An Introduction to Noncommutative Geometry
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Beschreibung
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 onexamples.
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 condiditons 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.
Autor*in
Joseph C Várilly
Themen in »An Introduction to Noncommutative Geometry«
Dirac operator Noncommutative geometry Spectral triple