Author: Alexander Engel

Catastrophe theory

This term I am teaching a course on catastrophe theory for 2nd year students. When you look up this topic on Wikipedia, you will see words like chaos and singularities mentioned, and you will be shown fancy pictures of the seven elementary catastrophes. But when you try to understand what the actual mathematical content of … Continue reading "Catastrophe theory"

Immersions of manifolds into Euclidean space

Recall the well-known result of Whitney that any (compact) smooth \(n\)-manifold admits an immersion into \(\mathbb{R}^{2n-1}\). Today there was a preprint posted on the arXiv (arXiv:2011.00974) which mentioned in its introduction the following result of Cohen, which strengthens Whitney’s result as follows: Any (compact) smooth \(n\)-manifold admits an immersion into \(\mathbb{R}^{2n-\alpha(n)}\), where \(\alpha(n)\) is the … Continue reading "Immersions of manifolds into Euclidean space"

Blockseminar on Dirac operators and scalar curvature

In mid-October we gathered for one week in Bollmannsruh (somewhat west of Berlin) to work our way through the seminal paper Positive scalar curvature and the Dirac operator on complete Riemannian manifolds by Gromov and Lawson. The hotel we stayed in lies directly at the beautiful lake Beetzsee. It was the perfect place to do … Continue reading "Blockseminar on Dirac operators and scalar curvature"

Seminar on soap bubbles and positive scalar curvature

Together with Rudolf Zeidler I am organizing a reading seminar this winter on generalized soap bubbles and positive scalar curvature. The goal of it is to read the corresponding preprint by Chodosh-Li (arXiv:2008.11888). The two main results we want to understand are the following: No closed aspherical manifold of dimension 4 or 5 admits a … Continue reading "Seminar on soap bubbles and positive scalar curvature"

Positive scalar curvature metrics on manifolds with boundary

Usually when I blog here about positive scalar curvature, the manifolds I consider are assumed to have no boundary. In this post I want to explain the basics of what happens when the manifolds actually do have a boundary. So first of all, one has to mention that any compact manifold with boundary admits a … Continue reading "Positive scalar curvature metrics on manifolds with boundary"

Akademisches und schulbezogenes Fachwissen

Das braucht man doch nie in der Schule! ist ein oft gehörtes Argument, wenn sich Studierende des gymnasialen Lehramts der Mathematik über die Inhalte ihrer Fachvorlesungen beschweren. Sie hören in den ersten Semestern oft dieselben Vorlesungen wie ihre Kommilitonen, welche die Mathematik auf einen Bachelor studieren, und in der Tat ist ein Teil des erlernten … Continue reading "Akademisches und schulbezogenes Fachwissen"