Miscellaneous – SPP 2026

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

Lotterien in Antragsverfahren

Heute kam die Ankündigung, dass die Stiftung Innovation in der Hochschullehre im Februar wieder eine Freiraum-Förderung ausschreiben wird (hier ein Link zu dem Blogbeitrag über die vorhergehende solche Förderung). Hierbei wird wieder, wie auch beim letzten Mal, zuerst eine Lotterie stattfinden, welche rein auf Glück eine gewisse Anzahl von Antragsskizzen durchlässt zum eigentlichen Verfahren. Die … Continue reading

Pseudo-isotopies of 4-manifolds

One of the landmark results about the topology of 4-manifolds is Freedman’s proof of the 4-dimensional Poincaré conjecture. The problem with this result was that after some time people claiming to understand its proof disappeared, so that in the last decade the validity of this result was openly questioned (e.g., link). In the end, a … Continue reading