Lectures: Eleny Ionel, Tue and Th 10:30am-11:50am in room 384-I

Office Hours: Tuesday 1-2pm and by appointment.

Course Description: This course is an introduction to symplectic geometry and topology. We will cover standard material and end with very brief introduction to more advanced topics (holomorphic curves techniques). The main topics we plan to cover in this course include:

• Symplectic manifolds, symplectomorphisms; Lagrangian submanifolds.
• Contact manifolds, Legendrian submanifolds; relations between contact and symplectic manifolds
• Moser theorem, Darboux theorem, Lagrangian tubular neighborhoods theorem.
• Hamiltonian diffeomorphisms, flux, Hamiltonian flows, isotopy.
• (Almost) Complex Structures, symplectic and complex vector bundles, Chern classes, Maslov index
• Hamiltonian group actions, moment maps and symplectic reductions
• Constructions of symplectic manifolds: symplectic fibrations, Lefschetz fibrations, symplectic blow-up, symplectic sums
• Pseudoholomorphic curves and Floer homology (a brief introduction)

This is meant to be mostly an introductory course to the subject, and there are no formal prerequisites for this class, although a good background in topology, geometry and analysis/PDEs is desirable.

Recommended reading: We will not follow any references too closely but here are some suggestions:

• Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, (second edition), Oxford, 1998.
• A. Cannas da Silva, Lectures on Symplectic Geometry, Springer, 2001
• Marsden and Ratiu, Introduction to Mechanics and Symmetry, Springer, 1999.

Further Readings/Classical papers:

• Gromov, "Pseudo holomorphic curves in symplectic manifolds".
• D. Salamon, "Lecture notes on Floer Homology".
  
• Atiyah, Bott, "The moment map and equivariant cohomology"; see also the survey "Morse Theory Indomitable" by Bott.
• Duistermaat-Heckman, "On the Variation in the Cohomology of the Symplectic Form of the Reduced Phase Space".
  
• Audin, Lalonde, Polterovich, "Symplectic rigidity: Lagrangian submanifolds"
  

Tentative Schedule:

• Week 1: Overview; symplectic manifolds, symplectic and Lagrangian submanifolds, symplectomorphisms: examples, obstructions and properties.
• Week 2: Moser's and Darboux theorem, tubular neighborhoods of symplectic/Lagrangian submanifolds.
• Week 3: Contact manifolds and Legendrian submanifolds; relations between contact and symplectic manifolds.
• Week 4: Symplectic geometry and Mechanics; variational principles, Lagrangian and Hamiltonian formalism, Legendre transform.
• Week 5: Hamiltonian diffeomorphisms, flux, Hamiltonian flows, isotopy.
• Week 6: Symplectic and Hamiltonian group actions, moment maps, symplectic reduction.
• Week 7: (Almost) Complex Structures, symplectic vector bundles, Chern classes, Maslov index.
• Week 8: Constructions of symplectic manifolds: symplectic fibrations, Lefschetz fibrations, symplectic blow-up, symplectic (fiber) sum.
• Week 9-10: Pseudo-holomorphic curves, Gromov Nonsqueezing Thm, discussion of the Weinstein conjecture, Arnold conjecture, Lagrangian intersections and Floer theory.