Triangulating real projective n-space

How many vertices do you need to triangulate the real projective n-space? From this blog post of Gil Kalai I learned about a recent preprint (arXiv:2009.02703) by Adiprasito-Avvakumov-Karasev where they construct triangulations with \[\exp\big((1/2 + \mathcal{o}(1))\sqrt{n}\log{n}\big)\text{-many}\] vertices, which is the first construction needing subexponentially-many vertices. More information, also about the history of this problem, may … Continue reading "Triangulating real projective n-space"

Resolution of Keller’s conjecture

Keller’s conjecture states that in any tiling of Euclidean space by identical hypercubes there are two cubes that meet face to face. (Consider the 2-dimensional picture on the right taken from Wikipedia. The squares share horizontal edges.) The conjecture is completely solved by now: it is true in dimensions 7 and less, but false in higher dimensions. The last missing part was … Continue reading "Resolution of Keller’s conjecture"

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"