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E-grāmata: Introduction to Symplectic Topology

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(Professor of Mathematics, ETH Zurich), (Professor of Mathematics, Columbia University)
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Introduction to Symplectic TopologyPalielināt
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Over the last number of 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. The first edition of Introduction to Symplectic Topology was published in 1995. The book was the first comprehensive introduction to the subject and became a key text in the area. A significantly revised second edition was published in 1998 introducing new sections and updates on the fast-developing area. This new third edition includes updates and new material to bring the book right up-to-date.

Recenzijas

A most welcome resource for graduate students entering the field... The book has been updated in many places, so that it also continues to serve as a reference work for specialists. The attentive reader will discover many other helpful details that have been added to this third edition. * Hansjorg Geiges *

Introduction 1(10)
I Foundations
1 From classical to modern
11(26)
1.1 Hamiltonian mechanics
11(17)
1.2 The symplectic topology of Euclidean space
28(9)
2 Linear symplectic geometry
37(57)
2.1 Symplectic vector spaces
38(5)
2.2 The symplectic linear group
43(7)
2.3 Lagrangian subspaces
50(5)
2.4 The affine nonsqueezing theorem
55(6)
2.5 Linear complex structures
61(18)
2.6 Symplectic vector bundles
79(6)
2.7 First Chern class
85(9)
3 Symplectic manifolds
94(58)
3.1 Basic concepts
94(14)
3.2 Moser isotopy and Darboux's theorem
108(4)
3.3 Isotopy extension theorems
112(4)
3.4 Submanifolds of symplectic manifolds
116(9)
3.5 Contact structures
125(27)
4 Almost complex structures
152(39)
4.1 Almost complex structures
153(8)
4.2 Integrability
161(6)
4.3 Kahler manifolds
167(5)
4.4 Kahler surfaces
172(8)
4.5 J-holomorphic curves
180(11)
II Symplectic Manifolds
5 Symplectic group actions
191(61)
5.1 Circle actions
192(10)
5.2 Moment maps
202(5)
5.3 Examples
207(11)
5.4 Symplectic quotients
218(11)
5.5 Convexity
229(11)
5.6 Localization
240(6)
5.7 Remarks on GIT
246(6)
6 Symplectic Fibrations
252(37)
6.1 Symplectic fibrations
252(5)
6.2 Symplectic 2-sphere bundles
257(5)
6.3 Symplectic connections
262(9)
6.4 Hamiltonian holonomy and the coupling form
271(11)
6.5 Hamiltonian fibrations
282(7)
7 Constructing Symplectic Manifolds
289(52)
7.1 Blowing up and down
289(27)
7.2 Connected sums
316(6)
7.3 The telescope construction
322(6)
7.4 Donaldson submanifolds
328(13)
III Symplectomorphisms
8 Area-preserving diffeomorphisms
341(15)
8.1 Periodic orbits
341(3)
8.2 The Poincare--Birkhoff theorem
344(6)
8.3 The billiard problem
350(6)
9 Generating functions
356(29)
9.1 Generating functions and symplectic action
356(7)
9.2 Discrete Hamiltonian mechanics
363(6)
9.3 Hamiltonian symplectomorphisms
369(9)
9.4 Lagrangian submanifolds
378(7)
10 The group of symplectomorphisms
385(32)
10.1 Basic properties
385(5)
10.2 The flux homomorphism
390(16)
10.3 The Calabi homomorphism
406(5)
10.4 The topology of symplectomorphism groups
411(6)
IV Symplectic Invariants
11 The Arnold conjecture
417(40)
11.1 Symplectic fixed points
418(7)
11.2 Morse theory and the Conley index
425(10)
11.3 Lagrangian intersections
435(11)
11.4 Floer homology
446(11)
12 Symplectic capacities
457(46)
12.1 Nonsqueezing and capacities
457(5)
12.2 Rigidity
462(3)
12.3 The Hofer metric
465(16)
12.4 The Hofer--Zehnder capacity
481(8)
12.5 A variational argument
489(14)
13 Questions of existence and uniqueness
503(46)
13.1 Existence and uniqueness of symplectic structures
503(4)
13.2 Examples
507(9)
13.3 Taubes--Seiberg--Witten theory
516(17)
13.4 Symplectic four-manifolds
533(16)
14 Open problems
549(25)
14.1 Symplectic structures
549(4)
14.2 Symplectomorphisms
553(5)
14.3 Lagrangian submanifolds and cotangent bundles
558(4)
14.4 Fano manifolds
562(1)
14.5 Donaldson hypersurfaces
563(2)
14.6 Contact geometry
565(2)
14.7 Continuous symplectic topology
567(2)
14.8 Symplectic embeddings
569(2)
14.9 Symplectic topology of Euclidean space
571(3)
A Smooth maps
574(9)
A.1 Smooth functions on manifolds with corners
574(4)
A.2 Extension
578(2)
A.3 Construction of a smooth function
580(3)
References 583(28)
Index 611
Dusa McDuff was born in London, UK, in 1945. She studied Mathematics at the University of Edinburgh (B. Sc. (Hon) in 1967), at Cambridge University (Ph. D. in 1971, supervised by G.A. Reid), and also at Moscow University (1968-69 with I.M.Gelfand). After a postdoc in Cambridge, she lectured at the Universities of York and Warwick, before taking up a position in SUNY, Stony Brook, USA in 1978. She moved to Barnard College in 2008.

Dietmar Salamon was born in Bremen, West Germany, in 1953. Studied Mathematics at the Universities of Hannover and Bremen from 1971 to 1978. PhD in Mathematics at the University of Bremen in 1982 under the supervision of Didi Hinrichsen. Postdoc positions at UW Madison and ETH Z urich from 1983 to 1986. Lecturer, Reader, and Professor at the University of Warwick from 1986 to 1998. Professor at ETH Zurich since 1998.
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