Dirac operators and spectral geometry

Abstract

This lecture course is an introduction to Dirac operators on spin manifolds and spectral triples in differential and noncommutative geometry. It goes beyond classical themes by recasting geometry in an operator-theoretic mould, with a view to reconciling ordinary geometry with quantum physics. This interplay of geometry and analysis demands the unification of several disparate strands of mathematics, going from classical geometrical topics to fully noncommutative cases, with emphasis on examples. The course starts with differential geometry: Clifford algebras and Clifford modules; spin structures and spin-c structures; Dirac operators, their geometric properties, and several examples. We then introduce the noncommutative toolbox: operator ideals and Dixmier traces; Wodzicki residues and Connes' trace theorem; pre-C*-algebras; Hochschild homology of algebras; culminating in the notion of a spectral triple, which provides an axiomatic framework for spin geometry. After reinprepreting spin manifolds in noncommutative terms, we move to fully noncommutative coordinate algebras: isospectral deformations of spin geometries, both compact and noncompact; and spectral triples based on spheres and quantum groups.