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"

Abel Prize 2021

The Abel Prize Laureates 2021 were announced today. They are László Lovász and Avi Wigderson for … their foundational contributions to theoretical computer science and discrete mathematics, and their leading role in shaping them into central fields of modern mathematics.

Hantzsche-Wendt manifolds

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"

Chern assembly maps

The Baum-Connes conjecture asserts that for a group G the analytic assembly map, which is nowadays usually defined using KK-theory, \[\mu_*^{K\!K}\colon RK_*^G(\underline{EG}) \to K_*^{top}(C^*_r G)\] is an isomorphism. This map can be factored through the algebraic K-theory of the group ring SG, where S denotes the Schatten-class operators on an \(\infty\)-dimensional, separable Hilbert space. The … Continue reading "Chern assembly maps"