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"

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"

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"

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"

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!"

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"