Course  Materials  

Dirac operators and spectral geometry
This lecture course will be an introduction to Dirac operators on spin manifolds and spectral triples in differential and noncommutative geometry. The essence of the matter is to go beyond classical themes by recasting geometry in an operatortheoretic mould, with a view to reconciling ordinary geometry with quantum physics. The interplay of geometry and analysis needed to achieve this demands the unification of several disparate strands of mathematics. We take a stepbystep approach, going from classical geometrical topics to fully noncommutative cases, with emphasis on examples. The course starts with some conventional differential geometry: Clifford algebras and Clifford modules; spin structures and spinc 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; preC*algebras; Hochschild homology of algebras; culminating in the notion of a spectral triple, which provides an axiomatic framework for spin geometry. Next, we reinperpret spin manifolds in noncommutative terms: showing how spectral triples are obtained from classical Dirac operators; reconstructing compact spin manifolds from spectral triples; and exploring the spectral aspect of spin geometry. Finally, we move to fully noncommutative coordinate algebras: isospectral deformations of spin geometries, both compact (toral deformations) and noncompact (Moyal planes); and spectral triples based on quantum groups and spheres. 
Dirac operators and spectral geometry
Joseph C. Varilly
Notes by P. Witkowski


Download  Last update  
24.01.2006 

Table of contents  

Exam  Exam questions  

The exam was on 31st January 2006. It consisted of the written part (six exercises) and oral part. In the oral part each student had to answer two questions: easy one and difficult one (chosen from the two difficult questions). Four students (on the graduate and undergraduate level) passed the exam. 
Exam, written and oral 