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Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · 19h

Right now I'm attending my first ever computer science conference (the CICM in Brasília), and I just delivered a talk about our Lie algebra formalization project!

The slides are available here: https://pschwahn.github.io/events/

Talks & Events

Hi, I am Paul Schwahn, a postdoctoral researcher at Unicamp.

pschwahn.github.io

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Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · 9d

@highergeometer good suggestion; done!

Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · 9d

My passion project (formalizing the basics of synthetic projective geometry in Lean) is now public on Github!

https://github.com/PSchwahn/IncidenceGeometry

Just finished the definition of the projective closure of an affine plane. Now on to proving that it satisfies the projective plane axioms...

Screenshot of VSCode containing Lean code.

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Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Sep 7

@highergeometer Anytime I try to read copyright agreements, my mind just goes blank. Even when I genuinely want to understand them! It's like back in school when I tried (and failed) to study for history exams.

As a consequence, I must admit I usually don't really read nor understand the […]

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Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Sep 4

@highergeometer what do the funny angular brackets mean?

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Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Aug 24

Why do there exist so many songs/albums/artists with the name "Calabi-Yau"?

I mean, yes, Calabi-Yau manifolds are cool as heck, but what are they doing in pop culture?

So far this is my favorite:
https://www.youtube.com/watch?v=4aEj-wKkMu0

Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Aug 4

@highergeometer @littmath @arXiv maybe it might be a good idea to allow users to flag articles in some way (perhaps with the option of adding a comment, only visible to moderators), so that the moderators can weed out nonsense much quicker and more reliably?

Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Jul 28

@highergeometer I think it's a standard fact that Λ²ℂⁿ is irreducible over SU(n), while Λ²ℂ²ⁿ=ℂω⊕Λ²₀ℂ²ⁿ over Sp(n).

If you need a reference for specific branchings anyway, try
"Tables of dimensions, indices, and branching rules for representations of simple Lie algebras" by McKay and Patera […]

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Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Jul 26

Took us long enough! Our survey article is finally on arXiv. It deals with two related topics:
(1) Einstein metrics as the critical points of the Einstein-Hilbert action, and what is known about their stability;
(2) The moduli space of Einstein metrics and the question whether a given Einstein […]

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Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Jul 25

@highergeometer Sorry for keeping you in suspense! My expertise on this stuff is rudimentary, so I don't think I can confidently write a quality MO answer here. In fact, I don't understand why π₃ should even be generated by a homomorphism from SU(2) - or whether every index 1 su(2)-subalgebra is […]

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Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Jul 10

@highergeometer Right - if 𝔥⊂𝔤 with index 1 such that 𝔤/𝔥≅𝔥⊕…, then one can also take 𝔰𝔲(2)⊂𝔥 with index 1 and has 𝔤/𝔰𝔲(2)≅𝔰𝔲(2)⊕…, and the index multiplies to 1*1=1.

However I don't know enough Lie theory to ascertain that the index 1 𝔰𝔲(2)-subalgebras of a given 𝔤 are always generated by some […]

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Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Jul 9

@highergeometer Maybe there really is a structural reason that this does not happen. On a related note, may I ask why π₃(𝐺/𝐻)=1 is important?

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Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Jul 9

@highergeometer Thanks! I couldn't track down the definition of index in the Russian version, but it seems very plausible that it is the same as the Dynkin index we are talking about.

Turns out Dynkin has also already computed the isotropy representations of \\(G/\widetilde G\\), which is […]

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Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Jul 7

@highergeometer do you have a reference for which subalgebras have index 1? It might be in a paper of Dynkin (https://doi.org/10.1090/trans2/006), but I can't access it right now.

American Mathematical Society

Advancing research. Creating connections.

www.ams.org

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Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Jul 7

@highergeometer Yep, that's a hurdle. But I'm always happy to procrastinate by calculating some branchings ;)

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Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Jul 7

@highergeometer Let's see... SU(3) is the stabilizer of a vector in the standard representation ℝ⁷ of 𝐺₂, and SU(2) is the stabilizer of a vector in the standard representation ℂ³ of SU(3). So under the restriction SU(2)₁⊂𝐺₂, the standard representation is ℝ⁷=3ℝ⊕ℂ². This may always serve as a […]

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Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Jul 7

@highergeometer [Not replying on MO because I don't have the energy to write a high-quality post]

I don't know any index-1 subgroups off the top of my head, but I don't think that this means that the (restriction of the) Killing forms coincide; rather, the invariant inner products determined by […]

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Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Jul 4

@highergeometer having varying choices of reductive complement always makes my head spin...
Do you do it with reference to a fixed 𝔪 and put in the 𝑃 somewhere, or do you work on the quotient space 𝔤/𝔥?

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Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Jun 30

@highergeometer Maybe there is indeed a way to relate this to some Yang-Mills functional.

Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Jun 30

@highergeometer Oh yes, absolutely! But for a fixed gauge group H, there are only so many interesting homogeneous spaces G/H one wants to do gauge theory on. The sphere is, as always, a special case.

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Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Jun 29

@highergeometer Hm, I've never seen people use H-principal connections on G/H which are not G-invariant. What would be the motivation for this?

But I think that given such a connection (equivalently, an H-right-invariant horizontal subbundle in TG), any H-invariant inner product on […]

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Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Jun 29

@highergeometer An interesting question! I don't know in general if there is a good choice of metric for a given connection. Do you want to find 𝑔 such that ∇𝑔=0? On compact homogeneous spaces, I'd try a Fourier series approach. Write \\(\nabla=\bar\nabla+\alpha\\), where \\(\bar\nabla\\) is the […]

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Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Jun 26

Finally managed to write down a rigorous argument for a theorem I've been using for years, but whose proof had never been written down properly (the treatments I've seen are either too specialized and misleading, or hand-wavy, or wrong): that the Casimir operator of a homogeneous vector bundle […]

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Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Jun 24

Got my first PR merged into mathlib 🥳

Paul Schwahn @pschwahn.mathstodon.xyz.ap.brid.gy · Jun 9

Now it's happened to me too: we asked a (tenured) colleague for some references on a particular question, and received an LLM-generated list, formatted in Markdown, complete with little "summaries" of the articles were relevant to the question.

Needless to say, none of the articles or books in […]

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