Introduction to symplectic topology / Dusa McDuff, and Dietmar SalamonType de document : MonographieCollection : Oxford mathematical monographsLangue : anglais.Pays : Grande Bretagne.Éditeur : Oxford : Clarendon Press, 1997Description : 1 vol. (VIII-425 p.) : ill. ; 24 cmISBN : 0198511779.ISSN : 0964-9174.Bibliographie : Bibliogr. p. -410. Index.Sujet MSC : 53-01, Differential geometry, Instructional exposition (textbooks, tutorial papers, etc.)
58E05, Global analysis, analysis on manifolds -- Variational problems in infinite-dimensional spaces, Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel'man) theory, etc.)
53C15, Differential geometry -- Global differential geometry, General geometric structures on manifolds (almost complex, almost product structures, etc.)
53D35, Differential geometry -- Symplectic geometry, contact geometry, Global theory of symplectic and contact manifolds
53D40, Differential geometry -- Symplectic geometry, contact geometry, Floer homology and cohomology, symplectic aspects
57R17, Manifolds and cell complexes -- Differential topology, Symplectic and contact topologyEn-ligne : Zentralblatt | MathSciNet
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Bibliogr. p. -410. Index
Part I: Basic background material. Chapter 1: Hamiltonian systems in Euclidean space, development of modern symplectic topology. Chapter 2: linear symplectic geometry, existence of the first Chern class. Chapter 3: symplectic forms on arbitrary manifolds, Darboux’s and Moser’s theorems, contact geometry (the odd-dimensional analogue of symplectic geometry). Chapter 4: almost complex structures, Kähler and Donaldson’s manifolds.
Part II: Examples of symplectic manifolds. Chapter 5: symplectic reduction, Atiyah-Guillemin-Sternberg convexity theorem. Chapter 6: different ways of constructing symplectic manifolds, by fibrations, symplectic blowing up and down, by fibre connected sum, Gompf’s result about the fundamental group of a compact symplectic 4-manifold. Chapter 7: existence and uniqueness of the symplectic structure, Gromov’s proof that every open, almost complex manifold has a symplectic structure.
Part III: Symplectomorphisms. Chapter 8: Poincaré-Birkhoff theorem (an area-preserving twist map of the annulus has two distinct fixed points), special case (strongly monotone twist maps). Chapter 9: generating functions (modern and classical guise), discrete-time variational problems. Chapter 10: structure of the group of symplectomorphisms, properties of the subgroup of Hamiltonian symplectomorphisms.
Part IV (the heart of the book): Finite-dimensional variational methods, full proofs of the simplest versions of important new results in the subject. Chapter 11: Arnold’s conjectures for the torus, Lysternik-Schnirelmann theory, Conley index. Chapter 12: non-squeezing theorem in Euclidean space, energy-capacity inequality for symplectomorphisms of Euclidean space.