Even with the current situation where editors are struggling to find reviewers for their math journals, I still think that it's a good thing referees aren't remunerated or otherwise rewarded. Otherwise by now 99% of referee reports would be written by LLMs.

Up to dimension seven, compact (simply connected) homogeneous Einstein manifolds had been almost completely classified. The only remaining case was the six-dimensional Lie group S³×S³ ≅ SU(2)×SU(2). The Einstein equations for left-invariant metrics on S³×S³ are extremely messy algebraic […]

The existence of a complex structure on S⁶ is now formalized in Lean. Measly 250k lines of code. So what do we do now? The formalization is even less comprehensible than the paper, mostly due to its length (in fact, the individual theorems are not the problem). Do we ask ChatGPT for summary […]

Uhh.. what? I already paid a 285 EUR on my visa application. Why does asking questions cost money?

"A metric on 𝑆²×𝑆² with positive sectional curvature", by S. Brendle and P. K. Hung. Almost a hundred years ago, Hopf asked whether such a metric exists. Now they found one! https://arxiv.org/abs/2608.19068

A metric on $S^2 \times S^2$ with positive sectional curvature

We construct a metric on $S^2 \times S^2$ with positive sectional curvature. Starting from the standard metric on $S^2 \times S^2$, we first perform a Cheeger deformation. The resulting metric has nonnegative sectional curvature. We refer to it as a Cheeger-Müter metric. We then consider a suitable third order perturbation of this Cheeger-Müter metric and show that the perturbed metrics have positive sectional curvature. The proof requires various calculations, some of which have been carried out with the help of MATHEMATICA. The MATHEMATICA code is attached to this submission.

arxiv.org

Ich bin immer noch auf Prokrastinieren und damit ich damit nicht allein bleibe teile ich die interaktive Online-Anwendung Eigendrum! 🙃 Man kann auf der Website beliebige Trommelformen zeichnen und dann anschlagen und lauschen. Im Hintergrund läuft keine KI, sondern eine mathematische Berechnung […]

From https://arxiv.org/abs/2608.16066

Kähler manifolds are the elite among the complex manifolds. They possess a Riemannian metric that is not only compatible with the complex structure (i.e. Hermitian), but also view the complex structure as parallel (i.e. the Riemannian holonomy group is contained in U(n)). So the Kähler condition […]

My colleague Misha Verbitsky over at IMPA, one of the top mathematicians of Brazil, is being detained in Armenia because Russia has designated him a terrorist after his comments on the war in Ukraine […]

Two exciting preprints on the arXiv this week, both about Einstein metrics! P.-A. Nagy has worked out the third variation of the Ricci tensor with respect to the metric (a long-standing computational problem). This enables us to study third order deformations of Einstein metrics (for up to […]

I must admit that I never understood the theorem statement of geometrization of 3-manifolds: every (smooth, compact, connected, orientable) 3-manifold (with boundary a disjoint union of tori) "decomposes" into "geometric pieces". But what exactly does "decompose" mean here, what are the 8 […]

I had never thought about the difference between matrices being conjugate via a matrix in G:=GL_n(F) or via a matrix in S:=SL_n(F). It turns out to depend on the determinants of matrices in the centralizer! More precisely, let both G and S act on S by conjugation. Each G-orbit is a disjoint of […]

"G₂-structures as Octonion Algebras", by Isak Sundelius https://arxiv.org/abs/2604.15966 In this article G₂-structures on a 7-manifold 𝑀 are interpreted as octonion algebras over the ring of smooth functions on 𝑀! More precisely, the category of G₂-structures on 𝑀 is isomorphic to a full […]

Because we care deeply about international mathematics and its mathematicians, we must recognize the threat to both that the upcoming ICM poses. Please read and consider signing: Move the 2026 ICM out of the United States […]

They made the isospectral drums https://prismika.github.io/2026/03/01/we-made-the-isospectral-drums.html

"The fundamental group of a spherical space form is not audible", by Mauro Colantonio and Emilio A. Lauret. A spherical space form is a complete Riemannian manifold of constant positive (sectional) curvature. The universal cover of a spherical space form is a round sphere (explaining the name) […]

Unlike their French colleagues, the German Mathematical Union (DMV) will not boycott the ICM 2026, as I was informed via email. They refer to a statement of the local organizing committee. I take the liberty to take one line out of context: "Gathering in Philadelphia is not only an academic […]

Life is going to hell in thousands of ways, but at least I got myself some nice drip from my university.

@johncarlosbaez Today I revisited our project on the subgroups of F₄. Recall, we were hoping to show that for each 𝕂=ℝ,ℂ,ℍ, all subalgebras of 𝔥₃(𝕆) which are isomorphic to 𝔥₃(𝕂) are F₄-related. We were almost done with this; in fact, I think I have shown that a subalgebra isomorphic to 𝔥₃(ℍ) […]

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

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.

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} […]

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 […]

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.

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/

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...

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