I'm interested in the higher category theory behind (fully extended) Topological Quantum Field Theory. I my first year my research focus was on the notion of 'reflection positivity' for TQFTs introduced by Freed and Hopkins in arXiv:1604.06527. This lead me to thinking about dualisable and hermitian objects in $Z/2$-equivariant $\infty$-categories. By the cobordism hypothesis the 1-dimensional cobordism category is the universal example for a category with dualisable objects. In arXiv:1811.11654 I show that this can be used to uniquely characterise the trace of endomorphisms in this context.

Currently I'm thinking about the classifying spaces of certain low dimensional cobordism categories, in particular the classifying space of the homotopy category of the one-dimensional bordism category. In particular I'm curious about how this relates to topological cyclic homology.