An Introduction to Noncommutative Geometry (original) (raw)
1997, Eprint Arxiv Physics 9709045
The lecture notes of this course at the EMS Summer School on Noncommutative Geometry and Applications in September, 1997 are now published by the EMS. Here are the contents, preface and updated bibliography from the published book. Contents 1 Commutative Geometry from the Noncommutative Point of View 1.1 The Gelfand-Naȋmark cofunctors 1.2 The Γ functor 1.3 Hermitian metrics and spin c structures 1.4 The Dirac operator and the distance formula 2 Spectral Triples on the Riemann Sphere 2.1 Line bundles and the spinor bundle 2.2 The Dirac operator on the sphere S 2 2.3 Spinor harmonics and the spectrum of D / 2.4 Twisted spinor modules 2.5 A reducible spectral triple 3 Real Spectral Triples: the Axiomatic Foundation 3.1 The data set 3.2 Infinitesimals and dimension 3.3 The first-order condition 3.4 Smoothness of the algebra 3.5 Hochschild cycles and orientation 3.6 Finiteness of the K-cycle 3.7 Poincaré duality and K-theory 3.8 The real structure Geometries on the Noncommutative Torus 4.1 Algebras of Weyl operators 4.2 The algebra of the noncommutative torus 4.3 The skeleton of the noncommutative torus 4.4 A family of spin geometries on the torus The Noncommutative Integral 5.1 The Dixmier trace on infinitesimals 5.2 Pseudodifferential operators 5.3 The Wodzicki residue 5.4 The trace theorem 5.5 Integrals and zeta residues Quantization and the Tangent Groupoid 6.1 Moyal quantizers and the Moyal deformation 6.2 Smooth groupoids 6.3 The tangent groupoid 6.4 Moyal quantization as a continuity condition 6.5 The hexagon and the analytical index 6.6 Quantization and the index theorem Equivalence of Geometries 7.1 Unitary equivalence of spin geometries 7.2 Morita equivalence and connections 7.3 Vector bundles over noncommutative tori 7.4 Morita-equivalent toral geometries 7.5 Gauge potentials Action Functionals 8.1 Algebra automorphisms and the metric 8.2 The fermionic action 8.3 The spectral action principle 8.4 Spectral densities and asymptotics Epilogue: New Directions 9.1 Noncommutative field theories 9.2 Isospectral deformations 9.3 Geometries with quantum group symmetry 9.4 Other developments
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