Paul Schwahn
@pschwahn.mathstodon.xyz.ap.brid.gy
Sometimes I do differential geometry.
[bridged from https://mathstodon.xyz/@pschwahn on the fediverse by https://fed.brid.gy/ ]
[bridged from https://mathstodon.xyz/@pschwahn on the fediverse by https://fed.brid.gy/ ]
This amogus surface in ℝ³ has principal curvatures contained in [-1, 1] and is homeomorphic to a sphere, but its enclosed volume is less than that of the unit ball.
https://arxiv.org/abs/2512.19659
https://arxiv.org/abs/2512.19659
December 23, 2025 at 3:40 PM
This amogus surface in ℝ³ has principal curvatures contained in [-1, 1] and is homeomorphic to a sphere, but its enclosed volume is less than that of the unit ball.
https://arxiv.org/abs/2512.19659
https://arxiv.org/abs/2512.19659
Does anyone here have access to the book by Ronveaux on Heun's Differential Equations?
https://academic.oup.com/book/54034
Asking for a friend.
https://academic.oup.com/book/54034
Asking for a friend.
December 15, 2025 at 7:04 PM
Does anyone here have access to the book by Ronveaux on Heun's Differential Equations?
https://academic.oup.com/book/54034
Asking for a friend.
https://academic.oup.com/book/54034
Asking for a friend.
Wait - am I reading this correctly that firearms are allowed on domestic flights in Brazil? The more you know.
December 9, 2025 at 3:48 PM
Wait - am I reading this correctly that firearms are allowed on domestic flights in Brazil? The more you know.
@johncarlosbaez
Coming back to our discussion about 𝔥₃(𝕂)-subalgebras of 𝔥₃(𝕆), and whether F₄ acts transitively on them:
I've written up a unified argument why on can always assume, up to the action of F₄, that the subalgebra consists of elements of the form
\\[\\{\begin{pmatrix} […]
Coming back to our discussion about 𝔥₃(𝕂)-subalgebras of 𝔥₃(𝕆), and whether F₄ acts transitively on them:
I've written up a unified argument why on can always assume, up to the action of F₄, that the subalgebra consists of elements of the form
\\[\\{\begin{pmatrix} […]
Original post on mathstodon.xyz
mathstodon.xyz
December 2, 2025 at 11:58 PM
@johncarlosbaez
Coming back to our discussion about 𝔥₃(𝕂)-subalgebras of 𝔥₃(𝕆), and whether F₄ acts transitively on them:
I've written up a unified argument why on can always assume, up to the action of F₄, that the subalgebra consists of elements of the form
\\[\\{\begin{pmatrix} […]
Coming back to our discussion about 𝔥₃(𝕂)-subalgebras of 𝔥₃(𝕆), and whether F₄ acts transitively on them:
I've written up a unified argument why on can always assume, up to the action of F₄, that the subalgebra consists of elements of the form
\\[\\{\begin{pmatrix} […]
If you have a parallel spinor ψ on a compact manifold, you can write down a parallel bundle map Sym²𝑇*𝑀→𝑇*𝑀⊗Σ𝑀 mapping a symmetric 2-tensor h to the spinor-valued one-form 𝑋↦h(𝑋)⋅ψ.
This allowed Dai, Wang & Wei to prove a lower bound on the Lichnerowicz Laplacian on Sym²𝑇*𝑀, by comparing it […]
This allowed Dai, Wang & Wei to prove a lower bound on the Lichnerowicz Laplacian on Sym²𝑇*𝑀, by comparing it […]
Original post on mathstodon.xyz
mathstodon.xyz
December 2, 2025 at 6:02 PM
If you have a parallel spinor ψ on a compact manifold, you can write down a parallel bundle map Sym²𝑇*𝑀→𝑇*𝑀⊗Σ𝑀 mapping a symmetric 2-tensor h to the spinor-valued one-form 𝑋↦h(𝑋)⋅ψ.
This allowed Dai, Wang & Wei to prove a lower bound on the Lichnerowicz Laplacian on Sym²𝑇*𝑀, by comparing it […]
This allowed Dai, Wang & Wei to prove a lower bound on the Lichnerowicz Laplacian on Sym²𝑇*𝑀, by comparing it […]
W from the Chair of Representation Theory at EPFL.
![Title: Representation theory studies symmetry
in mathematics, science and engineering
For example, the following is a Al-generated picture of Dynkin
diagrams, a fundamental concept in Lie theory: [AI slop image]
However, the pictures above are not only incorrect, but they make no
sense. As such, the Chair of Representation Theory seeks to unleash
human (underlined!) ingenuity and ambition to tackle problems from fields as
varied as geometry, low-dimensional topology, mathematical physics
and combinatorics](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:chspuljt4seuqplv2sdddi7r/bafkreid4hfwqs7c6frakazi7iqu5cn44lqp5a427xajzpjjs4w5cdryw2e@jpeg)
November 27, 2025 at 9:10 PM
W from the Chair of Representation Theory at EPFL.
Does anyone know a good website with info on tap water quality in places around the world?
All I've found on the internet are US-specific websites, or articles that look like they are LLM-generated.
All I've found on the internet are US-specific websites, or articles that look like they are LLM-generated.
November 17, 2025 at 5:51 AM
Does anyone know a good website with info on tap water quality in places around the world?
All I've found on the internet are US-specific websites, or articles that look like they are LLM-generated.
All I've found on the internet are US-specific websites, or articles that look like they are LLM-generated.
Take the adjoint representation of the Lie algebra 𝔰𝔬(7) (i.e. the representation of 𝔰𝔬(7) on itself). This is one of the fundamental representations of 𝔰𝔬(7) (together with ℝ⁷ and the spinor module) - all irreducible representations of 𝔰𝔬(7) can be obtained by taking Cartan products of these […]
Original post on mathstodon.xyz
mathstodon.xyz
November 7, 2025 at 4:27 AM
Take the adjoint representation of the Lie algebra 𝔰𝔬(7) (i.e. the representation of 𝔰𝔬(7) on itself). This is one of the fundamental representations of 𝔰𝔬(7) (together with ℝ⁷ and the spinor module) - all irreducible representations of 𝔰𝔬(7) can be obtained by taking Cartan products of these […]
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/
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
October 8, 2025 at 2:26 AM
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/
The slides are available here: https://pschwahn.github.io/events/
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...
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...
September 29, 2025 at 1:01 AM
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...
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...
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
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
August 24, 2025 at 10:19 PM
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
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
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 […]
(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 […]
Original post on mathstodon.xyz
mathstodon.xyz
July 26, 2025 at 2:35 PM
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 […]
(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 […]
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 […]
Original post on mathstodon.xyz
mathstodon.xyz
June 26, 2025 at 12:24 AM
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 […]
Got my first PR merged into mathlib 🥳
June 24, 2025 at 6:17 PM
Got my first PR merged into mathlib 🥳
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 […]
Needless to say, none of the articles or books in […]
Original post on mathstodon.xyz
mathstodon.xyz
June 9, 2025 at 11:36 PM
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 […]
Needless to say, none of the articles or books in […]
A small classification of Lie algebras, now formalized in Lean.
https://github.com/LieLean/LowDimSolvClassification
To be precisely, we formalized the complete classification of solvable Lie algebras in dimension ≤3 over arbitrary fields. This is only the beginning!
https://github.com/LieLean/LowDimSolvClassification
To be precisely, we formalized the complete classification of solvable Lie algebras in dimension ≤3 over arbitrary fields. This is only the beginning!
GitHub - LieLean/LowDimSolvClassification: A classification theorem in Lean of solvable Lie algebras of dimension zero to three
A classification theorem in Lean of solvable Lie algebras of dimension zero to three - LieLean/LowDimSolvClassification
github.com
May 30, 2025 at 9:57 PM
A small classification of Lie algebras, now formalized in Lean.
https://github.com/LieLean/LowDimSolvClassification
To be precisely, we formalized the complete classification of solvable Lie algebras in dimension ≤3 over arbitrary fields. This is only the beginning!
https://github.com/LieLean/LowDimSolvClassification
To be precisely, we formalized the complete classification of solvable Lie algebras in dimension ≤3 over arbitrary fields. This is only the beginning!
In the affine plane, lines are 1D affine subspaces, or zero sets of degree 1 polynomials.
In the projective plane, lines are zero sets of degree 1 *homogeneous* polynomials.
Is there an analogous class of functions in hyperbolic geometry? Consequentially, is there such a thing as "hyperbolic […]
In the projective plane, lines are zero sets of degree 1 *homogeneous* polynomials.
Is there an analogous class of functions in hyperbolic geometry? Consequentially, is there such a thing as "hyperbolic […]
Original post on mathstodon.xyz
mathstodon.xyz
May 14, 2025 at 7:59 PM
In the affine plane, lines are 1D affine subspaces, or zero sets of degree 1 polynomials.
In the projective plane, lines are zero sets of degree 1 *homogeneous* polynomials.
Is there an analogous class of functions in hyperbolic geometry? Consequentially, is there such a thing as "hyperbolic […]
In the projective plane, lines are zero sets of degree 1 *homogeneous* polynomials.
Is there an analogous class of functions in hyperbolic geometry? Consequentially, is there such a thing as "hyperbolic […]
Reposted by Paul Schwahn
Specifically the petition to ban conversion therapy practices is lacking enough votes from:
- #germany (49.9%)
- #sweden (49.96%)
- #Netherlands (94%)
- #austria (25.5%)
- #slovenia (83.14%)
- #portugal (24.46%)
If you're a citizen of these countries you can help advance a ban on conversion […]
- #germany (49.9%)
- #sweden (49.96%)
- #Netherlands (94%)
- #austria (25.5%)
- #slovenia (83.14%)
- #portugal (24.46%)
If you're a citizen of these countries you can help advance a ban on conversion […]
Original post on hachyderm.io
hachyderm.io
May 13, 2025 at 5:45 PM
Alright, I must concur, index notation is better than whatever this is.
May 13, 2025 at 7:24 PM
Alright, I must concur, index notation is better than whatever this is.
low-effort meme cooked up after collaborating with complex geometers (this is a joke)
April 26, 2025 at 9:56 PM
low-effort meme cooked up after collaborating with complex geometers (this is a joke)
"A piecewise-linear isometrically immersed flat Klein bottle in Euclidean 3-space", or, in other words, an Origami Klein bottle with self-intersections:
https://arxiv.org/abs/2504.08826
https://arxiv.org/abs/2504.08826
April 15, 2025 at 6:33 PM
"A piecewise-linear isometrically immersed flat Klein bottle in Euclidean 3-space", or, in other words, an Origami Klein bottle with self-intersections:
https://arxiv.org/abs/2504.08826
https://arxiv.org/abs/2504.08826
Found this on arXiv today, and it evoked imagery of cosmic terror in my mind.
April 15, 2025 at 5:02 PM
Found this on arXiv today, and it evoked imagery of cosmic terror in my mind.
According to the Iwasawa decomposition G=KAN for semisimple Lie groups, every symmetric space of noncompact type G/K is isometric to the solvable Lie group AN with some invariant metric.
Is there a standard notation for this Lie group (and its Lie algebra) when, say, G/K is (real, complex, ...) […]
Is there a standard notation for this Lie group (and its Lie algebra) when, say, G/K is (real, complex, ...) […]
Original post on mathstodon.xyz
mathstodon.xyz
April 14, 2025 at 2:45 AM
According to the Iwasawa decomposition G=KAN for semisimple Lie groups, every symmetric space of noncompact type G/K is isometric to the solvable Lie group AN with some invariant metric.
Is there a standard notation for this Lie group (and its Lie algebra) when, say, G/K is (real, complex, ...) […]
Is there a standard notation for this Lie group (and its Lie algebra) when, say, G/K is (real, complex, ...) […]
Just learned that the Mathematical Congress of the Americas 2025 in taking place in Miami, Florida, of all places. Perhaps not the most auspicious location nowadays.
April 9, 2025 at 11:15 PM
Just learned that the Mathematical Congress of the Americas 2025 in taking place in Miami, Florida, of all places. Perhaps not the most auspicious location nowadays.