Topological quantum field theory is a part of algebraic topology related to finding geometric invariants using a category of manifolds and cobordisms between them.  The manifolds do not have extra structure.

Turaev has proposed a variant, Homotopy Quantum Field Theories, in which the manifolds come with a characteristic map to a `background space, B. This gives the manifolds some `background' structure. When B is an Eilenberg-Maclane space (so higher homotopy groups are trivial), he classified HQFTs in which the manifolds are 1-dimensional, in terms of some graded algebras generalising the group algebra of the fundamental group of B. The calculations are elegant and fun! If B has trivial fundamental group, but has a non-trivial second homotopy group, then Brightwell and Turner have a similar classification, but what if B has both of these groups non-trivial?

Crossed modules classify such spaces.  They are not that well known as such yet underly lots of other structures. They are easy to play with and generalise groups in their theory and applications. Turaev and T.P. have introduced a notion of formal map on a cobordism (surface), which is a map on the surface with edges and faces labelled by different part of a crossed module.  The talk will try to explain how one can `play' with the resulting diagrams to get a classification result for the general (1+1) dimensional  HQFTs.