Ph.D Thesis
Ph.D Student Prokhorova Marina Spectral Flow and Family Index for Self-Adjoint Elliptic Local Boundary Value Problems on Compact Surfaces Department of Mathematics Professor Emeritus Simeon Reich

Abstract

The thesis presents a first step towards a family index theorem for classical elliptic self-adjoint boundary value problems. I address here the simplest non-trivial case of manifolds with boundary, namely the case of two-dimensional manifolds. Over such a manifold, that is, a smooth compact surface with non-empty boundary, I consider first order self-adjoint elliptic differential operators with self-adjoint elliptic local boundary conditions.

The first part of my results concerns paths in the space of such boundary value problems connecting two boundary value problems conjugated by a unitary automorphism. I compute the spectral flow for such paths in terms of the topological data over the boundary. In addition, I show that the spectral flow is a universal additive invariant for such paths if the vanishing on paths of invertible operators is required.

The second part of my results concerns families of such boundary value problems parametrized by points of an arbitrary compact topological space X. I prove a family index theorem for such families, namely, I compute the K1(X)-valued analytical index of a family in terms of the topological data over the boundary. In addition, I show that the index is a universal additive invariant for such families if the vanishing on families of invertible operators is required.