Lecture 1 - Linear algebra, definition of a symplectic manifold.
Lecture 2 - Basic properties, submanifolds, symplectic vector bundles, examples of symplectic manifolds: surfaces, Kähler manifolds (in particular the Fubini-Study form on CPn) and cotangent bundles.
Lecture 3 - Certain Lagrangian submanifolds of cotangent bundles, flows of vector fields and the Lie derivative, Moser's theorem.
Lecture 4 - Moser's stability theorem, symplectic tubular neighborhoods, Darboux's theorem.
Lecture 5 - Lagrangian tubular neighborhood theorem, symplectic tubular neighborhood theorem, symplectic and Hamiltonian vector fields, Hamilton's equations.
Lecture 6 - Conservation of Hamiltonian, Hamiltonian symplectomorphisms, flux of a path of symplectomorphisms, Arnold conjectures.