U. Haifa Topology & Geometry seminar: Wednesday, Mar. 19, 2025. Speaker: Marina Prokhorova (Haifa). Title: “Index theory for unbounded Fredholm operators”.
A linear operator on a Hilbert space is called Fredholm if its kernel and cokernel are finitedimensional. As was shown in classical works of Atiyah, J¨anich, and Singer, the space of bounded Fredholm operators represents even K-theory; its subspace consisting of self-adjoint operators has three connected components, one of which represents odd K-theory. The index theory of elliptic differential operators on closed manifolds is based on these classical results.However, in some situations (e.g., for elliptic operators on manifolds with boundary) one needs to deal with families of unbounded operators. A proper notion of continuity for such families of operators is continuity of their graphs. My talk is devoted to an index theory of such families. I will explain how the relevant spaces of unbounded operators are related to classical spaces of bounded Fredholm operators and show that natural maps between them are homotopy equivalences. The proof is based on a theorem of tom Dieck: a map is a homotopy equivalence if it is locally a homotopy equivalence.The talk is based on my preprint arXiv:2110.14359. All operator-theoretical notions will be explained in the talk.
