Abstract: The spectral flow is a priori an integer invariant of a path of unbounded self-adjoint Fredholm operators. Its origin lies in Atiyah-Patodi-Singers famous work on index theory for manifolds with boundary. In this overview talk I will first address at least two rigorous definitions. I will also explain the relation to the homotopy theory of the classifying spaces of the K-functor. Finally, I will hint at recent joint work with Bourne, Carey, and Rennie on a KO-valued version of spectral flow. The latter finds its explanation in the aforementioned relation between the spectral flow and the homotopy of the classifying spaces for the K-functor.
The talk will be non-technical and address a general math/physics audience with a certain functional analytic interest.