Heat kernels and the index theorems on even and odd dimensional manifolds

TL;DR

This paper reviews heat kernel methods for Atiyah-Singer and Atiyah-Patodi-Singer index theorems on closed and boundary manifolds, extending results to odd dimensions and including a joint theorem for Toeplitz operators.

Contribution

It introduces a new index theorem for Toeplitz operators on odd-dimensional manifolds with boundary, expanding the scope of index theory.

Findings

Heat kernel approach effectively proves classical index theorems.

Extension of index theorems to odd-dimensional manifolds.

New joint result on Toeplitz operators on odd-dimensional manifolds with boundary.

Abstract

In this talk, we review the heat kernel approach to the Atiyah-Singer index theorem for Dirac operators on closed manifolds, as well as the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary. We also discuss the odd dimensional counterparts of the above results. In particular, we describe a joint result with Xianzhe Dai on an index theorem for Toeplitz operators on odd dimensional manifolds with boundary.