Introduction to Symplectic Topology

by ;
Edition: 2nd
Format: Paperback
Pub. Date: 1999-07-29
Publisher(s): Oxford University Press
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Summary

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 ofsymplectic 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 editionhas 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

Author Biography


Dusa McDuff is one of the world's leading researchers in this field, and has been invited to speak at the International Congress of Mathematicians 1998.

Table of Contents

Introduction 1(10)
I FOUNDATIONS
From classical to modern
11(26)
Hamiltonian mechanics
12(16)
The symplectic topology of Euclidean space
28(9)
Linear symplectic geometry
37(44)
Symplectic vector spaces
38(5)
The symplectic linear group
43(7)
Lagrangian subspaces
50(5)
The affine nonsqueezing theorem
55(6)
Complex structures
61(7)
Symplectic vector bundles
68(13)
Symplectic manifolds
81(36)
Basic concepts
81(12)
Isotopies and Darboux's theorem
93(6)
Submanifolds of symplectic manifolds
99(6)
Contact structures
105(12)
Almost complex structures
117(34)
Almost complex structures
117(6)
Integrability
123(7)
Kahler manifolds
130(11)
J-holomorphic curves
141(10)
II SYMPLECTIC MANIFOLDS
Symplectic group actions
151(46)
Circle actions
151(10)
Moment maps
161(4)
Examples
165(8)
Symplectic quotients
173(6)
Convexity
179(12)
Localization
191(6)
Symplectic Fibrations
197(36)
Symplectic fibrations
197(5)
Symplectic 2-sphere bundles
202(5)
Symplectic connections
207(8)
Hamiltonian holonomy and the coupling form
215(11)
Hamiltonian fibrations
226(7)
Constructing Symplectic Manifolds
233(32)
Blowing up and down
233(18)
Connected sums
251(6)
The telescope construction
257(8)
III SYMPLECTOMORPHISMS
Area-preserving diffeomorphisms
265(15)
Periodic orbits
265(4)
The Poincare-Birkhoff theorem
269(6)
The billiard problem
275(5)
Generating functions
280(31)
Generating functions of type S
280(8)
Discrete Hamiltonian mechanics
288(5)
Hamiltonian symplectomorphisms
293(10)
Lagrangian submanifolds
303(8)
The group of symplectomorphisms
311(28)
Basic properties
311(4)
The flux homomorphism
315(13)
The Calabi homomorphism
328(5)
The topology of symplectomorphism groups
333(6)
IV SYMPLECTIC INVARIANTS
The Arnold conjecture
339(32)
Symplectic fixed points
340(6)
Morse theory and the Conley index
346(11)
Lagrangian intersections
357(9)
Floer homology
366(5)
Symplectic capacities
371(46)
Nonsqueezing and capacities
371(6)
Rigidity
377(3)
The Hofer metric
380(14)
The Hofer-Zehnder capacity
394(7)
A variational argument
401(16)
New directions
417(41)
Various examples
419(15)
Symplectic structures on closed manifolds
434(5)
Symplectic 4-manifolds
439(11)
Symplectic submanifolds
450(8)
References 458(15)
Index 473

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