General information:

• 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:
  • symplectic manifolds: brief introduction
  • local behavior of J-holomorphic curves
  • moduli space of J-holomorphic curves
  • Gromov-Witten invariants: construction, properties and applications
  • Brief overview of Hamiltonian Floer theory and the Arnold conjecture (fixed points of hamiltonian vector fields)
  • Brief overview of Lagrangian Floer theory (Lagrangian intersections)

  For more details see tentative weekly Course Outline.

• 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.