Lectures: TTh 9:50 AM – 11:25 AM, McHenry Clrm 4130

Instructor: Viktor Ginzburg; office: McHenry 4124; email: ginzburg(at)ucsc.edu

Office Hours: TBA or by appointment

Text: There will be no “official” textbook in this course. Suggested reading:

• Introduction to Symplectic Topology by Dusa McDuff and Dietmar Salamon,
• Lectures on Symplectic Geometry by Ana Canas da Silva,
• Morse Theory and Floer Homology by Michelle Audin and Mihai Damian

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 make sure 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.

Lecture notes (pdf files) The entire set (60MB)

• Week 1 (Lectures 1-3): Definitions, Examples; the Homotopy Method: Darboux’ and Moser’s Theorems
• Week 2 (Lectures 4-5): Hamilton Equations and Vector Fields and Flows; Examples; Energy and Symplectic Form Conservation
• Week 3 (Lectures 6-7): Time change, the Group of Hamiltonian Diffeomorphisms; Submanifolds of Symplectic Manifolds – Some Linear Algebra; the Lagrangian Grassmannian
• Week 4 (Lectures 8-9): More Linear Algebra; the Lagrangian Grassmannian (continued); Examples of Lagrangian Submanifolds; Weinstein’s Tubular Neighborhood Theorem
• Week 5 (Lectures 10-11): Lagrangian Submanifolds of R ²ⁿ; Contact Manifolds – A Brief Look
• Week 6 (Lectures 12-13): Elements of Morse Theory Continued: Motivation; the Morse Lemma; the Morse Complex
• Weeks 7-8 (Lectures 14-16): Morse Theory Continued: the Morse Differential and Morse homology; Some Applications
• Weeks 9-10 (Lectures 17-19): Arnold’s Conjectures; the Least Action Principle; a Quick Look at Hamiltonian Floer Homology