Alain Connes

Noncommutative Geometry (eBook, ePUB)

• Format: ePub

Produktbeschreibung

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

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Produktdetails

• Verlag: Elsevier Science & Techn.
• Seitenzahl: 661
• Erscheinungstermin: 17. Januar 1995
• Englisch
• ISBN-13: 9780080571751
• Artikelnr.: 47273712

Inhaltsangabe

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.