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This book covers the basics of noncommutative geometry (NCG) and its applications in topology, algebraic geometry, and number theory. The author takes up the practical side of NCG and its value for other areas of mathematics. A brief survey of the main parts of NCG with historical remarks, bibliography, and a list of exercises is included. The presentation is intended for graduate students and researchers with interests in NCG, but will also serve nonexperts in the field.
Contents
Part I: Basics
Model examples
Categories and functors
C -algebras
Part II: Noncommutative invariants
Topology
Algebraic geometry
Number theory
Part III: Brief survey of NCG
Finite geometries
Continuous geometries
Connes geometries
Index theory
Jones polynomials
Quantum groups
Noncommutative algebraic geometry
Trends in noncommutative geometry
Report
"The book is a good survey of noncommutative geometry, and is an excellent starting point for doing good research in the field. Each section of the book ends with a list of references for going deeply into each subject, and so this book gives a framework for the field of noncommutative geometry."
Arvid Siqveland in: Zentralblatt für Mathematik 1388, 1-2
"In any case, it is true with a vengeance that NCG is a big and important subject, and the book we are looking at here, with its subtitle of "A functorial approach" is doubtless a very valuable contribution on any number of counts, not the least of which being that the more functorial we get, the more generality we reap, and the clearer the overarching structure becomes. [...] In point of fact, the book under review is the first book on NCG to take a functorial approach as its Leitmotiv, and that is in itself a very solid contribution, pedagogically and mathematically."
Michael Berg in: MAA Reviews (27.07.2018), https://www.maa.org/press/maa-reviews/noncommutative-geometry-a-functorial-approach
