### 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"
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### 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"
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### 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!"

### 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"

### Gegenseitige Korrektur in Lehrveranstaltungen

Dieses Semester halte ich eine Vorlesung für Studierende im dritten Semester (ich habe hier über den mathematischen Inhalt gebloggt). Zu dieser Vorlesung gibt es offiziell keinen Übungsbetrieb, aber ich gebe den Studierenden trotzdem hin und wieder (freiwillige) Übungsaufgaben auf, denn wie wir alle wissen lernt man Mathematik am besten wenn man sich selbst an ihr … Continue reading "Gegenseitige Korrektur in Lehrveranstaltungen"
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