Contents

• Clifford algebras and spinor representations
  • Clifford algebras
  • The universality property
  • The trace
  • Periodicity
  • Chirality
  • $\Spinc$ and $\Spin$ groups
  • The Lie algebra of $\Spin(V)$
  • Orthogonal complex structures
  • Irreducible representations of $\bCl(V)$
  • Representations of $\Spinc(V)$
  
  
• Spinor modules over compact Riemannian manifolds
  • Remarks on Riemannian geometry
  • Clifford algebra bundles
  • The existence of $\Spin^c$ structures
  • Morita equivalence for (commutative) unital algebras
  • Classification of spinor modules
  • The spin connection
  • Epilogue: counting the spin structures
  
  
• Dirac operators
  • The metric distance property
  • Symmetry of the Dirac operator
  • Selfadjointness of the Dirac operator
  • The Schrödinger-Lichnerowicz formula
  • The spectral growth of the Dirac operator
  
  
• Spectral Growth and Dixmier Traces
  • Definition of spectral triples
  • Logarithmic divergence of spectra
  • Some eigenvalue inequalities
  • Dixmier traces
  
  
• Symbols and Traces
  • Classical pseudodifferential operators
  • Homogeneity of distributions
  • The Wodzicki residue
  • Dixmier trace and Wodzicki residue
  
  
• Spectral Triples: General Theory
  • The Dixmier trace revisited
  • Regularity of spectral triples
  • Pre-C*-algebras
  • Real spectral triples
  • Summability of spectral triples
  
  
• Spectral Triples: Examples
  • Geometric conditions on spectral triples
  • Isospectral deformations of commutative spectral triples
  • The Moyal plane as a nonunital spectral triple
  • A geometric spectral triple over $SU_q(2)$
  
  
• Exercises
  • Examples of Dirac operators
    • The circle
    • The (flat) torus
    • The Hodge-Dirac operator on $\bS^2$
  • The Dirac operator on the sphere $\bS^2$
    • The spinor bundle $S$ on $\bS^2$
    • The spin connection $\nabla^S$ over $\bS^2$
    • Spinor harmonics and the Dirac operator spectrum
  • Spin$^c$ Dirac operators on the 2-sphere
  • A spectral triple on the noncommutative torus
  
  
• Bibliography