This week, Thursday at 11:00, U. Haifa Topology & Geometry seminar on November 7, 2019
Geometry & Topology Seminar
Speaker: Marina Prokhorova (Technion)
Topic: Family index for self-adjoint elliptic boundary value problems
Place: Room 614 in the Science & Education Building
Date: Thursday, November 7, 2019
An index theory for elliptic operators on a closed manifold was developed by Atiyah and Singer. For a family of such operators parametrized by points of a compact space X, they computed the K^0(X)-valued analytical index in purely topological terms. An analog of this theory for self-adjoint elliptic operators on closed manifolds was developed by Atiyah, Patodi, and Singer; the analytical index of a family in this case takes values in the K^1 group of a base space.
If a manifold has non-empty boundary, then boundary conditions come into play, and situation becomes much more complicated. The integer-valued index of a single boundary value problem was computed by Boutet de Monvel. This result was recently generalized to K^0-valued family index by Melo, Schrohe, and Schick. The case of self-adjoint operators, however, remained open; it seems that Boutet de Monvel’s calculus is not adapted to it.
In the talk I present a family index theorem for self-adjoint elliptic operators on a surface with boundary. I compute the K^1(X)-valued analytical index in terms of the topological data of the family over the boundary. The talk is based on my preprint arXiv:1809.04353.