Duistermaat J.J. Symplectic geometry

Utrecht University, 2004. — 42 p.Symplectic Linear Algebra
Symplectic Forms
Orthogonal Complements in the Dual Space
Orthogonal Complements for a Bilinear Form
Isotropic Subspaces
Standard Form of the Sympctic Form
The Lagrangian Grassmannian
The Symplectic Linear Group
Exterior Algebra
Hermitian Forms
Historical Remarks
Exercises
Symplectic Manifolds
Definition
The Cotangent Bundle
Reduction
Complex Projective Varieties
Almost Complex Structure
Cohomology Classes
Exercises
Hamiltonian Systems
Flows of Vector Fields
Lie Derivatives
Hamiltonian Vector Fields
The Legendre Transform
Poisson Brackets
Darboux’s Lemma
Hamiltonian Group Actions
Poisson Structures
Exercises
Hamilton-Jacobi Theory
Lagrange Manifolds
Lie’s View on First Order PDE
An Initial Value Problem
Ray Bundles
High Frequency Waves and Fourier Integral Operators
Some History
Exercises