Tentative Syllabus:
The course will cover fundamentals from symplectic geometry and touch upon
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 very brief
review of Morse theory. In contrast with previous iterations of this course, this time I plan to focus more on Lagrangian submanifolds -- one of the most fundamental objects in symplectic geometry. Time permitting, we will touch upon symplectic topological methods (e.g., Lagrangian and 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.
COVID-19 Information:
Please take care to comply with all university guidelines about masking in indoor settings, performing daily symptom and badge checks, testing as required by the campus vaccine policy, self-isolating in the event of exposure, and respecting others’ comfort with distancing. Please do not come to class if your badge is not green. If you are ill or suspect you may have been exposed to someone who is ill, or if you have symptoms that are in any way similar to those of COVID-19, please err on the side of caution and stay home until you are well or have tested negative after an exposure.