Shaw Prize 2021

The Shaw Prize 2021 in the Mathematical Sciences goes to Jean-Michel Bismut and Jeff Cheeger for their remarkable insights that have transformed, and continue to transform, modern geometry. Bismut […] imported ideas from probability into index theory, reproving all the main theorems and vastly extending them, which enabled him to link index theory to other parts … Continue reading "Shaw Prize 2021"

Characterizing Euclidean 3-space

In my previous post about the recent preprint of Jiang Wang we saw the following characterization of Euclidean 3-space among all contractible 3-manifolds: It is the only one that admits a complete Riemannian metric of non-negative scalar curvature. Today a preprint was posted on the arXiv:2105.09035 by Bargagnati and Frigerio providing another characterization: The only … Continue reading "Characterizing Euclidean 3-space"

Online Fernklausuren in Mathematik

Im letzten Semester mussten aufgrund der Pandemie viele Prüfungen in anderen (oft digitalen) Formaten abgehalten werden. Für das laufende Sommersemester – auch wenn sich die Lage derzeit bessert – scheint es mir wahrscheinlich, dass zumindest Prüfungen mit vielen Teilnehmern weiterhin Einschränkungen unterworfen sind. Daher will ich die Gelegenheit nutzen, um ein paar Überlegungen zu sammeln, … Continue reading "Online Fernklausuren in Mathematik"

Contractible 3-manifolds and positive scalar curvature, III

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"

Topological complexity

Rummaging in the arXiv I ran across this article and the notion of topological complexity which is really appealing. The idea of topological complexity isn’t quite new, it was developed by M. Farber in a short article published in 2003 in Discrete & Computational Geometry. The article doesn’t even have a proper review on MathSciNet, … Continue reading "Topological complexity"

Extensions, coarse embeddability and the coarse Baum-Connes conjecture

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"

Multiplying matrices

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"