Unoriented cobordism categories and Klein TQFTs

The mid 1980s saw a shift in the nature of the relationship between mathematics and physics. Differential equations and geometry applied in a classical setting were no longer the principal players; in the quantum world topology and algebra had come to the fore. In this talk we discuss a method of classifying 2-dim invertible Klein topological quantum field theories (KTQFTs). A key object of study will be the unoriented cobordism category $\mathscr{K}$, whose objects are closed 1-manifolds and whose morphisms are surfaces (a KTQFT is a functor $\mathscr{K}\rightarrow\operatorname{Vect}_{\mathbb{C}}$). Time permitting, the open-closed version of the category will be considered, yielding some surprising results.