This volume covers a wide range of topics including sources of noncommutative geometry; fundamentals of noncommutative topology; K-theory and Morita equivalance; noncommutative integrodifferential calculus; noncommutative Riemannian spin manifolds; commutative geometrics; tori; proof of Connes' Riemannian spin theorem; second quantization; noncommutative quantum field theory; C*-algebras; Hopf algebras; Clifford algebras; Moyal algebras; pseudodifferential operators; the commutative Chern character theorem.
Elements of Noncommutative Geometry
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"In collecting the material in a coherent form, the authors have performed a valuable service and this book will no doubt open noncommutative geometry to a wider audience.... [A] large number of metaphorical and mathematical remarks embedded in the text...highlight the plethora of connections between the material of the book and other areas of mathematics." —Mathematical Reviews "The material is collected from an enormous number of sources.... All of it is carefully worked out and serves as a highly recommendable introduction to the vital field of noncommutative geometry." —Zentralblatt Math "The style of the book is lively, exempt of the stereotypes one encounters rather often in the mathematical literature. The authors know how to tell stories and enjoy doing it. The friendly tone could be of some help for an hypothetical unexperienced reader confronted with almost 700 pages of rather difficult and delicate mathematics." —Journal of Operator Theory "The present book is a systematic course in noncommutative differential geometry and operator theory, with applications to guantum physics. Its topics cover C*-algebras, vector bundles and C*-modules, K-theory, Fredholm operators, Clifford algebras, Spin representations, noncommutative integration and differential calculus, spectral triples and Connes spin manifold theorem. As applications, noncommutative tori, quantum theory and Kreimer-Connes-Moscovici algebras are discussed. The book will be helpful to all mathematicians and mathematical physicits who wish to learn about noncommutative geometry and tis ramifications." ---Matheamtica Bohemica
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6 of 14 people found the following review helpful:
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right complement,
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This review is from: Elements of Noncommutative Geometry (Birkhäuser Advanced Texts Basler Lehrbücher) (Hardcover)
Pepe and Joe have been involved in NCG researchsince the first physical calculations. And the book happened to be published in a very adequate time range, so it contains up-to-date information on all the developments, including the finding of Connes-Kreimer renormalization algebra (hopf algebras and butcher groups). It is a compulsory complement to Connes's book.
1 of 10 people found the following review helpful:
4.0 out of 5 stars
right complement,
By A Customer
This review is from: Elements of Noncommutative Geometry (Birkhäuser Advanced Texts Basler Lehrbücher) (Hardcover)
Pepe and Joe have been involved in NCG researchsince the first physical calculations. And the book happened to be published in a very adequate time range, so it contains up-to-date information on all the developments, including the finding of Connes-Kreimer renormalization algebra (hopf algebras and butcher groups). It is a compulsory complement to Connes's book.
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Inside This Book
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First Sentence:
The geometrical study of quadratic curves or surfaces, i.e., zero sets of second-degree polynomials, proceeds by examining points of intersection or tangent lines directly; but already for cubic curves it pays to examine first the ideal of all polynomials that vanish on the curve: in this way the study of an algebraic variety (the zero set of a given finite collection of polynomials) is replaced by the study of the corresponding polynomial ideal. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
quantum scattering matrix, noncommutative spin geometry, spectral triple, noncommutative integral, spin theorem, asymptotic morphism, spinc structure, nonunital case, noncommutative tori, grading operator, entire cyclic cohomology, noncommutative manifolds, prehilbert space, spinor module, noncommutative topology, classical pseudodifferential operator, logarithmic elements, chirality element, unital morphism, graded differential algebra, compact spin manifold, counit property, noncommutative residue, periodic cyclic cohomology, orthogonal complex structure Key Phrases - Capitalized Phrases (CAPs): (learn more)
Proof Let, Standard Model, Proof First, Proof Suppose, Proof Choose, Hopf H-module New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
The geometrical study of quadratic curves or surfaces, i.e., zero sets of second-degree polynomials, proceeds by examining points of intersection or tangent lines directly; but already for cubic curves it pays to examine first the ideal of all polynomials that vanish on the curve: in this way the study of an algebraic variety (the zero set of a given finite collection of polynomials) is replaced by the study of the corresponding polynomial ideal. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
quantum scattering matrix, noncommutative spin geometry, spectral triple, noncommutative integral, spin theorem, asymptotic morphism, spinc structure, nonunital case, noncommutative tori, grading operator, entire cyclic cohomology, noncommutative manifolds, prehilbert space, spinor module, noncommutative topology, classical pseudodifferential operator, logarithmic elements, chirality element, unital morphism, graded differential algebra, compact spin manifold, counit property, noncommutative residue, periodic cyclic cohomology, orthogonal complex structure Key Phrases - Capitalized Phrases (CAPs): (learn more)
Proof Let, Standard Model, Proof First, Proof Suppose, Proof Choose, Hopf H-module New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 75
books:
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16
books
cite this book:
See all 16 books citing this book
- Noncommutative Geometry by Alain Connes on 4 pages
- Perturbation Theory for Linear Operators (Classics in Mathematics) by Tosio Kato in Back Matter
- An Introduction to Homological Algebra (Cambridge Studies in Advanced Mathematics) by Charles A. Weibel in Back Matter
- An Introduction to Noncommutative Differential Geometry and its Physical Applications (London Mathematical Society Lecture Note Series) by J. Madore in Back Matter
- Calculus on Heisenberg Manifolds (Annals of Mathematics Studies) by Richard Beals in Back Matter
See all 75 books this book cites
- Geometric and Topological Methods for Quantum Field Theory - Proceedings of the Summer School by A. Cardona on page 81, and page 225
- Quantum Stochastic Processes and Noncommutative Geometry (Cambridge Tracts in Mathematics) by Kalyan B. Sinha in Back Matter
- Noncommutative Geometry: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 3-9, 2000 (Lecture Notes in Mathematics / C.I.M.E. Foundation Subseries) by Alain Connes in Front Matter
- An Introduction to Noncommutative Differential Geometry and its Physical Applications (London Mathematical Society Lecture Note Series) by J. Madore in Back Matter
- String Phenomenology 2003: Proceedings of the 2nd International Conference Durham, UK, 4 July- 4 August, 2003 by Veronica Sanz on page 21
See all 16 books citing this book
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