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: Oriented cohomology theories and the associated formal groups laws have been a subject of intensive investigations since 60's, mostly inspired by the theory of complex cobordism. In the present talk we discuss several recent developments in the study of algebraic analogues of such theories, e.g. algebraic cobordism of Levine-Morel or algebraic elliptic cohomology, of projective homogeneous spaces.