Recall the following conjecture about aspherical manifolds (i.e., manifolds whose universal cover is contractible): If M is a closed, aspherical manifold, then M does not admit any Riemannian metric of positive scalar curvature. In January I saw a preprint being posted on the arXiv (2001.02644) claiming to have resolved this conjecture. If it turns out … Continue reading "Aspherical manifolds and positive scalar curvature"
The Corona Crisis is throwing off our lives. I was also not spared, though this is now probably complaining on a high comfort level: As some of you know, I was on a sabbatical since last fall traveling with my wife through Southeast Europe (check our travel blog Engel auf Reisen), but due to the … Continue reading "Remote teaching"
It seems that at the beginning of this year a major breakthrough on prime numbers was achieved. I learned about it from this blog: link. Let me summarize the result quickly for you if you don’t want to read the other blog post. Almost a hundred years ago Jensen and Pólya proved that the Riemannian … Continue reading "A Prime Breakthrough"
Let M be a closed Riemannian manifold and denote by X its universal covering space equipped with the pulled back Riemannian metric. There is an intimate relation between the Laplace operator on X and the fundamental group of M. One example of this is the result of Brooks from 1981: the fundamental group of M … Continue reading "Laplace operator and covering spaces"
Today I stumbled across a news article ( link ) written about someone (actually, Athanassios Fokas – a known mathematician) having announced progress on the Lindelöf hypothesis ( wikipedia ). The Lindelöf hypothesis is related to the Riemannian hypothesis and actually also follows from it – progress on the Lindelöf hypothesis would also mean progress … Continue reading "Lindelöf hypothesis"
This is actually already old news (it is from September 2018), but since then it was on my list of things to blog about and only now I found the time to actually do it. There was a meta-study published in Nature Communications (doi:10.1038/s41467-018-06292-0) about the performance of girls and boys in STEM fields. I … Continue reading "Girls and boys performing in STEM fields"
This year’s Shaw Prize in Mathematical Sciences goes to Michel Talagrand (link to the press release, link to his Wikipedia page).
Recently a preprint was posted on the arXiv ( arXiv:1904.12720 ) claiming to have constructed the first examples of compact orientable hyperbolic non-spin manifolds (in every dimension at least 4). I was a bit surprised by this, since I thought that such examples should be already known. Apparently not … One reason why constructing such … Continue reading "Compact hyperbolic non-spin manifolds"