Math 248, Symplectic Geometry, Winter 2010

• Lectures: TTh 2:00-3:45pm, JBE 360

• Instructor: Viktor Ginzburg; office: 353C JBE,
  email: ginzburg(at)math.ucsc.edu, phone: 459-2218

• Office Hours: Tu 4:00-5:30pm, Th 11:00-12:00pm 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,
  • Symplectic Invariants and Hamiltonian Dynamics by Helmut Hofer and Eduard Zehnder

• 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 Floer homology, action selectors, and other tools modern approaches to these problems rely on.

  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. However, the course is coordinated with Math 249A, Mechanics I (Debra Lewis, MW 2:00-3:45pm) and some of these notions, such as, e.g., symplectic reduction, are covered in Mechanics.