The unitary cobordism hypothesis

The cobordism hypothesis classifies extended topological quantum
field theories (TQFTs) in terms of algebraic information in the target
category. One of the core principles in quantum field theory - unitarity -
says that state spaces are not just vector spaces, but Hilbert spaces.
Recently in joint work with many others, we have defined unitarity for
extended TQFTs, inspired by Freed and Hopkins. Our main technical tool is a
higher-categorical version of dagger categories; categories $C$ equipped
with a strict anti-involution $\dagger: C \to C^{op}$ which is the identity
on objects. I explain joint work in progress with Theo Johnson-Freyd,
Cameron Krulewski and Lukas Müller in which we prove a version of the
cobordism hypothesis for unitary TQFTs. The main observation is that the
\emph{stably} framed bordism n-category is freely generated as a symmetric
monoidal dagger n-category with unitary duals by a single object: the point.