Symplectic Geometry

Symplectic linear algebra; symplectomorphisms and symplectic manifolds, symplectic structure of the cotangent bundle. Submanifolds (Lagrangian, isotropic, co-isotropic) and generating functions, Moser’s method, Darboux-Weinstein theorems (Lagrangian neighborhood theorem and applications). Compatible almost complex structures and Kähler manifolds. Hamiltonian vector fields and Hamiltonian systems, Poisson brackets, variational principles; Poisson manifolds. Integrable systems: Liouville Arnold theorem and the existence of action-angle variables. Geometry of Hamiltonian actions: actions of the group of symplectomorphisms, momentum maps (obstructions to their existence, uniqueness). Symplectic reduction and applications. Other topics: Duistermaat-Heckman theorem, convexity theorem of Atiyah-Guillemin-Sternberg, Delzant’s theorem, introduction to symplectic topology and global invariants.

CANNAS DA SILVA, A. – Lectures on Symplectic Geometry , Lectures Notes in Mathematics 1764, Springer-Verlag, 2001.
MCDUFF, D., SALAMON, D. – Introduction  to  Symplectic  Topology ,  Oxford Math. Monographs, Oxford Univ. Press,1995.
GUILLEMIN,  V.,  STERNBERG,  S. –  Symplectic  Techniques  in  Physics, Cambridge University Press, 1990.