### 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!"

### Database of online seminar talks

Due to the current situation there are numerous online seminar talks organized all over the world. I was quite happy to discover this week that the website researchseminars.org maintains a useful database of online seminars and conferences, focusing on mathematics and related fields. The website is supported by the American Mathematical Society, the MIT and … Continue reading "Database of online seminar talks"

### Tetrahedra

Consider the following three basic questions about tetrahedra: Does a given tetrahedron tile space? Which tetrahedra are scissors-congruent to a cube? Can one describe the tetrahedra all of whose six dihedral angles are a rational number of degrees? The first question goes back to Aristotle, the second is from Hilbert’s list of problems, and the … Continue reading "Tetrahedra"

### The commutator subgroups of surface groups

Every subgroup of infinite index in a surface group is a free group. There are many ways to see this, for instance, using a bit of topology. An infinite index subgroup corresponds to a covering space with infinitely many sheets and this covering space is a non-compact surface. By a result of Whitehead it deformation … Continue reading "The commutator subgroups of surface groups"