Miscellaneous – SPP 2026

W-Triviality of 9-Fold Suspensions

Here is a result that I randomly stumbled upon. Let us call a space W-trivial if every real vector bundle over it has trivial total Stiefel-Whitney class. Atiyah and Hirzebruch now proved in the 60s: If Y is a finite CW-complex, then its 9-fold suspension is W-trivial. Why did they prove such a result? They … Continue reading

Optimality of Gerver’s Sofa

The moving sofa problem is one of those deceptively simple yet incredibly difficult “real-world” math problems. Despite its straightforward formulation, it has remained unsolved for roughly 60 years—until recently, when Jineon Baek announced a complete solution (arXiv:2411.19826; hopefully, this time the topic I’m blogging about won’t turn out to be incorrect in the end). In … Continue reading

K-homology class of the Euler characteristic operator

When studying the Atiyah-Singer index theorem one usually sees four main examples. Atiyah-Singer operator: Its topological index is the \(\hat{A}\)-genus and its analytical index can be related to scalar curvature. This shows that the \(\hat{A}\)-genus of a manifold is an obstruction to the existence of a Riemannian metric of positive scalar curvature on it. Signature … Continue reading

Polynomial and metallic manifolds

I recently came across the notion of a polynomial, resp. metallic manifold. Since I have never heard of it before and I assume that the same applies to most of you, I wanted to share its definition with you. Let \(M\) be a (smooth) manifold. A polynomial structure on \(M\) is a \((1,1)\)-tensor field \(\Phi\) … Continue reading

The Probability of the Law of Excluded Middle

This is basically a repost of this blog post by John Carlos Baez: link. I wanted to share it with you since I find the result unexpectedly interesting. The Law of Excluded Middle states that for any statement P we have that either P or not P is true. Almost all of us work under … Continue reading

Kervaira invariant one problem

Last month a solution to the Kervaire invariant one problem in the final remaining dimension 126 was announced (link to a mathstodon post). I have already mentioned this problem in an earlier post here: link (2nd paragraph about the 2-line). edit (17th Dec 2024): The preprint is now on the arXiv:2412.10879.

A Mathematical Conventions Survey

I wanted to share the results of this funny survey with you: 100 Questions About Mathematical Conventions. But beware, if you click on the link you will loose an hour of your life time. 😉

Isoclinism of groups

Do you know when two groups \(G\) and \(H\) are called isoclinic to each other? I have never heard of this notion before, but there was a paper on the arXiv listing today using it in its abstract; so I looked in this paper, not because I was interested in its content, but just because … Continue reading