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"