@johncarlosbaez@mathstodon.xyz
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johncarlosbaez

@johncarlosbaez@mathstodon.xyz

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

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johncarlosbaez, (edited ) to random
@johncarlosbaez@mathstodon.xyz avatar

About half the periodic tables you see are wrong. The only question is: which half?

Check out this one from Encyclopædia Britannica. See the row of elements in yellow-green near the bottom? They start with element 58, cerium and end with element 71, lutetium. There are 14 of them. They're called 'lanthanoids'.

Okay. But note that lanthanum itself, element 57, is up somewhere else. It's also called a lanthanoid, and it's under two other elements in yellow-green called 'rare earths'.

Next compare the periodic table on Wikipedia.

(1/4)

johncarlosbaez, (edited )
@johncarlosbaez@mathstodon.xyz avatar

On Wikipedia, the top row of elements in green start with element 57, lanthanum, and end with element 70, ytterbium. Now lutetium, element 71, is up near hafnium!

Which periodic table is right? Or is it just a matter of taste?

(2/4)

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

I have strong opinions about this, explained in this blog article:

https://johncarlosbaez.wordpress.com/2022/01/21/the-periodic-table/

It's been a heated issue for years.

(3/4)

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@Pashhur - that's right, but a more chemical way to say it is: Brittanica says that the elements cerium-lutetium are a family of 14 similar elements, with lanthanum being an outlier, while Wikipedia says that lanthanum-ytterbium are a family of 14 similar elements, with lutetium being an outlier.

The importance of the number 14 is that the f subshell contains 14 electrons, and the familiy of similar elements called 'lanthanides' are the first elements to contain electrons in the f subshell.

johncarlosbaez, (edited )
@johncarlosbaez@mathstodon.xyz avatar

The only reason we draw the lanthanoids (and actinoids) in a separate row below the rest of the table is that including it makes the table very long.

If we draw the table in its long form, the choice between approaches can be made very stark, as shown below!

This table is from here:

• Eric Scerri, Provisional report on discussions on group 3 of the periodic table, https://www.degruyter.com/document/doi/10.1515/ci-2021-0115/html

Scerri argues for including lanthanum with the lanthanides.

On the other hand, this detailed article gives 10 reasons for including lutetium with the lanthanides:

• René E. Vernon,The location and composition of group 3 of the periodic table, https://link.springer.com/article/10.1007/s10698-020-09384-2

(4/4)

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@maz - truth and justice may be starting to win in this one case!

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@mattmcirvin - hydrogen is certainly another ambiguous case! Is it an element with its outermost shell missing just one electron, or an element with just one electron in its outermost shell? BOTH!

It's not like an ultra-virulent halogen, even nastier than fluorine, that's for sure. Is it like a ultra-benign alkali metal, even more gentle than lithium? Not so much at standard temperature and pressure! I don't know what solid hydrogen is like. But it's probably not shiny like a metal, not until it's ultra-compressed like in Jupiter's core.

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@BartoszMilewski - I'm intrinsically more optimistic about the power of math than you are; I think nature has the nice property that simple approximate theories like the shell model of atomic orbitals get us quite far, but break down in ways that point us toward better theories, e.g. in this case Schrodinger's equation. The periodic table is based on the idea of shells and subshells, which would be valid if electron-electron interactions were almost negligible - just big enough to make the Madelung rules hold.

https://johncarlosbaez.wordpress.com/2021/12/08/the-madelung-rules/

johncarlosbaez, to random
@johncarlosbaez@mathstodon.xyz avatar

Guilty! This man is a felon.

vacuumbubbles, to random
@vacuumbubbles@mathstodon.xyz avatar

It's interesting how expressive Feynman diagram notation can get if you just think of them as tensor contractions instead of particles bouncing all around the place.

Or in other words, I'd like to enter the "unhinged notation" contest please

image/png
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johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@vacuumbubbles - have you seen Cvitanovic's book 𝘎𝘳𝘰𝘶𝘱 𝘛𝘩𝘦𝘰𝘳𝘺: 𝘉𝘪𝘳𝘥𝘵𝘳𝘢𝘤𝘬𝘴, 𝘓𝘪𝘦'𝘴, 𝘢𝘯𝘥 𝘌𝘹𝘤𝘦𝘱𝘵𝘪𝘰𝘯𝘢𝘭 𝘎𝘳𝘰𝘶𝘱𝘴? It's a treatment of group representation theory based on some of the ideas you're pointing out, developed from his extensive work on quantum field theory. Even if it's too much to read the whole thing, you'd probably enjoy a look at the pictures:

https://birdtracks.eu/version8.9/GroupTheory.pdf

johncarlosbaez, to random
@johncarlosbaez@mathstodon.xyz avatar

Given how the Catholic Church pressured Galileo to give up his belief that the Earth goes around the Sun, and the Inquisition held him in house arrest for the last 9 years of his life, it's somehow fitting that now his middle finger is a relic.

I'm not kidding, there's even a Wikipedia article just about this:

https://en.wikipedia.org/wiki/Galileo%27s_middle_finger

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@MartinEscardo - not quite the same, but Alexandra Elbakyan is the "Pirate Queen of Science" who broke the rules and made scientific knowledge free to the masses.

https://www.sci-hub.st/alexandra

gregeganSF, to random
@gregeganSF@mathstodon.xyz avatar

Gram negative bacteria have been laughing at us:

https://pubmed.ncbi.nlm.nih.gov/20419407/

but now we might finally have our revenge, with lolamicin:

https://www.nature.com/articles/d41586-024-01566-8

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@gregeganSF - so much to learn about here!

vacuumbubbles, to random
@vacuumbubbles@mathstodon.xyz avatar

Right now I'm having the problem that I vaguely remember a (seemingly) useful equation, but can't find the reference. Now I'm questioning whether I actually saw this equation in reality or if it was just a nonsensical dream. Does anyone else here have this problem?

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@vacuumbubbles - hmm, I forget lots of stuff but I don't remember a case of this particular scenario.

andrewstroehlein, to random
@andrewstroehlein@mastodon.social avatar

Why are authoritarian-minded rulers so thin-skinned?

In Uzbekistan, for example, authorities are increasingly imprisoning people for “insulting the president online.” The criminal charge could get you 5 years in prison. It’s absurd.
https://www.hrw.org/news/2024/05/30/uzbekistan-imprisoned-insulting-president-online

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@rancoisse @andrewstroehlein - indeed, these massively repressive systems are the opposite of stable: they're like sinking ships where everyone in charge has to keep bailing.

hanse_mina, to hungary
@hanse_mina@mastodon.social avatar

’s prime minister is often the odd man out in the and the NATO military alliance. Since ’s invasion of , Viktor has been essentially alone in pushing back against Western efforts to come to ’s aid. With the war now in its third year and Ukraine in dire need of assistance, the nationalist leader’s stonewalling is exasperating Hungary’s allies and raising questions about his government’s true allegiances.

https://www.bloomberg.com/news/articles/2024-05-29/why-hungary-s-orban-is-stonewalling-eu-nato-allies-over-ukraine-aid

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@hanse_mina - The EU should quickly find a way kick out Hungary.

joey, to random
@joey@mathstodon.xyz avatar

Dumb question: what's geometry?

I'm interested in categories and toposes from a logic and PL viewpoint. But people keep taking about"geometric intuitions".

The last time I studied geometry was in high school and it was things like "area of a triangle" and "sum of internal angles of an n-gon".

I've heard of Algebraic Geometry, but I'm not sure what makes it "Geometry". What's the link? Is it topology, or metric spaces, or what? How do we get from there to here?

johncarlosbaez, (edited )
@johncarlosbaez@mathstodon.xyz avatar

@antidote @joey wrote: "Or even what people do with these roots once they’ve found them."

Algebraic geometry is the study of shapes you can describe with polynomials, e.g. this shape

𝑥²+𝑥𝑦+𝑦²=5

is an ellipse. Here's a theorem from algebraic geometry: if you take any 6 points on an ellipse and connect them by lines as shown here, these lines meet in 3 points (G, H, K) which always lie on a line. This was discovered by Pascal in 1639 when he was 16 years old.

Algebraic geometry has progressed a lot in the last ~400 years, so the current work is harder to quickly describe, and a lot of it concerns higher-dimensional shapes or shapes described using number systems that are lot stranger than the real numbers, but it keeps building on the old stuff.

julesh, to random
@julesh@mathstodon.xyz avatar

Vaguely get the feeling that (multivariate) Gaussians are like the linear functions of probability

In which case the central limit theorem kinda looks like differentiation as in the best linear approximation

Is this a thing?

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@julesh - multivariate Gaussians are the probability measures that maximize entropy subject to constraints on the expected values of linear and quadratic functions (e.g. mean and variance). So they're an example of how quadratics are much nicer than higher-degree polynomials.

Just trying to sharpen your first comment.

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@dougmerritt @julesh - in 1d there are lots of functions preserved by the Fourier transform. There's an othonormal basis of functions ψₙ equal to a specific Gaussian times the nth Hermite polynomial. These are called 'energy eigenstates of the harmonic oscillator'. To take the Fourier transform of ψₙ you just multiply it by 𝑖ⁿ. So every fourth one is preserved by the Fourier transform, and taking linear combinations you get all functions preserved by the Fourier transform.

Something similar works in higher dimensions too.

johncarlosbaez, (edited ) to random
@johncarlosbaez@mathstodon.xyz avatar

The Kervaire Invariant Problem is a famous problem in topology that remains open in just one case: the case of 126 dimensions! See the Wikipedia article for an explanation.

Apparently this one remaining case has now been solved by Zhouli Xu, Weinan Lin and Guozhen Wang. Xu will reveal the answer tomorrow, Thursday May 30th, at the Princeton Algebraic Topology Seminar! I think a video will appear later online.

.....

Computing differentials in the Adams spectral sequence
May 30, 2024 - 01:00 - May 30, 2024 - 02:00
Zhouli Xu, University of California, San Diego

Online Talk

I will review classical methods computing differentials in the Adams spectral sequence, and then discuss some recent progress in joint work with Weinan Lin and Guozhen Wang. In particular, I will discuss the fate of h_6^2, resolving the Last Kervaire Invariant Problem in dimension 126.

https://en.wikipedia.org/wiki/Kervaire_invariant

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@highergeometer - It looks like he's heading toward an abstract proof using purely algebraic considerations, but he hasn't revealed yet whether there is or is not a 126-manifold with nonzero Kervaire invariant. It sounds like you've seen my guess:

https://categorytheory.zulipchat.com/#narrow/stream/274877-community.3A-our-work/topic/John.20Baez/near/441291190

but that guess seems more plausible if he's showing there is such a manifold.

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@MotivicKyle @highergeometer - what does "synthetic" mean here?

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@MotivicKyle @highergeometer - thanks so much for this overview!

pschwahn, to random German
@pschwahn@mathstodon.xyz avatar

The compact Lie group 𝐺₂, usually defined as automorphism group of the octonion algebra 𝕆 has (up to conjugacy) three maximal connected subgroups:

  • the subgroup preserving the algebra of quaternions ℍ⊂𝕆, which is isomorphic to SO(4),
  • the subgroup preserving some imaginary element like i, which is isomorphic to SU(3),
  • the subgroup SO(3)ᵢᵣᵣ given by the image of the irreducible, faithful 7-dimensional real representation of SO(3). This representation may be realized as the space of harmonic cubic homogeneous polynomials on ℝ³, or if you are a chemist, the space of f-orbital wavefunctions.

Now I wonder whether SO(3)ᵢᵣᵣ also has some interpretation in terms of the octonions. What irreducible action of SO(3) on the imaginary octonions is there?

@johncarlosbaez , do you perhaps have an idea?

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@pschwahn - can you get ahold of this paper?

Judd, B. R. "Selection rules within atomic shells." Advances in atomic and molecular physics. Vol. 7. Academic Press, 1971. 251-286.

I'm having trouble. But I see it contains a passage

"For electrons, Racah showed that the exceptional group G2 of Cartan (1894) can be …. It is nevertheless true to say that the sequence of groups established by Racah …"

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@pschwahn - thanks! This article is much clearer than Racah's when it comes to explaining how G₂ gets into the game. Unfortunately that's a low bar. In Section 2B he seems to give an explicit formula for generators of SO(3) and G₂ as subgroups of SO(7) (or SU(7)). Unfortunately I don't yet understand his notation! But it seems that it should be simple.

Are you making progress on your related problem: trying to relate the 7d cross product to the 3d one via a "cubing" operation? I really want to see how that goes.

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