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An Introduction to Noncommutative Geometry
Joseph C. Várilly, University of Costa Rica, San Jose, Costa Rica
A publication of the European Mathematical Society.
EMS Series of Lectures in Mathematics
2006; 121 pp; softcover
Volume: 4
ISBN-10: 3-03719-024-8
ISBN-13: 978-3-03719-024-1
List Price: US$34
Member Price: US$27.20
Order Code: EMSSERLEC/4
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Noncommutative geometry, inspired by quantum physics, describes singular spaces by their noncommutative coordinate algebras and metric structures by Dirac-like operators. Such metric geometries are described mathematically by Connes' theory of spectral triples. These lectures, delivered at an EMS Summer School on noncommutative geometry and its applications, provide an overview of spectral triples based on examples.

This introduction is aimed at graduate students of both mathematics and theoretical physics. It deals with Dirac operators on spin manifolds, noncommutative tori, Moyal quantization and tangent groupoids, action functionals, and isospectral deformations. The structural framework is the concept of a noncommutative spin geometry; the conditions on spectral triples which determine this concept are developed in detail. The emphasis throughout is on gaining understanding by computing the details of specific examples.

The book provides a middle ground between a comprehensive text and a narrowly focused research monograph. It is intended for self-study, enabling the reader to gain access to the essentials of noncommutative geometry. New features since the original course are an expanded bibliography and a survey of more recent examples and applications of spectral triples.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.


Graduate students and researchers in mathematics and theoretical physics interested in noncommutative geometry.

Table of Contents

  • Commutative geometry from the noncommutative point of view
  • Spectral triples on the Riemann sphere
  • Real spectral triples: the axiomatic foundation
  • Geometries on the noncommutative torus
  • The noncommutative integtral
  • Quantization and the tangent groupoid
  • Equivalence of geometries
  • Action functionals
  • Epilogue: new directions
  • Bibliography
  • Index
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