Books

Masoud Khalkhali Basic Noncommutative Geometry-2nd Edition EMS Series of Lectures in Mathematics, Published by the European Mathematical Society Publishing House, December 2013. ISBN: 978-3-03719-061-6 Introduction | Contents | webpage on EMS Here is a Free Copy of the first edition of my book, Basic Noncommutative Geometry. Extra material is added in the second edition. You can order the new edition from EMS website. This text provides an introduction to noncommutative geometry and some of its applications. The book can be used either as a textbook for a graduate course on the subject 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.

Masoud Khalkhali Perspectives on Noncommutative Geometry Coedited with Guolinag Yu Published by the Fields Institute and the American Mathematical Society, Sept. 2011. ISBN-13: 978-0-8218-4849-4 Contents | webpage on AMS This volume represents the proceedings of the Noncommutative Geometry Workshop that was held as part of the thematic program on operator algebras at the Fields Institute in May 2008. Pioneered by Alain Connes starting in the late 1970s, noncommutative geometry was originally inspired by global analysis, topology, operator algebras, and quantum physics. Its main applications were to settle some long-standing conjectures, such as the Novikov conjecture and the Baum-Connes conjecture. Next came the impact of spectral geometry and the way the spectrum of a geometric operator, like the Laplacian, holds information about the geometry and topology of a manifold, as in the celebrated Weyl law. This has now been vastly generalized through Connes' notion of spectral triples. Finally, recent years have witnessed the impact of number theory, algebraic geometry and the theory of motives, and quantum field theory on noncommutative geometry. Almost all of these aspects are touched upon with new results in the papers of this volume. This book is intended for graduate students and researchers in both mathematics and theoretical physics who are interested in noncommutative geometry and its applications. Titles in this series are co-published with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

Masoud Khalkhali Quanta of Maths, a festshrift in honor of Alain Connes' 60 th birthday Coedited with Etienne Blanchard, David Ellwood, Matilde Marcolli, and Henri Moscovici Published jointly by the Clay Mathematics Institute and the American Mathematical Society, December 2010. ISBN-13: 978-0-8218-5203-3 Contents | Webpage on AMS The work of Alain Connes has cut a wide swath across several areas of math- ematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics. Specific themes covered by the articles are as follows: entropy in operator algebras, regular C*-algebras of integral domains, properly infinite C*-algebras, representations of free groups and 1-cohomology, Leibniz seminorms and quantum metric spaces; von Neumann algebras, fundamental Group of II1 factors, subfactors and planar algebras; Baum-Connes conjecture and property T, equivariant K-homology, Hermitian K-theory; cyclic cohomology, local index formula and twisted spectral triples, tangent groupoid and teh index theorem; noncommutative geometry and space-time, spectral action principle, quantum gravity, noncommutative ADHM and instantons, non-compact spectral triples of finite volume, noncommutative coordinate algebras; Hopf algebras, Vinberg algebras, renormalization and combinatorics, motivic renormalization and singularities; cyclotomy and analytic geometry over F1, quantum modular forms; differential K-theory, cyclic theory and S-cohomology.

Masoud Khalkhali Basic Noncommutative Geometry EMS Series of Lectures in Mathematics, Published by the European Mathematical Society Publishing House, December 2009. ISBN: 978-3-03719-061-6 Introduction | Contents | webpage on EMS This text provides an introduction to noncommutative geometry and some of its applications. The book can be used either as a textbook for a graduate course on the subject 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.

Masoud Khalkhali An Invitation to Noncommutative Geometry Coedited with Matilde Marcolli Published by World Scientific, Spring 2008. ISBN-13: 978-981-270-616-4 Preface | Contents | webpage on worldscience This is the first existing volume that collects lectures on this important and fast developing subject in mathematics. The lectures are given by leading experts in the field and the range of topics is kept as broad as possible by including both the algebraic and the differential aspects of noncommutative geometry as well as recent applications to theoretical physics and number theory.

M. Khalkhali Very Basic Noncommutative Geometry IPM Lecture Notes Series 5, Published by the Institute for Advanced Studies in Theoretical Physics and Mathematics (IPM), May 2005. Preface | Introduction | Contents

Masoud Khalkhali Cyclic Cohomology and Noncommutative Geometry Coedited with Joachim Cuntz Fields Institute Communications 17, Copublished by American Mathematical Society and the Fields Institute, 1997. ISBN-13: 978-0-8218-0823-8 webpage on AMS Noncommutative geometry is a new field that is among the great challenges of present-day mathematics. Its methods allow one to treat noncommutative algebras--such as algebras of pseudodifferential operators, group algebras, or algebras arising from quantum field theory--on the same footing as commutative algebras, that is, as spaces. Applications range over many fields of mathematics and mathematical physics. This volume contains the proceedings of the workshop on "Cyclic Cohomology and Noncommutative Geometry" held at The Fields Institute (Waterloo, ON) in June 1995. The workshop was part of the program for the special year on operator algebras and its applications. Features: Contributions by originators of the subject who are leaders in the field. Survey articles not previously available. Expository articles geared toward the larger mathematical community.