Topology

The topics studied in Münster in topology include K-Theory, geometric topology, higher algebra (in the sense of Waldhausen and Lurie), index theory, and Riemannian geometry.

Trace methods, including Topological Hochschild homology and topological cyclic homology, are used to study K-theory. They also have applications to $p$-adic Hodge theory and chromatic homotopy theory.
K-theory of group rings is studied via the Farrell—Jones Conjecture and has applications to the classification of manifolds, for instance to Borel’s conjecture on the topological rigidity of aspherical manifolds.
The methods here are also applied to smooth representation theory of reductive $p$-adic groups.

The study of the scalar curvature intertwines topology with Riemannian geometry. Index theory and K-theory yield obstructions to the existence of Riemannian metrics with positive scalar curvature on certain manifolds. These obstructions can be enhanced to index maps that, together with tools from the study of moduli spaces of manifolds, detect features of the homotopy type of (moduli) spaces of positive scalar curvature metrics. Index-theoretic obstructions are also being used in novel ways to study the geometry of spaces with lower bounds on the scalar curvature in the spirit of comparison geometry.