@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
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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 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, (edited ) to random
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Using his periodic table, Mendeleev predicted the properties of many elements that hadn't been discovered yet. But he missed others - and more than half of his predicted elements were never found!

Two of the most interesting are coronium and newtonium. He predicted both to have atomic masses lighter than hydrogen!

Mysterious green spectal lines had been seen in the Sun's corona in 1869. Mendeleev was not the only one who thought it was a new element: coronium. He estimated its atomic mass as 0.4. It later turned out to be a highly ionized form of iron, Fe¹³⁺, meaning that 13 electrons get knocked off by the extremely high temperatures.

Even more interesting is newtonium. Back then, a lot of people thought light consisted of vibrations in a mysterious substance called the aether. In 1904, in a 50-page paper called “An attempt towards the chemical conception of the aether”, Mendeleev hypothesized that the aether was a noble gas! A bunch of new noble gases had just been discovered, forcing a new column in his table. He dubbed this one newtonium, and he estimated its atomic mass as 0.17.

For Mendeleev's other never-found elements, and how he estimated the masses of coronium and newtonium, read this:

• Gábor Lente, Where Mendeleev was wrong: predicted elements that have never been found, https://link.springer.com/article/10.1007/s40828-019-0092-5

Moral: you don't have to be right all the time.

johncarlosbaez,
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@nyrath - I guess he's making a nod to this elements fascinating history!

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

gregeganSF, to random
@gregeganSF@mathstodon.xyz avatar

“exeme” sounds like some new-fangled technical term from the cutting edge of genetics or information theory ... but actually it’s just an archaic Scottish word meaning to release or exempt.

https://en.wiktionary.org/wiki/exeme

johncarlosbaez,
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@gregeganSF - next book title?

dmm, to math
@dmm@mathstodon.xyz avatar

Here I tried to prove the Existence Theorem for Laplace Transforms. I don't know what the/a "conventional proof" looks like, but this is what I came up with.

A few of my notes on this and related topics are here: https://davidmeyer.github.io/qc/dirac_delta.pdf

As always, questions/comments/corrections/* greatly appreciated.

johncarlosbaez,
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@dmm - Your theorem says the Laplace transform exists if some condition holds, but at the end of your proof you say that the Laplace transform does not exist if some condition does not hold. So you seem to be proving the converse of the theorem statement.

Also, are you assuming 𝑓 is nonnegative? Your inequalities seem insufficient otherwise.

johncarlosbaez,
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@dmm - By your definition, to say f is of exponential order is the same as saying

f(t) ≤ Me^{ct}

and also

f(t) ≥ -Me^{ct}

The first inequality says f doesn't get too big in the positive direction, while the second says f doesn't get too big in the negative direction. You'll need both to show the Laplace transform is well-defined: after all,

-e^{e^t}

obeys the first inequality but not the second, and its Laplace transform is ill-defined. But your argument is only making use of the first inequality, so something is wrong. You either need to use both inequalities directly, or assume f is positive, prove the theorem given f(t) ≤ Me^{ct}, and then do the general case by splitting f into its positive and negative part.

More importantly, your argument seems to be proving the converse of the claimed theorem.

johncarlosbaez, (edited )
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@dmm - you're saying what I said, or more precisely part of what I said. Everything you just said is correct. But you're having trouble understanding why I said what I said. And I think the reason is that you haven't made up your mind what you're proving! The last sentence in your proof doesn't match the result claimed in your theorem. Which one are you actually trying to prove here?

johncarlosbaez,
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@dmm - sure. If it's stressful to do this publicly we can do this by DMs or not at all.

johncarlosbaez,
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@dmm - you can prove that the Laplace exists under the conditions you've just listed, by showing that the integral in the definition of Laplace transform converges. There are various standard ways to show convergence of integrals that will do the job here.

johncarlosbaez,
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@dmm - it's not work for me, it's fun!

julesh, to random
@julesh@mathstodon.xyz avatar

Old people, not content with merely destroying the economy for young people, are now trying to actively murder young people

image/png

johncarlosbaez,
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@julesh - the Conservatives are in a bubble so large that they think young people are the ones in the bubble.

johncarlosbaez, (edited ) to random
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In the real world, the rope in a knot has some nonzero thickness. In math, knots are made of infinitely thin stuff. This lets mathematical knots be tied in infinitely complicated ways - ways that are impossible for knots with nonzero thickness! These are called 'wild' knots.

See the wild knot here? There's just one point where the stuff it's made of needs to have zero thickness. So we say it's wild at just one point. But some knots are wild at many points.

There are even knots that are wild at every point! To build these you need to recursively put in wildness at more and more places, forever. This is hard to draw. I'd really like to see a good try.

Wild knots are extremely hard to classify. This is not just a feeling - it's a theorem. Vadim Kulikov showed that wild knots are harder to classify than any sort of countable structure that you can describe using first-order classical logic with just countably many symbols!

Very roughly speaking, this means wild knots are so complicated that we can't classify them using anything we can write down. This makes them very different from 'tame' knots - knots that aren't wild. Yeah, tame knots are hard to classify, but nowhere near that hard.

(1/3)

https://www.youtube.com/watch?v=o7U3yvMF8Sw

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

For the hard-core math lovers out there, let me say a bit more about this paper:

• Vadim Kulikov, A non-classification result for wild knots, https://arxiv.org/abs/1504.02714.

As I mentioned, he proved wild knots are harder to classify than any sort of countable structure describable using first-order classical logic with countably many symbols. And it's interesting how he proved this. He proved it by studying the space of all knots.

So he used logic to prove a topology problem is hard - but he also used topology to study logic!

More precisely:

Kulkov studied the topological space of all knots, which are topological embeddings K of the circle in the 3-sphere. He also studied the equivalence relation on knots saying K ∼ K' if there's a homeomorphism of the 3-sphere mapping K to K'.

This is an example of a 'Borel relation on a Polish space'. A Polish space is a reasonably nice sort of topological space, and a Borel relation is a reasonably nice sort of relation on such a space. I don't want to stun you with the definitions - they're easy to look up on Wikipedia.

A lot of classification problems can be thought of this way: you give a Polish space of things you're trying to classify, and an equivalence relation saying when two count as 'the same', which is a Borel relation. There's a theory of when you can reduce one such classification problem to another. This is what Kulikov used to state and prove his result.

At this point you start noticing that the word 'logic' is hiding inside the word 'topology'. Probably not a coincidence.

(2/3)

johncarlosbaez, (edited )
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To see even more on this - including all the definitions I just left out - read my blog article:

• Wild knots are wildly difficult to classify, https://golem.ph.utexas.edu/category/2024/05/wild_knots_are_wildly_difficul.html

I've got a bunch of questions for category theorists and logicians at the end - especially ones who like the connections between logic and topology. (You know who you are.)

Here's the best picture I've seen so far of a knot that's wild at every point, called the 'Bing sling'. It's not a very good picture (since some bits of the knot just fizzle out in mid-air), but maybe you can guess how it should work.

I want better pictures of everywhere wild knots!

(3/3)

johncarlosbaez,
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@j5v - I used to work on knot theory and quantum gravity, and wrote a book called Gauge Fields, Knots and Gravity. So I know Lou, though our paths have diverged since then.

Your problem sounds tough!

johncarlosbaez,
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@leemph - I guess it's adjacent but I just wanted to write something about @henryseg's new video on wild knots, and quickly bumped into Kulikov's paper. I've been interested in Borel reducibility ever since watching a great series of talks about it at an algebra and logic conference called BLAST at Chapman University. I forget who gave those talks!

My interested was later rekindled by @hallasurvivor.

I think studying Borel reducibility on a space of algorithms would be a way of studying how hard it is to classify of algorithms, which is indeed a bit 'meta' compared to ordinary algorithmic complexity.

johncarlosbaez, (edited ) to random
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Robert Recorde introduced the equal sign in 1557. He used parallel lines because "no two things can be more equal". And his equal sign was hilariously looooooong.

This is from @mjd's excellent blog article:

https://blog.plover.com/math/recorde.html

and I recommend following him here on Mastodon.

It's fun to fight your way through Recorde's text, with its old font and spellings. But if you give up, @mjd has transliterated it:

Howbeit, for easie alteration of equations. I will propounde a fewe exanples, bicause the extraction of their rootes, maie the more aptly bee wroughte. And to avoide the tediouse repetition of these woordes "is equalle to" I will sette as I doe often in woorke use, a pair of paralleles, or Gemowe lines of one lengthe, thus: =====, bicause noe 2 thynges, can be moare equalle.

The only real mystery here is "Gemowe", which means "identical" and comes from the same root as "Gemini": twins.

In the same book Robert Recorde introduced the mathematical term "zenzizenzizenzike", but I'm afraid for that you'll have to read @mjd's article!

johncarlosbaez,
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@highergeometer - is that stuff you already knew???

highergeometer, to random
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Two thematically related papers

Felix Cherubini, Thierry Coquand, Matthias Hutzler, David Wärn, "Projective Space in Synthetic Algebraic Geometry"

Abstract: Working in an abstract, homotopy type theory based axiomatization of the higher Zariski-topos called synthetic algebraic geometry, we show that the Picard group of projective n-space is the integers, the automorphism group of projective n-space is PGL(n+1) and morphisms between projective standard spaces are given by homogenous polynomials in the usual way.

https://arxiv.org/abs/2405.13916

=====

Matías Menni, "Bi-directional models of `Radically Synthetic' Differential Geometry", Theory and Applications of Categories, Vol. 40, 2024, No. 15, pp 413-429.

Abstract: The radically synthetic foundation for smooth geometry formulated in [Law11] postulates a space T with the property that it has a unique point and, out of the monoid T^T of endomorphisms, it extracts a submonoid R which, in many cases, is the (commutative) multiplication of a rig structure. The rig R is said to be bi-directional if its subobject of invertible elements has two connected components. In this case, R may be equipped with a pre-order compatible with the rig structure. We adjust the construction of `well-adapted' models of Synthetic Differential Geometry in order to build the first pre-cohesive toposes with a bi-directional R. We also show that, in one of these pre-cohesive variants, the pre-order on R, derived radically synthetically from bi-directionality, coincides with that defined in the original model.
http://www.tac.mta.ca/tac/volumes/40/15/40-15abs.html

johncarlosbaez,
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@highergeometer - nice! One charm of Blechschmidt's proof is that it's apparently simpler than the usual proof, and is perfectly readable by people who don't know what a topos is, and don't even want to learn that.

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,
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@pschwahn - wow, that's amazing! I really wonder how Racah applied G₂ to spectra of rare earth elements! This would be add one to the extremely short list of concrete applications of exceptional Lie groups to real world physics.

johncarlosbaez,
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@pschwahn - more recently we've seen people on MathOverflow wondering about reports that 𝐸₈ was detected experimentally:

https://mathoverflow.net/questions/32315/has-the-lie-group-e8-really-been-detected-experimentally

johncarlosbaez,
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@pschwahn - Skip Garibaldi is an expert on exceptional Lie groups who teamed up with Jacques Distler to attack Garrett Lisi's E8-based "theory of everything".

johncarlosbaez, (edited )
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@pschwahn - here is a hint of 𝐺₂ showing up in studies of the f shell:

https://link.springer.com/article/10.1007/s10910-007-9288-9

"For the atomic g shell the group L₂(9) is isomorphic with the alternating group A₆ on six objects of order 360 or the symmetry group of the 5-dimensional simplex, a 5-dimensional analogue of the tetrahedron with 6 vertices and 15 edges. This leads to the subgroup chain SO(9) ⊃ SO(5) ⊃ L₂(9) for the atomic g shell analogous to the subgroup chain SO(7) ⊃ G₂ ⊃ L₂(7) for the atomic f shell".

L₂(q) = PSL₂(q) is the group of 2×2 matrices with determinant 1 over the field with q elements, mod multiples of the identity. It's a finite group. L₂(7) is famous among lovers of the octonions since it's the automorphism group of the Fano plane.

I haven't gotten the actual paper yet. The study of the g shell is deeply unpopular in chemistry since it only applies to elements of atomic number 124 and higher!

johncarlosbaez,
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@pschwahn - okay, now we're making progress! In that paper the author says:

"The detailed study of the atomic d shell was initiated by Condon and Shortley [1] in 1935 following earlier work by Slater [2] in 1929. In 1949 Racah [3] developed group-theoretical methods for study of both the atomic d and f shells..."

and [3] is

G. Racah, Phys. Rev. 76, 1352 (1949).

But the paper also has interesting stuff on f orbitals, and refers to this for more:

R.B. King, Mol. Phys. 104, 3261 (2006)

johncarlosbaez,
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@pschwahn - yes, Racah's paper discusses the f-shell, G₂ and an antisymmetric trlinear form!

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@pschwahn - yes, Racah's paper is very computational, and since it's part 4 of a 4-part paper he doesn't define a bunch of his terms.

It's annoying that King doesn't explain how L₂(7) and L₂(9) sit inside SO(7) and SO(9). He claims it's part of a general pattern.

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