MAT 645, Spring 2002

Monday, Wednesday 10:30-12:00 a.m. (tentatively), Math Tower  P 131

Instructor: Klaus Mohnke
                        klaus@math.sunysb.edu
                        Office: 4-109 Math Tower
                        phone: (631)632-8271

Office Hours: Wednesday 2-4 p.m. in  Math Tower   4-109
                              You are welcome at any other time!
 
 

In the  sixties V.I . Arnold raised a couple of inspiring problems in symplectic topology. In the end these questions and their (partial) solutions led to a whole new area of active research. The breakthrough came with Gromov's seminal work on
(pseudo)holomorphic curves in symplectic manifolds. He had pushed the  usability of "soft" techniques to attack these problems to the limit and was the first to recognize the need for new "hard" ones.  Among the questions he could
answer with his new inventions were

  • Does the three-sphere, S3, admit a Lagrangian embedding into C3?
  • Do there exist other symplectic structures on C2 which are standard outside a compact set?
  • Can a ball, B2n(R), in R2n be symplectically compressed into a cylinder, B2(r)xR2n-2, when r<R?

    • The course aims to show how to use Gromov's holomorphic curves to attack problems in symplectic topology and Hamiltonian dynamics. We will merely discuss properties of these, many of which have been proven in Dusa McDuff's
    class in Fall 2001.

    Topics:

  • Applications to Sympletcic Topology
    • -symplectic structures on R4
    -Lagrangian embeddings and holomorphic disks
    -uniqueness of the symplectic structure on CP2
    -Gromov width
  • Applications to Hamiltonian Dynamics
    • -Hofer's norm
    -Arnold's fixed point conjecture
    -Floer homology
    -Reeb chords
  • Symplectic Field Theory
    • -holomorphic curves in symplectic manifolds with ends
    -Weinstein conjecture
    -contact homology for Legendrian knots and links
    -Lagrangian embeddings
    Prerequisites: Some knowledge about symplectic geometry and holomorphic curves, as presented in Dusa McDuff's course in Fall 2001 would be helpful. In any case, I will always list the facts needed, often without proving them (again). So, if you are new in the subject and did not attend Dusa's calls - this should not be a problem.

    Literature:
    (1) D.McDuff, Dietmar Salamon, Introduction to symplectic topology, Clarendon Press, Oxford
    (2) Holomorphic curves in symplectic geometry, Michéle Audin, Jacques Lafontaine (Editors), Birkhaeuser
    (3) H.Hofer and E.Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhaeuser
    (4) A. Floer's original papers
    (5) K. Mohnke, Holomorphic disks and teh chord conjecture, Annals of Math. 154 (2001),  electronic version
    (6) Y.Eliashberg, A.Givental, H.Hofer, Introduction to Symplectic Field Theory, GAFA 2000 (Tel Aviv, 1999),
           Geom. Funct. Anal. 2000, Special Volume, Part II,  electronic version
    (7)  Joshua Sabloff, Symplectic Field Theory webpage
    (8) K. Mohnke, How to (symplecticcally) thread a (Lagrangian) needle, electronic version
     
     

    Klaus Mohnke
    Wed Jan 23  09:30 a.m. EST 2002