to do with bromine. It comes from Theobroma, the genus of the cacao tree, whose name comes from the Greek roots theo ("god") and broma ("food"), meaning "food of the gods"!
🍫

2026-02-08 22:49:36 UTC

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@nprofile…8qh7 - the use of honey suggests they weren't insane.

2026-02-08 22:42:15 UTC

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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 UTC

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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 UTC

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.
(1/3)
https://media.mathstodon.xyz/media_attachments/files/116/037/400/946/848/573/original/2a0a486577f4affe.jpg

2026-02-08 22:22:31 UTC

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@nprofile…l7vu - thanks, that was a great listen. I'm planning to do a lot of study before visiting various sites this spring. This video was a good springboard toward digging deeper into these matters, because it situated Chaco in an interacting network of different cultures.

2026-02-08 03:54:59 UTC

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@nprofile…t5a9 - I said it would be fun, but I don't know if I'll ever do it: I have too many project half-finished. Right now my main passion is the mathematics of tuning systems.

2026-02-08 01:14:00 UTC

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@nprofile…t5a9 - thanks! I try to b careful, but I'm also indebted to a referee who found more typos.
It would be fun to take that paper on motives, expand it, and add an introduction to the Tate conjecture, Hodge conjecture and standard conjectures - for people who don't know much algebraic geometry!
For now these expository papers are my favorites. They touch on some of these conjectures and why they matter:
James Milne, Motives: Grothendieck’s dream, in Open Problems and Surveys
of Contemporary Mathematics, eds. Lizhen Ji, Yat-Sun Poon and
Shing-Tung Yau, International Press, Somerville, Massachusetts, 2013, pp.
325–342. Available at https://www.jmilne.org/math/xnotes/mot.html
James Milne, The Riemann Hypothesis over finite fields: from Weil to the
present day, in The Legacy of Bernhard Riemann after One Hundred and
Fifty Years, vol. II, eds. Lizhen Ji, Frans Oort and Shing-Tung Yau, International
Press, Somerville, Massachusetts, 2015, pp. 487–565. Available at
https://www.jmilne.org/math/xnotes/pRH.html
James Milne, Motives over finite fields, Proc. Sympos. Pure Math. 55, Part
1, AMS, Providence, 1994, pp. 401–459. Available at https://www.jmilne.
org/math/articles/1994a.html

2026-02-07 17:29:06 UTC

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@nprofile…t5a9 - I'm not sure I learn very fast, but if I do it's probably by ignoring details until they seem to become necessary.
If you want to learn about motives, you might try my talk "Motivating motives", or the slides for that, or the paper I wrote based on those:
https://math.ucr.edu/home/baez/motives/
They have an intimidating reputation, but I blundered in and tried to understand them. The idea is that any variety can be chopped into basic 'pieces' in a very abstract sense, which are not usually subsets, since they can have a negative number of points.

2026-02-07 00:44:01 UTC

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@nprofile…t5a9 - Does a typical good, meaty first course on algebraic geometry give some intuition for the bigrading on the cohomology of a smooth complex variety? I don't know. I never took such a course, but I think there are lots of ways such a course could go. This stuff is in Griffiths and Harris, but that book says a lot more about complex manifolds than some intros to algebraic geometry. (That's why I like it: I'm really more of an analyst.)
I feel I understand why the Hodge conjecture says "the world is nice and simple". You take the two most obvious properties of a cohomology class that comes from a rational combination of algebraic cycles and say "that's it - that's all we need!" So in other words, if it's true a mysterious gap between algebraic topology and algebraic geometry is gone.
Looking at Deligne's intro, I see he makes this vague feeling much more precise: he says the Hodge conjecture implies that
the category of motives over ℂ is a full subcategory of the category of Hodge structures. I hadn't known that, but this is a big deal. It turns this category of motives, which is important but very elusive, into something far more concrete that can be described using linear algebra. So, thanks for pressing me on this point! I've spent a bunch of time trying to understand the bare basics of motives, and this is a step forward.
I don't think a good meaty first course in algebraic would explain motives or Hodge structures, though!

2026-02-06 17:23:32 UTC

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@nprofile…t5a9 - what do you want to know by asking why the Hodge conjecture is a "big deal"? Different people have different ideas of what counts as a big deal.
Are you trying to understand why the Hodge conjecture is a natural question? Or are you wondering what consequences it would have?
It seems a very natural question to me, and I could explain it at various levels of depth. One problem is that it's less natural than the "integral" Hodge conjecture - the original Hodge conjecture, which was disproved by Atiyah and Hirzebruch in 1961. The current version, the "rational" version, is a fallback, so it's less natural.
The original version says that every cohomology class in a smooth complex variety that could possibly come from an integral linear combination of subvarieties actually does. That's false. The current version replaces the word "integral" by "rational".

2026-02-05 23:13:31 UTC

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@nprofile…xuq7 - I think that first section was really just a rapid attempt to convince people that there's more to algebraic geometry than a pile of slick abstract definitions. It may have scared as many students as it intrigued.

2026-02-05 23:11:11 UTC

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@nprofile…rgsr - I didn't actually ever sleep through a class. I just never took a course on commutative algebra.

2026-02-05 17:34:08 UTC

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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 UTC

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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 UTC

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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 UTC

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!

2026-02-05 07:26:32 UTC

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@nprofile…a3ey - it's a great read, though not all at once.

2026-02-04 23:06:55 UTC

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@nprofile…jrcx - it looks like an error-ridden text. Throw it out and get a better book on number theory!

2026-02-04 23:02:31 UTC

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@nprofile…0g3j - you can change the conclusion, right?

2026-02-04 22:59:30 UTC

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@nprofile…sz8l @nprofile…u3x7 - It's pretty obvious that having Terry Tao use their software is better than any advertisement that money could buy.

2026-02-04 22:46:28 UTC

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@nprofile…az9a - I was explaining the situation to my followers around the world, many of whom don't know the intricacies of the US federal system. I don't think Trump cares about laws and the constitution, but there's a whole network of people who do care, so when he breaks the laws they get energized to push back, and this does have an effect.

2026-02-04 08:17:30 UTC

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@nprofile…ezv6 - there's no link there.

2026-02-04 02:16:32 UTC

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@nprofile…az9a - it would be illegal, indeed against the US Constitution, for Trump to attempt to take over the election process. In fact I recently added a remark about this to my original post, for people unfamiliar with the US system.

2026-02-03 19:12:06 UTC

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@nprofile…5ahq - I'll check it out.

2026-02-03 18:22:52 UTC

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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.