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Title: Noncommutative Geometry by Alain Connes ISBN: 0-12-185860-X Publisher: Academic Press Pub. Date: 22 November, 1994 Format: Hardcover Volumes: 1 List Price(USD): $81.95 |
Average Customer Rating: 4.25 (4 reviews)
Rating: 5
Summary: The place to find it!
Comment: This is *the book* on noncommutative geometry. Alain Connes created the main parts of the theory, and paved the way for
its many exciting applications. As most would expect, the book
covers operator algebras, quantum theory, and other areas of mathematical physics; --but, in addition, there is a lovely treatment of tiling theory[Penrose tilings, cartwheel tilings and the like], of the Dixmier trace, of groupoids, and of foliations.-- All new subjects: intriguing and penetrating!
Rating: 4
Summary: A beautiful subject
Comment: Even though detailed proofs are omitted for most of the major results, the book is an excellent overview of a beautiful subject that the author has made substantial contributions to. The subject of noncommutative geometry has recently made its way into theoretical physics, and so a perusal of this book would be of interest to individuals working in string theory or quantum field theory.
The main idea of this book is to generalize measure and operator theory to non-commutative situations. In the usual operator theory, von Neumann algebras serve as a generalization of "classical" measure theory. Commutative von Neumann algebras, or W*-algebras as they are sometimes called, are essentially bounded meausurable functions, and have measure spaces as their dual. These facts and a fine movtivation for the subject appear in the introduction to the book. The author shows with great clarity what is involved in extending measure theory to the non-commutative case. What is most interesting about the extensions is that they involve ideas from quantum physics. In addition, readers familiar with K-theory will see some brilliant uses of it in the book, particularly in the extension of BDF-theory to noncommutative situations, namely the KK-theory of Kasparov. The author also gives a taste of physics applications in the very last section of the book. He shows, interestingly, that when space-time is replaced by a product with a certain finite space, the Lagrangian of quantum electrodynamics becomes that of the Standard Model. Although such "add-ons" to space-time are not uncommon in physics (Kaluza-Klein theories being one example), the author's strategy is unique in its use of bimodules, and gives the three lepton generations.
There are also many other interesting topics as well in the book, such as how to deal with non-Hausdorff quotient spaces using noncommutative C*-algebras, deformation theory and the Kasparov group, the notion of Morita equivalence, leaf spaces of foliations, the E-theory of morphisms of separable C*-algebras, the extension of de Rham cohomology to a noncommutative framework (cyclic cohomology) and its relation to K-theory, the noncommutative torus and the quantum Hall effect.
The book is an excellent source of information on noncommutative geoemtry and with the many references given one can find more detailed proofs. It is a subject that will no doubt continue to make its presence known in mathematics (and physics) in years to come.
Rating: 3
Summary: Does not stand on its own
Comment: I must depart a bit from the previous breathless outpourings about this book. The fact is, it is a whirlwind TOUR (or travellog) of noncommutative geometry, not anything like a handbook of it, or even an atlas of detailed maps of it. I say this because theorems are asserted but almost never proved, no 'problems' are worked, and it is my experience and universally that of all mathematicians/physicists I know, no matter how gifted, that one cannot really understand the subject matter without doing problems! The previous reviewer does hint: "even if you do not know the subject matter" - aye, there's the rub! I appreciate the author's gifts, and I can compass his vision of how useful his approach might be, but between rather trivial points in quantum mechanics to very abstruse theorems in abstract harmonic analysis there is no bridge provided, and the original literature, either by Connes or his predecessor Dixmier, is practically all in French. If you already know this stuff, it might be useful to have all the relevant topics gathered together in one place, but if you don't already know it, you are going to be disappointed. Customers should be aware of this fact before they shell out the bucks. A much better book covering similar ground, but at a more directly physical and more elementary level, is Souriau's: "Structure of Dynamical Systems. A Symplectic View of Physics" Although that work again doesn't have worked exercises. I wonder, is it just the STYLE in French literature nowadays to DISCOURSE about mathematics, instead of DEMONSTRATING it? Rather like a self-fulfillment of the Derrida-Lacan-Latour-deconstructionist position about scientific communities and their provenance, I'd say.
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Title: Elements of Noncommutative Geometry (Birkhauser Advanced Texts) by Joseph C. Varilly, Hector Figueroa, Jose M. Gracia-Bondia ISBN: 0817641246 Publisher: Birkhauser Boston Pub. Date: 23 October, 2000 List Price(USD): $69.95 |
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Title: Analysis, Manifolds and Physics, Part II - Revised and Enlarged Edition by Yvonne Choquet-Bruhat, Cecile Dewitt-Morette, Y. Choquet-Bruhat ISBN: 0444504737 Publisher: North-Holland Pub. Date: 01 November, 2000 List Price(USD): $88.50 |
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Title: Counterexamples in Topology by Lynn Arthur Steen, J. Arthur Seebach ISBN: 048668735X Publisher: Dover Pubns Pub. Date: 22 September, 1995 List Price(USD): $11.95 |
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Title: Principles of Algebraic Geometry by Phillip Griffiths, Joseph Harris ISBN: 0471050598 Publisher: Wiley-Interscience Pub. Date: 02 August, 1994 List Price(USD): $94.50 |
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Title: Analysis, Manifolds and Physics : Part I by Yvonne Cgiqyet-Bruhat, Cecile Dewitt-Morette, Yvonne Choquet-Bruhat, C. Dewitt-Morette ISBN: 0444860177 Publisher: North-Holland Pub. Date: 01 January, 1982 List Price(USD): $63.50 |
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