Scientific seminar on the differential and functional differential equation under the guidance of prof. Sternin B. Yu. and prof. Savin A.Yu

Speaker: Maxim Braverman (Northeastern University, USA).

Topic: Тhe spectral Flow of a family of Toeplitz operators.

Abstract:  We show that the (graded) spectral flow of a family of Toeplitz operators on a complete Riemannian manifold is equal to the index of a certain Callias-type operator. When the dimension of the manifold is even this leads to a cohomological formula for the spectral flow. As an application, we compute the spectral flow of a family of Toeplitz operators on a strongly pseudoconvex domain in \mathbb{C}^n. This result is similar to the Boutet de Monvel's computation of the index of a single Toeplitz operator on a strongly pseudoconvex domain. Finally, we show that the bulk-boundary correspondence in a tight-binding model of topological insulators is a special case of our result. In the appendix, Koen van den Dungen reviewed the main result in the context of (unbounded) KK-theory.