Alexander Engel – Page 15

Local-to-global principles for the topology of boundaries of hyperbolic groups

Two weeks ago a paper was posted (by Benjamin Barrett) on the arXiv (arXiv:2004.11650) proving the following theorem about Gromov boundaries of word hyperbolic groups: Let \(G\) be a one-ended hyperbolic group. Then \(\partial G\) is locally simply-connected if and only if for every point \(\xi\in \partial G\) the space \(\partial G \setminus \xi\) is … Continue reading

Remote teaching for the working mathematician

Ich bekam als Reaktion auf meinen Blogbeitrag zu Remote teaching einige eMails, in denen nach konkretem Rat und praktischen Tips gefragt wurde für das kommende Semester. Vorlesungen Präsentiert man den Studierenden den Stoff mit Hilfe eines ausführlichen Skriptes oder aufgezeichneten Videos, so sollte man ihnen meiner Meinung nach konkrete Arbeitsanweisungen zum Durcharbeiten des Materials geben. … Continue reading

Aspherical manifolds and positive scalar curvature

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

Remote teaching

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

A Prime Breakthrough

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

Laplace operator and covering spaces

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

Lindelöf hypothesis

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

Girls and boys performing in STEM fields

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