2005/06, 3rd semester

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 operator-theoretic 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 step-by-step 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 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. 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
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Table of contents
  1. Clifford algebras and spinor representations
    1. Clifford algebras
    2. Universality
    3. The trace
    4. Periodicity
    5. Chirality
    6. Spin^c and Spin groups
    7. The Lie algebra of Spin(V)
    8. Orthogonal complex structures
    9. Irreducible representations of Cl(V)
    10. Representations of Spin^c(V)
  2. Spinor modules over compact Riemannian manifolds
    1. Remarks on Riemannian geometry
    2. Clifford algebra bundles
    3. The existence of Spin^c structures
    4. Morita equivalence for (commutative) unital algebras
    5. Classification of spinor modules
    6. The Spin connection
    7. Epilogue: counting the spin structures
  3. Dirac operators
    1. The metric distance property
    2. Symmetry of the Dirac operator
    3. Selfadjointness of the Dirac operator
    4. The Schrodinger-Lichnerowicz formula
    5. The spectral growth of the Dirac operator
  4. Spectral growth and Dixmier traces
    1. Definition of spectral triples
    2. Logarithmic divergence of spectra
    3. Some eigenvalue inequalities
    4. Dixmier traces
  5. Symbols and traces
    1. Classical pseudodifferential operators
    2. Homogenity of distributions
    3. The Wodzicki residue
    4. Dixmier trace and Wodzicki residue
  6. Spectral triples: general theory
    1. The Dixmier trace revisited
    2. Regularity of spectral triples
    3. Pre-C*-algebras
    4. Real spectral triples
    5. Summability of spectral triples
  7. Spectral triples - examples
    1. Geometric conditions on spectral triples
    2. Isospectral deformations of commutative spectral triples
    3. The Moyal plane as a nonunital spectral triple
    4. A geometric spectral triple over SU_q(2)
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

Designed by: Pawel Witkowski