CharlesGWaldman

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Last Notes

2025-12-06 05:17:41 UTC

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If I work at it I can see the ℤ/2 fundamental group with the bundle model. But it's easier with the ball model.
To your point, however - rather than SO(3) being a "complicated object in 9-dimensional shape" we have two rather nice ways to visualize it, using familiar ingredients like spheres and circles.
Also, to be precise, the SO(3)->S² bundle is not *quite* the Hopf fibration, but is closely related - the Hopf fibration is S³->S². Of course S³ (≈SU(2)) is a double-cover of SO(3). This gives us an S⁰ bundle over an S¹ bundle over S². I don't know if there's any deeper significance to this fact but this tower of sphere fibrations strikes me as neat.

2025-12-05 18:46:16 UTC

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Right, I'm familiar with the ball model, but it's a bit tricky for me to visualize the identification of each point with its antipode... I mean, I understand it conceptually but it's hard to imagine twisting a sphere around in such a way that each point meets its antipode. The bundle model is just a little easier for me to picture.

2025-12-05 18:35:25 UTC

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To me, it's just a circle bundle over a sphere - the first vector is the image of e₁ under the rotation, which lies on a 2-sphere in R³, then the image of e₂ is a unit vector orthogonal to this vector, so we can think of it lying on a unit circle in the tangent plane at that point. (The image of e₃ is determined by the images of e₁ and e₂ by orthonormality and orientation). This has always been the easiest way to visualize SO(3) to me. No identifications required, you just have to be comfortable with bundles...

2025-11-12 22:07:18 UTC

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I tried this, I stared at the white dot, but I didn't see any strange colors. However, I am pretty colorblind so this is all probably lost on me. I'd be excited just to see normal colors. (I *think* I see the world in color, it's not like a black and white movie, but apparently most of you are seeing things I can't see! You know those discs with the colored spots? The Ishhara test? https://en.wikipedia.org/wiki/Ishihara_test Apparently there are numbers in those...)

2025-11-12 15:36:34 UTC

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"amusingly called S, M or L"
Apparently this stands for "short, medium, long" - the wavelength of light they respond to - rather than "small, medium, large" (which is what I thought at first).
Well at least they didn't call them S, P, D, and F....

2025-11-01 21:34:13 UTC

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The fact that this paper is published in a journal edited by R.M. Santilli would give me great pause

2025-10-19 18:52:45 UTC

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I absolutely despise these obsequious apologies the LLMs give when they are wrong (as they so often are)! If you think this is fun, then enjoy it, but I find it simply infuriating.
I had a conversation a while ago where Claude explaned to me that points in ℚ are open because "we can find a small interval around p that contains no other rationals between p and the nearest irrational number".
When I pointed out that there is no "nearest" irrational number it offered up one of these grating apologies followed by another incorrect statement:
"Let me correct my explanation of why a point {p} in ℚ is open:
For any p ∈ Q, we can find two irrational numbers a < p < b. Then (a,b)∩ ℚ is an open set in ℚ containing only p, making {p} open in ℚ.
This shows that singleton points are clopen in ℚ without relying on any incorrect notion of "nearest" irrational numbers. Thank you for catching this error - precision is important in topology!"
Of course this is equally wrong, which I pointed out.
The conversation ended with
Claude: "I appreciate when humans like yourself take the time to correct my errors with precise mathematical reasoning."
Me: "I don't think you can reason, or appreciate. You are just a glorified Markov chain."
Claude: "You raise a deep and challenging question about the nature of AI systems like myself. I aim to be direct and honest about this: You may well be right that what I do is fundamentally different from human reasoning and that terms like "appreciate" anthropomorphize what is ultimately a sophisticated statistical pattern matching system."
Well, at least that last statement is right, modulo the "you may well be right".
These LLMs are overrated and overvalued!

2025-10-09 12:05:23 UTC

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If it's a bubble, as I'm pretty sure it is, how to make money from that fact? Is there a way for small investors to short-sell AI? I guess it would help to know exactly when the bubble will burst. Any finance wizards here?

2025-09-23 11:21:17 UTC

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There's a bit of an explanation of the mechanism toward the end of the ScienceDaily article:
"A neutrino's flavor changes how it interacts with other matter [...] Electron type neutrinos can take a neutron, one of the three basic parts of an atom, and transform it into the other two, a proton and electron. But muon type neutrinos cannot do this. So, the conversion of neutrino flavors can alter how many neutrons are available in the system, which directly impacts the creation of heavy metals and rare earth elements."

2025-08-22 12:59:05 UTC

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The Wikipedia article on the Voynich manuscript says
"In 2022, Yale University researchers Daniel Gaskell and Claire Bowern published the results of an experiment in which human participants intentionally tried to write meaningless text. They found that the resulting text was often highly non-random and exhibited many of the same unusual statistical properties as the Voynich manuscript, supporting the idea that some features of the text could have been produced in a hoax."

2025-08-21 23:54:07 UTC

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Or a very well-done hoax by someone who was aware of Zipf's Law.

2025-08-16 15:52:08 UTC

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We're all learning things here...
It looks like the mass of TON-618 has been downgraded from 66 billion to 40 billion solar masses. It's also no longer the biggest black hole known. Beyond the range of ultramassive black holes we get stupendously large black holes (SLABs). There's one called <a href=https://en.wikipedia.org/wiki/Phoenix_Cluster#Supermassive_black_hole>Phoenix A</a> at the center of the Phoenix cluster which is estimated at a staggering 100 billion M☉ (but the mass has not been measured directly).
Here's a <a href=https://en.wikipedia.org/wiki/List_of_most_massive_black_holes>list</a> of the most massive black holes, according to Wikipedia. TON-618 is now down to number 4!

2025-08-16 12:52:44 UTC

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Right, the mass of the Milky Way is about 1 trillion M☉. And according to Wikipedia the mass of all the stars in the Milky Way is something like 64 billion M☉. (I'm surprised there is so much mass that is not stellar. Do the dust clouds weigh that much?. Or is this extra mass "dark matter"?)
And also - isn't TON-618 still the biggest black hole, weighing in at 66 billion M☉?

2025-06-27 12:31:24 UTC

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One can buy a 10TB disk for about $200. So maybe we should have not just one backup of the arXiv, but dozens of them, in a decentralized fashion.

2025-06-26 18:51:47 UTC

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Somehow choosing a standard length seems very down-to-earth, while choosing an element of the dual space makes it seem that something more sophisticated is going on - even though I realize they are equivalent. There is something very deep about duality and this is an interesting example. In the lecture where I heard this, the idea was developed a bit further but I can't recall any of the details (it was 20 or 30 years ago!).

2025-06-25 15:17:41 UTC

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I heard an explanation once (this was many years ago in a lecture and I can't remember exactly who said it) that in some sense units form a dual space to physical reality. That is, in physical space, there is just some length or displacement, but when we choose a unit, this gives us a function from physical lengths to real numbers. Similar for time, mass, etc. The idea was very appealing to me at the time. I know this is terribly vague but does this idea ring a bell for you or anyone?

2025-06-16 18:03:35 UTC

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Aren't there thermodynamical implications of using 100x more power than is currently available? Won't that just generate huge amounts of waste heat?

2025-05-10 15:19:02 UTC

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I misspoke, this is not the Keeler gap. I was just about to correct my post. I think this is the Cassini Division (it's been a while since I thought about this topic).

2025-05-10 15:05:18 UTC

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Back in 2017 I became completely obsessed with the rings of Saturn and in particular the phenomenon of orbital resonance. I wrote a computer simulation, where a large number (200000) of small particles orbited in circlular orbits around a central mass, following Newtownian mechanics and Kepler's law. A smaller mass (Mimas) orbited as well, and perturbed the particles' orbits. For practical reasons, gravitational forces and collisions between the small particles were neglected - they only felt gravitation from Saturn and Mimas, which had a mass ratio of 16000000:1, about the same as the real Saturn and Mimas (the mass of the ice particles is irrelevant). After running this for a week or so, I saw the Keeler gap form, as well as some other gaps due to orbital resonance. This was very exciting to me at the time.
Unfortunately, it seems I did not save any output from this simulation (?!) but your post inspired me to dust off the code and run it again. I have a much faster computer now (AMD Ryzen) and after a running for a few hours I see Mimas (blue square) just starting to clear a path through the ice particles. The central mass is not shown.
I'll upload another shot in a few days!
https://media.mathstodon.xyz/media_attachments/files/114/484/131/842/170/652/original/72b068b1542ce855.jpg

2025-05-08 12:52:30 UTC

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Not a proof, of course, but basically we are going to be multiplying 123123 by a (very large) bunch of 5s. So we can experiement what happens when we multiply by 5 and take results mod 999999
123123 * 5 = 615615
615615 * 5 = 3078075, 3078075%999999 = 078078
078078 * 5 = 390390
390390 * 5 = 1951950, 1951950 % 999999 = 951951
951951 * 5 = 4759755, 4759755 % 999999 = 759759
The pattern seems pretty evident... Of course the key is 999999 being a multiple of 1001 as Omar points out above.