I have already blogged several times about complete Riemannian metrics of positive scalar curvature on 3-manifolds: here, here and here. Now it seems that finally Jian Wang proved the result that he was working on for quite some time now (arXiv:2105.07095): Any contractible 3-manifold admitting a complete Riemannian metric of non-negative scalar curvature is homeomorphic to Euclidean … Continue reading "Contractible 3-manifolds and positive scalar curvature, III"

A few days ago I was reading with Rudolf Zeidler the recent preprint by Zhizhang Xie (arXiv:2103.14498). We were especially focusing on the proof of the relative index theorem therein and the global analytic lemmas leading to it. While doing this, we realized that we can not come up on the spot with a counter-example … Continue reading "A relative index theorem too good to be true?"

One of the pinnacle results so far about the (strong) Novikov conjecture is Guoliang Yu’s proof that it holds for groups which are coarsely embeddable into a Hilbert space. In fact, he first proved that under this assumption the coarse Baum-Connes conjecture holds, and then one can invoke the descent principle to get to the … Continue reading "Extensions, coarse embeddability and the coarse Baum-Connes conjecture"

Two years ago I blogged about recent developments about multiplying integers. The next most important operation in (applied) mathematics is multiplying matrices. The usual way of doing this requires \(n^3\) multiplications (and some additions) for multiplying two \((n\times n)\)-matrices. But there is actually a way of doing it with less than this: the current record … Continue reading "Multiplying matrices"

Recently a preprint was put on the arXiv:2103.01051 about Hantzsche-Wendt manifolds. I did not know what these manifolds are, got interested and started reading the introduction. By definition, which goes back to arXiv:math/0208205, a Hantzsche-Wendt manifold is an orientable, n-dimensional flat manifold whose holonomy group is an elementary abelian 2-group of rank n-1, i.e., isomorphic to \((\mathbb{Z}/2\mathbb{Z})^{n-1}\). Every … Continue reading "Hantzsche-Wendt manifolds"