1 Basic notions of differential topology
De Rham cohomology
Frobenius theorem and Lie derivative
Poisson’s bracket
Equivalence of de Rham cohomology to singular cohomology
2 Basic notions of symplectic topology
Symplectic geometry of vector space
Topology of Lagrangian Grassmannian
Examples of symplectic manifolds
Darboux’s theorem
Symplectic bundles
Constructions of symplectic manifolds
Symplectomorphisms and generating functions
3 Structural theorem of symplectic topology
Liouville’s theorem
Almost complex structure
Lagrangian submanifolds
Symplectic capacities and Gromov’s non-squeezing theorem
Literature
Students talks:
Legendre transformation and Hamilton equations
Holonomy and Noether Theorem
Contact structures
Hydrodynamic lemma, Poincare-Cartan’s invariant
Periodic orbits on energy surfaces
Arnold’s theorem on fixed points of a symplectomorphisms
Moment map
Toric symplectic manifolds, Delzant theorem
Embeddings of lagrangians to symplectic manifolds
Deformations of a symplectic structure, Moser’s theorem