# Basic Noncommutative Geometry: Second Edition

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*Masoud Khalkhali*

A publication of the European Mathematical Society

This text provides an introduction to noncommutative geometry and some
of its applications. It can be used either as a textbook for a graduate course
or for self-study. It will be useful for graduate students and researchers in
mathematics and theoretical physics and all those who are interested in gaining
an understanding of the subject.

One feature of this book is the wealth of examples and exercises
that help the reader to navigate through the subject. While
background material is provided in the text and in several appendices,
some familiarity with basic notions of functional analysis, algebraic
topology, differential geometry and homological algebra at a first
year graduate level is helpful.

Developed by Alain Connes since the late 1970s, noncommutative
geometry has found many applications to long-standing conjectures in
topology and geometry and has recently made headways in theoretical
physics and number theory. The book starts with a detailed description
of some of the most pertinent algebra geometry correspondences by
casting geometric notions in algebraic terms, then proceeds in the
second chapter to the idea of a noncommutative space and how it is
constructed. The last two chapters deal with homological tools: cyclic
cohomology and Connes–Chern characters in \(K\)-theory
and \(K\)-homology, culminating in one commutative diagram
expressing the equality of topological and analytic index in a
noncommutative setting. Applications to integrality of noncommutative
topological invariants are given as well.

Two new sections have been added to the second edition: the first
new section concerns the Gauss–Bonnet theorem and the definition
and computation of the scalar curvature of the curved noncommutative
two torus, and the second new section is a brief introduction to Hopf
cyclic cohomology. The bibliography has been extended and some new
examples are presented.

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

#### Readership

Graduate students and research mathematicians interested in mathematics and theoretical physics.