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Basic Noncommutative Geometry
EMS Series of Lectures in Mathematics

Masoud Khalkhali (The University of Western Ontario, London, Canada)

Basic Noncommutative Geometry

Second edition

ISBN print 978-3-03719-128-6, ISBN online 978-3-03719-628-1
DOI 10.4171/128
December 2013, 257 pages, softcover, 17 x 24 cm.
38.00 Euro

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

Keywords: Noncommutative space, noncommutative quotient, groupoid, cyclic cohomology, Connes–Chern character, index formula

Further Information

MAA Reviews (1st ed.)

Review of 1st ed. in MR 2567651 (2011d:58017)

Review of 1st ed. in Zentralblatt MATH 1210.58006

Review of 1st ed. in Bull. London Math. Soc. 43 (2011), no. 1, 206–208

Review of 2nd ed. in The Mathematical Intelligencer, March 2015, Volume 37, Issue 1, pp 93-94

Review of 2nd ed. in Bulletin of the AMS