Tentative Syllabus:
The course will cover fundamental from symplectic geometry and Morse theory with an eye
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. 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)
studied by symplectic techniques. Then we turn to a
review of Morse theory with applications to Hamiltonian circle-actions
and homology calculations. Time permitting, we will touch upon
symplectic topological methods (e.g., Hamiltonian Floer homology)
and/or conclude the course with student presentations.
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.