Math 257B: Symplectic Geometry and Topology
- Lectures: Tue and Th 1:30- 3:00pm, room 381T
- Professor: Eleny Ionel, office 383L,
ionel "at" math.stanford.edu
Office Hours: Tue after class (3-4pm) and by appointment
Course Description: This is a graduate level course meant to be mostly an introduction/outline of the use of the moduli space of J-holomorphic curves in symplectic topology.
After a brief introduction to symplectic manifolds, we focus on the moduli space of closed J-holomorphic curves and the construction of the corresponding Gromov-Witten invariants. We then move to the "open" case, and the construction of Floer theories,
following the outline of Morse theory. In this case one also needs to
impose a boundary condition for the holomorphic curves, and they come in two
basic flavors: asymptotic boundary conditions or Lagrangian boundary
conditions. We will discuss both types and the corresponding construction
of the two basic Floer theories: the Hamiltonian Floer theory and the
Lagrangian Floer theory.
Prerequisites:While there are no formal prerequisites for this class, a good background in topology, geometry and analysis/PDEs is desirable.
- Brief Course Outline: The topics we tentatively plan to cover are:
- Homework: There will be 4 homework assignments, about one every 2 weeks. You are encouraged to discuss and work together on your homework, but everyone must write up their own solutions.
- Recommended References: We will not follow any references too closely, but there are a few standard ones:
- Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology (for background on symplectic topology)
- Ana Cannas da Silva, Lectures on Symplectic Geometry (for background on symplectic topology)
- Dusa McDuff and Dietmar Salamon, J-holomorphic curves and quantum cohomology (this is the "thin version", available here)
- Dusa McDuff and Dietmar Salamon, J-holomorphic curves and Symplectic Topology (this is the greatly expanded version)
- Chris Wendl, Lectures on Holomorphic Curves in Symplectic and Contact Geometry, available here
- Dietmar Salamon, Lectures on Floer homology, available here
- A. Floer, Morse theory for Lagrangian intersections.
Michele Audin and Mihai Damian, Morse theory and Floer homology.