Research areas

My research areas are the differential geometry of higher-categorical structures, and mathematical aspects of classical and quantum field theories. I am also interested in the geometry of loop spaces, Lie theory, and homotopy theory.

Central in my research is the notion of a gerbe: a generalization of a fibre bundle over a manifold to a structure whose fibres are categories. Gerbes form a higher-categorical structure und yet allow to study classical differential-geometric aspects such as connections, curvature, and holonomy. Applications of gerbes in the area of 2-dimensional field theories arise because their holonomy can be understood as a coupling between higher-dimensional elementary particles (string) to a gauge field. Their relation to the geometry of loop spaces is established by a procedure called transgression, which transforms higher-categorical geometry over a manifold into ordinary geometry over mapping spaces.

An interesting aspect of the theory of gerbes is that often Lie-theoretical problems arise. This comes essentially from the fact that compact Lie groups carry canonical gerbes, which encode part of the geometry and representation theory of the group. Homotopy theory is relevant because higher-categorical structures can be seen as an instance of infinity-categorical structures, which in turn constitute an algebraic formulation of topology.

Specifically, I work about the following topics: