# An Introduction to Noncommutative Geometry

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*Joseph C. Várilly*

A publication of the European Mathematical Society

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.

#### Readership

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