Lectures on families of Dirac operators and applications

TL;DR

This paper introduces families of Dirac operators and explores their applications in geometry and topology, including index theorems, positive scalar curvature metrics, and the Weinstein conjecture.

Contribution

It provides an accessible overview of families of Dirac operators and demonstrates their use in solving key problems in geometry and topology.

Findings

Index theorem for families of twisted Dirac operators

Applications to metrics of positive scalar curvature

Insights into the three-dimensional Weinstein conjecture

Abstract

These are the notes for a minicourse taught at the 2022 ICTP summer school `Frontiers in Geometry and Topology'. The goal is to introduce families of Dirac operators and how they can be used to study interactions between geometry and topology. In particular, we discuss the index theorem for families of twisted Dirac operators, and its applications to metrics of positive scalar curvature and the three-dimensional Weinstein conjecture.