Course information

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

• A. Cannas da Silva, Lectures on symplectic geometry, Lecture Notes in Mathematics, 1764, Springer-Verlag, Berlin, 2001 and 2008 (corrected printing).
• D. McDuff and D.A. Salamon, Introduction to symplectic topology, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998.

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

Lecturers

Teaching assistants