Symplectic Geometry (London Mathematical Society Lecture Note Series, Series Number 192)

This Volume Is Based On Lectures Given At A Workshop And Conference On Symplectic Geometry At The University Of Warwick In August 1990. The Area Of Symplectic Geometry Has Developed Rapidly In The Past Ten Years With Major New Discoveries That Were Motivated By And Have Provided Links With Many Other Subjects Such As Dynamical Systems, Topology, Gauge Theory, Mathematical Physics And Singularity Theory. The Conference Brought Together A Number Of Leading Experts In These Areas Of Mathematics. The Contributions To This Volume Reflect The Richness Of The Subject And Include Expository Papers As Well As Original Research. They Will Be An Essential Source For All Research Mathematicians In Symplectic Geometry. A Variational Interpretation Of Melnikov's Function And Exponentially Small Separatrix Splitting / S. Angenent -- Global Darboux Theorems And A Linearization Problem / E. Ciriza -- Complex Cobordism, Ashtekar's Equations And Diffeomorphisms / S. Donaldson -- Instanton Homology And Symplectic Fixed Points / S. Dostoglou And D. Salamon -- An Energy-capacity Inequality For The Symplectic Holonomy Of Hypersurfaces Flat At Infinity / Y. Eliashberg And H. Hofer -- Caustics D[subscript K] At Points Of Interface Between Two Media / M. Kazarian -- Examples Of Singular Reduction / E. Lerman, R. Montgomery And R. Sjamaar -- Remarks On The Uniqueness Of Symplectic Blowing Up / D. Mcduff -- The 4-dimensional Symplectic Camel And Related Results / D. Mcduff And L. Traynor -- Differential Forms And Connections Adapted To A Contact Structure, After M. Rumin / P. Pansu -- The Maslov Class Rigidity And Non-existence Of Lagrangian Embeddings / L. Polterovich -- Phase Functions And Path Integrals / J. Robbin And D. Salamon -- Symplectic Mappings Which Are Stable At Infinity / E. Zehnder. Edited By Dietmar Salamon. Includes Bibliographical References. A Proof of Lemma 2.3References; An energy-capacity inequality for the symplectic holonomy of hypersurfaces flat at infinity; 1 Introduction; 2 Functional analysis of the action integral; 3 A weak ps'-condition for a class of functionals; 4 Some estimates for max-min-levels; 5 Proof of the main result; References; Caustics Dk at points of interface between two media; 1 Lagrangian manifolds at points of refraction; 2 Initial data for propogating of waves in the second medium; 3 Lagrangian mappings with fixed boundary conditions; References; Examples of singular reduction; Introduction 1 A simple example1.1 Digression: Smooth structures on reduced spaces; 1.2 The Reduced space (T*R2)0 as an orbifold; 1.3 Reduction via Invariants; 2 A summary of the general theory; 2.1 Stratifications; 2.2 Hamiltonian mechanics on a stratified symplectic space; 2.3 Orbit types; 2.4 The closure of a coadjoint orbit as a stratified symplectictic space; 3 Reduction of cotangent bundles; 3.1 The cotangent bundle of a quotient variety; 3.2 Cross-sections; 3.3 Row, row, row your boat; 3.4 Reduction of the cotangent bundle of a symmetric space; 4 Poisson embeddings of reduced spaces 1 Submanifolds of Kähler manifolds of non-positive curvature2 The local structure of a Liouville vector field; References; Complex cobordism, Ashtekar's equations and diffeomorphisms; 1. Introduction; 2. Diffeomorphisms of a 3-manifold and complex cobordisms; 3. Nahm's equations, hyperkahler metrics and other topics.; References; Instanton homology and symplectic fixed points; 1 Introduction; 2 Instanton homology; 3 Floer homology for symplectic fixed points; 4 Flat connections over a Riemann surface; 5 Mapping cylinders; 6 Instantons and holomorphic curves; 7 Perturbations 5 Reduced space at angular momentum zero for n particles in d-spaceReferences; Remarks on the uniqueness of symplectic blowing up; 1 Introduction; 2 Blowing up and down in the symplecticcategory; 3 Uniqueness of blow ups of CP2; (3.6) Embeddings of more than two balls.; References; The 4-dimensional symplectic camel and related results; 1 Introduction; 2 Basic definitions; 3 Properties of j-holomorphic a-discs; 4 Filling the sphere; 5 The camel theorem; 6 Embeddings of balls; References; Differential forms and connections adapted to a contact structure, after M. Rumin Cover; Title; Copyright; Contents; List of Participants; Introduction; References; Acknowledgements; About this volume; Short description; A variational interpretation of Melnikov's function and exponentially small separatrix splitting; 1. Introduction; 2. A variational account of the melnikov function; 3. Rapidly oscillating perturbations; 4. Separatrix splitting; 5. Holomorphic contraction mapping lemma; 6. Proof of lemma 4.1; 7. Proof of lemma 4.5; 8. The forced duffing equation; References; Global Darboux theorems and a linearization problem