**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. -
**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.

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.