Lectures on Symplectic Geometry

Produktbeschreibung

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. 

Inhaltsverzeichnis

Symplectic Manifolds.- Symplectic Forms.- Symplectic Form on the Cotangent Bundle.- Symplectomorphisms.- Lagrangian Submanifolds.- Generating Functions.- Recurrence.- Local Forms.- Preparation for the Local Theory.- Moser Theorems.- Darboux-Moser-Weinstein Theory.- Weinstein Tubular Neighborhood Theorem.- Contact Manifolds.- Contact Forms.- Contact Dynamics.- Compatible Almost Complex Structures.- Almost Complex Structures.- Compatible Triples.- Dolbeault Theory.- Kähler Manifolds.- Complex Manifolds.- Kähler Forms.- Compact Kähler Manifolds.- Hamiltonian Mechanics.- Hamiltonian Vector Fields.- Variational Principles.- Legendre Transform.- Moment Maps.- Actions.- Hamiltonian Actions.- Symplectic Reduction.- The Marsden-Weinstein-Meyer Theorem.- Reduction.- Moment Maps Revisited.- Moment Map in Gauge Theory.- Existence and Uniqueness of Moment Maps.- Convexity.- Symplectic Toric Manifolds.- Classification of Symplectic Toric Manifolds.- Delzant Construction.- Duistermaat-Heckman Theorems.