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Noncommutative Geometry

1st Edition - November 22, 1994

Author: Alain Connes

Hardback ISBN:9780121858605

9 7 8 - 0 - 1 2 - 1 8 5 8 6 0 - 5

eBook ISBN:9780080571751

9 7 8 - 0 - 0 8 - 0 5 7 1 7 5 - 1

This English version of the path-breaking French book on this subject gives the definitive treatment of the revolutionary approach to measure theory, geometry, and mathematical… Read more

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This English version of the path-breaking French book on this subject gives the definitive treatment of the revolutionary approach to measure theory, geometry, and mathematical physics developed by Alain Connes. Profusely illustrated and invitingly written, this book is ideal for anyone who wants to know what noncommutative geometry is, what it can do, or how it can be used in various areas of mathematics, quantization, and elementary particles and fields.

First full treatment of the subject and its applications

Written by the pioneer of this field

Broad applications in mathematics

Of interest across most fields

Ideal as an introduction and survey

Examples treated include:

the space of Penrose tilings

the space of leaves of a foliation

the space of irreducible unitary representations of a discrete group

the phase space in quantum mechanics

the Brillouin zone in the quantum Hall effect

A model of space time

Graduate students and researchers in mathematics and mathematical physics. Particular appeal for analysts, geometers, and differential geometers; those interested in quantization in physics and mathematics; particle physicists,functional analysts; and algebraists.

Noncommutative Spaces and Measure Theory: Heisenberg and the Noncommutative Algebra of Physical Quantities Associated to a Microscopic System. Statistical State of a Macroscopic System and Quantum Statistical Mechanics. Modular Theory and the Classification of Factors. Geometric Examples of von Neumann Algebras: Measure Theory of Noncommutative Spaces. The Index Theorem for Measured Foliations. Topology and K-Theory:C*-Algebras and their K-Theory. Elementary Examples of Quotient Spaces. The Space X of Penrose Tilings. Duals of Discrete Groups and the Novikov Conjecture. The Tangent Groupoid of a Manifold. Wrong-way Functionality in K-Theory as a Deformation. The Orbit Space of a GroupAction. The Leaf Space of a Foliation. The Longitudinal Index Theorem for Foliations. The Analytic Assembly Map and Lie Groups. Cyclic Cohomology and Differential Geometry: Cyclic Cohomology. Examples. Pairing of Cyclic Cohomology with K-Theory. The Higher Index Theorem for Covering Spaces. The Novikov Conjecture for Hyperbolic Groups. Factors of Type III, Cyclic Cohomology and the Godbillon-Vey Invariant. The Transverse Fundamental Class for Foliations and Geometric Corollaries. QuantizedCalculus: Quantized Differential Calculus and Cyclic Cohomology. The Dixmier Trace and the Hochschild Class of the Character. Quantized Calculus in One Variable and Fractal Sets. Conformal Manifolds. Fredholm Modules and Rank-One Discrete Groups. Elliptic Theory on the Noncommutative Torus (NOTE: See book for proper symbol. Math T with a 2 over () and the Quantum Hall Effect. Entire Cyclic Cohomology. The Chern Character of (-Summable Fredholm Modules. (-Summable K-Cycles, Discrete Groups, and Quantum Field Theory. Operator Algebras: The Papers of Murray and von Neumann. Representations of C*-Algebras. The Algebraic Framework for Noncommutative Integration and the Theory of Weights. The Factors of Powers, Araki and Woods,and of Krieger. The Radon-Nikodom Theorem and Factors of Type III(. Noncommutative Ergodic Theory. Amenable von Neumann Algebras. The Flow of Weights: mod(M). The Classification of Amenable Factors. Subfactors of Type II1 Factors. Hecke Algebras ,Type III Factors and Statistical Theory of Prime Numbers. The Metric Aspect of Noncommutative Geometry: Riemannian Manifolds and the Dirac Operator. Positivity in Hochschild Cohomology and the Inequalities for the Yang-Mills Action. Product of the Continuum by the Discrete and the Symmetry Breaking Mechanism. The Notion of Manifold in Noncommutative Geometry. The Standard U (1) x SU (2) x SU (3) Model. Bibliography. Notation and Conventions. Index.CONTENTS (Chapter Headings): Noncommutative Spaces and Measure Theory. Topology and K-Theory. Cyclic Cohomology and Differential Geometry. Quantized Calculus. Operator Algebras. The Metric Aspect of Noncommutative Geometry. Bibliography. Notation and Conventions. Index.