Geometric TQFTs and parallel transport

I will discuss a definition of bordism categories where bordisms are equipped with general geometric structures. This is motivated by applications such as the representation of cohomology theories through field theories. Then I will discuss the case of 1-dimensional TQFTs over a manifold X. It turns out, as one would hope, that these are nothing but vector bundles with connection over X. I'll explain how the difficulties entailed in this statement and their resolution are related to the problem of representing cohomology theories. This is joint work with Matthias Ludewig and Stephan Stolz.