### A relative index theorem too good to be true?

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?"

### 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"

### Unit conjecture disproved!

There are three conjectures about group rings of torsion-free groups that are attributed to Kaplansky. To state them, let $$K$$ be a field, $$G$$ be a torsion-free group and denote by $$K[G]$$ the corresponding group ring. The unit conjecture states that every unit in $$K[G]$$ is of the form $$kg$$ for $$k \in K\setminus\{0\}$$ and … Continue reading "Unit conjecture disproved!"