An Introduction to Noncommutative Geometry

TL;DR

This paper introduces the fundamental concepts and developments in noncommutative geometry, covering spectral triples, noncommutative spaces like the torus, and recent advances in the field.

Contribution

It provides an overview of noncommutative geometry, including new perspectives, axiomatic foundations, and recent research directions from a course given in 1997 with updates through 2006.

Findings

Spectral triples on the Riemann sphere analyzed.

Noncommutative integral and quantization methods discussed.

Recent developments and new references included.

Abstract

This is the introduction and bibliography for lecture notes of a course given at the Summer School on Noncommutative Geometry and Applications, sponsored by the European Mathematical Society, at Monsaraz and Lisboa, Portugal, September 1-10, 1997. In the published version, an epilogue of recent developments and many new references from 1998-2006 have been added. 1. Commutative geometry from the noncommutative point of view. 2. Spectral triples on the Riemann sphere. 3. Real spectral triples, the axiomatic foundation. 4. Geometries on the noncommutative torus. 5. The noncommutative integral. 6. Quantization and the tangent groupoid. 7. Equivalence of geometries. 8. Action functionals. 9. Epilogue: new directions.