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Noncommutative Geometry de Alain Connes

Noncommutative Geometry

Autor Alain Connes
en Limba Engleză Hardback – 17 ian 1995

Notăm cu interes prezența acestei lucrări fundamentale semnate de Alain Connes, personalitatea centrală a matematicii contemporane și laureat al Medaliei Fields, care a pus bazele disciplinei pe care o descrie aici. Această ediție în limba engleză, publicată de ELSEVIER SCIENCE, reprezintă tratamentul definitiv al abordării sale revoluționare asupra teoriei măsurii și geometriei diferențiale. Autorul nu doar că sintetizează decenii de cercetare proprie, dar propune o schimbare de paradigmă în modul în care înțelegem structura spațiului la nivel microscopic și macroscopic. Comparabil cu An Introduction to Noncommutative Spaces and Their Geometries de Giovanni Landi în rigoarea definițiilor, volumul de față se distinge prin faptul că este sursa originală a conceptelor, oferind o profunzime teoretică superioară și o conexiune directă cu fizica matematică. În contextul operei sale, lucrarea face tranziția de la cercetările pur analitice către interogații fundamentale despre realitate, teme explorate ulterior și în On Space and Time. Structura celor 680 de pagini este organizată progresiv: începe cu fundamentele algebrelor von Neumann și teoria modulară, avansează către topologie și K-teorie, culminând cu coomologia ciclică și aplicațiile în modelarea spațiu-timpului. Putem afirma că rigoarea cu care sunt tratate exemplele, de la spațiul pavajelor Penrose la faza în mecanica cuantică, transformă acest text într-un instrument de lucru indispensabil pentru orice cercetător care dorește să înțeleagă cuantificarea dincolo de metodele clasice.

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Specificații

ISBN-13: 9780121858605
ISBN-10: 012185860X
Pagini: 680
Ilustrații: references, indices
Dimensiuni: 175 x 245 x 35 mm
Greutate: 1.37 kg
Editura: ELSEVIER SCIENCE

Public țintă

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.

De ce să citești această carte

Această carte este esențială pentru studenții doctoranzi și cercetătorii în matematică sau fizică teoretică. Cititorul câștigă o înțelegere profundă a geometriei necomutative direct de la fondatorul domeniului, Alain Connes. Este o resursă rară care unifică analiza funcțională, algebra și geometria, oferind un model matematic coerent pentru fenomenele fizice complexe, de la efectul Hall cuantic la structura fină a spațiului.

Descriere scurtă

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

Cuprins

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.

Recenzii

"...A milestone for mathematics. Connes has created a theory that embraces most aspects of 'classical' mathematics and sets us out on a long and exciting voyage into the world of noncommutative mathematics.
"The book contains a colourful account of the meaning of the term 'non-commutative space,' based on an extraordinary wealth of examples, including the set of all Penrose tilings, the space of leaves of a foliation, the quantum Hall effect and an intriguing non-commutative model of four-dimensional space-time that reproduces the standard model of elementary particles from quite general considerations...
"The reader of the book should not expect proofs of theorems. This is much more a tapestry of beautiful mathematics and physics which contains material to intrigue readers with any mathematical background. At the same time there is a comprehensive bibliography that will lead the reader straight to the sources and proofs of the results."
--VAUGHAN F.R. JONES, University of California, Berkeley
"This beautiful, ambitious, and erudite book explains, through many examples, the phenomena, tools, and some of the applications of noncommutative geometry...The book is written in a way that anyone can get some of the feeling and ideas of the subject...Connes has accomplished the wonderful feat of explaining in a simple and coherent way 20 years (or so) of his impressive work. I recommend this book most highly." --Jonathan Block, THE MATHEMATICAL INTELLIGENCER, Vol. 20, No. 1, 1998