Abstract: In the context of B(H) (the set of bounded operators on a separable Hilbert space), the spectral flow counts the net number of eigenvalues which change sign as one travels along a path of self-adjoint Fredholm operators. The ability to calculate the spectral flow allows one to calculate the Fredholm index of some operators, making it of interest in the study of non-commutative geometry. It is possible to generalize the concept of spectral flow to a semifinite von Neumann algebra, as we can use a trace on the algebra to measure the amount of spectrum which changes sign. During the course of the two talks, I will start by giving a detailed introduction to spectral flow (for both bounded and unbounded operators), followed by an overview of some important results for the B(H) case, including a characterization of spectral flow due to Lesch, integral formulas for spectral flow, and geometric interpretations (e.g. spectral flow as an intersection number). I will give sketches of some of the more illuminating proofs, and conclude by discussing some of the changes required for the generalization to semifinite von Neumann algebras.

Abstract: We will resume next week.