Book Description
Symplectic structures underlie the equations of classical mechanics and their properties are reflected in the behaviour of a wide range of physical systems. Over the last few years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results. At its publication in 1995, Introduction to Symplectic Topology was the first comprehensive introduction to the subject, and has since become an established text in this fast-developing area of mathematics. This second edition has been significantly revised and expanded, with new references and examples added and theorems included or revised. A section has been included on new developments in the subject, and there is a more extensive discussion of Taubes and Donaldson's recent contributions to the subject. From reviews of the first edition: '...an authoritative and comprehensive reference...McDuff and Salamon have done an enormous service to the symplectic community: their book greatly enhances the accessibility of the subject to students and researchers alike.' Book Reviews, AMS
Customer Reviews:
Perfect.......2007-05-09
That book is in perfect condition. I got it in just 1 week (free shipping). I just hope the price could be a little cheaper.
A must for researchers new to the field.......2000-09-14
An authoritative and comprehensive reference...McDuff and Salamon have done an enormous service to the symplectic community: their book greatly enhances the accessibility of the subject to students and researchers alike.
The discussion begins with classic topology and cover a variety of final year undergraduate topics such as complex manifolds and inverse differential techniques before moving into the vastly complex world of Symplectic Topology.
A must for researchers new to the field
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Symplectic Geometry and Quantum Mechanics (Operator Theory: Advances and Applications / Advances in Partial Differential Equations)
Maurice de Gosson
Manufacturer: Birkhäuser Basel
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ASIN: 3764375744 |
Book Description
This book is devoted to a rather complete discussion of techniques and topics intervening in the mathematical treatment of quantum and semi-classical mechanics. It starts with a very readable introduction to symplectic geometry. Many topics are also of genuine interest for pure mathematicians working in geometry and topology.
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Torus Actions on Symplectic Manifolds (Progress in Mathematics)
Michèle Audin
Manufacturer: Birkhäuser Basel
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ASIN: 3764326026 |
Book Description
This is an extended second edition of "The Topology of Torus Actions on Symplectic Manifolds" published in this series in 1991. The material and references have been updated. Symplectic manifolds and torus actions are investigated, with numerous examples of torus actions, for instance on some moduli spaces. Although the book is still centered on convexity theorems, it contains much more results, proofs and examples.
Chapter I deals with Lie group actions on manifolds. In Chapters II and III, symplectic geometry and Hamiltonian group actions are introduced, especially torus actions and action-angle variables. The core of the book is Chapter IV which is devoted to applications of Morse theory to Hamiltonian group actions, including convexity theorems. As a family of examples of symplectic manifolds, moduli spaces of flat connections are discussed in Chapter V. Then, Chapter VI centers on the Duistermaat-Heckman theorem. In Chapter VII, a topological construction of complex toric varieties is presented, and the last chapter illustrates the introduced methods for Hamiltonian circle actions on 4-manifolds.
Book Description
The theory of $J$-holomorphic curves has been of great importance since its introduction by Gromov in 1985. Its mathematical applications include many key results in symplectic topology. It was also one of the main inspirations for the creation of Floer homology. In mathematical physics, it provides a natural context in which to define Gromov-Witten invariants and quantum cohomology--two important ingredients of the mirror symmetry conjecture.
This book establishes the fundamental theorems of the subject in full and rigorous detail. In particular, the book contains complete proofs of Gromov's compactness theorem for spheres, of the gluing theorem for spheres, and of the associativity of quantum multiplication in the semipositive case. The book can also serve as an introduction to current work in symplectic topology: There are two long chapters on applications, one concentrating on classical results in symplectic topology and the other concerned with quantum cohomology. The last chapter sketches some recent developments in Floer theory. The five appendices of the book provide necessary background related to the classical theory of linear elliptic operators, Fredholm theory, Sobolev spaces, as well as a discussion of the moduli space of genus zero stable curves and a proof of the positivity of intersections of $J$-holomorphic curves in four dimensional manifolds.
Book Description
The papers, some of which are in English, the rest in French, in this volume are based on lectures given during the meeting of the Seminare Sud Rhodanien de Geometrie (SSRG) organized at the Mathematical Sciences Research Institute in 1989. The SSRG was established in 1982 by geometers and mathematical physicists with the aim of developing and coordinating research in symplectic geometry and its applications to analysis and mathematical physics. Among the subjects discussed at the meeting, a special role was given to the theory of symplectic groupoids, the subject of fruitful collaboration involving geometers from Berkeley, Lyon, and Montpellier.
Book Description
The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text covers symplectomorphisms, local forms, contact manifold, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moments maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster.
Book Description
Symplectic geometry is very useful for formulating clearly and concisely problems in classical physics and also for understanding the link between classical problems and their quantum counterparts. It is thus a subject of interest to both mathematicians and physicists, though they have approached the subject from different viewpoints. This is the first book that attempts to reconcile these approaches. The authors use the uncluttered, coordinate-free approach to symplectic geometry and classical mechanics that has been developed by mathematicians over the course of the past thirty years, but at the same time apply the apparatus to a great number of concrete problems. Some of the themes emphasized in the book include the pivotal role of completely integrable systems, the importance of symmetries, analogies between classical dynamics and optics, the importance of symplectic tools in classical variational theory, symplectic features of classical field theories, and the principle of general covariance.
Book Description
This volume is an attempt to provide an overview of some of the recent advances in representation theory from a geometric standpoint. A geometrically-oriented treatment is very timely and has long been desired, especially since the discovery of D-modules in the early '80s and the quiver approach to quantum groups in the early '90s.
The first half of the book fills the gap between the standard knowledge of a beginner in Lie theory and the much wider background needed by the working mathematician. Thus, Chapters 1-3 and 5-6 provide some basics in symplectic geometry, group actions on Kahler manifolds and Borel--Moore homology, geometry of semisimple groups, equivariant algebraic K-theory "from scratch," topology and algebraic geometry of flag varieties and conjugacy classes, respectively.
The material covered by Chapters 5 and 6 (as well as most of Chapter 3) has never been presented in book form. Chapters 3-4 and 7-8 form the heart of the book, presenting a uniform approach to representation theory of three quite different objects: (1) Weyl groups; (2) Lie algebra sln; (3) Iwahori--Hecke algebra. The results of Chapters 4 and 8 are new, with complete proofs, not to be found elsewhere in the literature. The techniques developed are quite general and can be successfully applied to such other areas of mathematics, as Quantum groups, affine Lie algebras, and quantum field theory. The exposition is practically self-contained and each chapter potentially serving as a basis for a graduate course.
Customer Reviews:
a nice blend of mathematics.......2002-01-06
it's not at all easy-going, but admittedly, it is probably the best way to learn some of the most stimulating and illuminating interactions between representation theory, symplectic geometry, algebraic geometry, and algebra;
a nice compliment to this book is 'symplectic fibrations and mutliplicity diagrams' by Guillemin et al.
other perspectives in ring theory from a geometric point of view (which could serve as yet another compliment) is Broho's 'nilpotent orbits, primitive ideals and characteristic classes'
no doubt, the advanced graduate student and professional mathematician would do well in, at least, taking a peek at the contents and the extensive introduction to whet his appetite and peak his curiousity!
Book Description
The discoveries of the last decade have shed new light on the old field of Hamiltonian systems and have led to the creation of a new field: symplectic topology. Surprising rigidity phenomena demonstrate that the nature of symplectic mappings is very different from that of volume preserving mappings. This raises new questions, many of them still unanswered. On the other hand, analysis of an old variational principle in classical mechanics has established global periodic phenomena in Hamiltonian systems. As it turns out, these seemingly different phenomena are mysteriously related. One of the links is a class of symplectic invariants, called symplectic capacities. These invariants are the main theme of this book, which includes such topics as basic symplectic geometry, symplectic capacities and rigidity, periodic orbits for Hamiltonian systems and the least action principle, a bi-invariant metric on the symplectic diffeomorphism group and its geometry, symplectic fixed point theory and first order elliptic systems, the Arnold conjectures and a survey on Floer- and symplectic homology. The exposition is self-contained and addressed to researchers and students from the graduate level onwards.
Customer Reviews:
Very well written.......2001-03-21
This monograph gives a good overview of a very important subject in both mathematics and physics. The phase space formulation of classical mechanics via the Hamiltonian formalism is represented throughout this book in the language of symplectic manifolds. The geometry of symplectic manifolds has many surprises considering the very simple definition of a symplectic form. One would not have suspected that they would imply the Gromov squeezing theorem and other interesting results. The authors do an excellent job of explaining this theorem and other intricacies of symplectic geometry, in particular the idea of symplectic capacities, Floer-Morse theory, and the Arnold conjecture. The Gromov squeezing theorem has an analogue in quantum physics. Jokingly called "quantum claustrophobia" by physicists, this represents the idea that particles in the quantum realm tend to avoid small regions of phase space. The book should be of great assistance to the mathematically-inclined physicist interested in the properties of nonlinear dynamical systems in phase space. Even though the KAM theory is only briefly discussed in the book, the presentation should prepare the physicist reader for additional reading on the subject. The most interesting and important part of the book is the discussion on how symplectic capacities are related to volumes and Lebesgue measures in ordinary Euclidean space. The capacity and volume agree as invariants only for two-dimensional symplectic manifolds. In addition, in four dimensions and higher, open sets can be very different in terms of shape, size, measure, and topology, and still have the same capacity. The author's presentation of these facts is very well-written and thought provoking. Probably the most difficult part of the book is the justification of the squeezing theorem using variational principles. The technical details of the proof are very intricate and take lots of time to digest. Everything about the symplectic category is interesting and this book is a fine compendium of modern results in the field.
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Cohomology of Quotients in Symplectic and Algebraic Geometry. (MN-31) (Mathematical Notes)
Frances Clare Kirwan
Manufacturer: Princeton University Press
ProductGroup: Book
Binding: Paperback
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ASIN: 0691083703 |
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