Tentative Syllabus: The course will focus on applications
of modern symplectic topological techniques to Hamiltonian
dynamics. We will begin with an (ideally, brief) discussion of
basic concepts of symplectic geometry: symplectic manifolds,
Hamiltonian diffeomorphisms and flows, Lagrangian submanifolds, the
least action principle, etc. Here we will also introduce several
classes of dynamical systems of interest, such as geodesic flows
and twisted geodesic (or magnetic) flows, and formulate the main
problems in dynamics (e.g., Arnold's and Weinstein's conjectures,
i.e., the existence of fixed points and periodic orbits) addressed
by symplectic topological techniques. Then we turn to a detailed
discussion of symplectic topological methods, starting with a
review of Morse theory and including an introduction to Hamiltonian
Floer homology, spectral invariants, and other modern tools from
symplectic topology and dynamics.
It should be said that this is not a comprehensive course in
symplectic geometry and many important concepts (mainly those concerning
symmetries) will be entirely omitted or just briefly
mentioned.