Programme on Lie Groups 2001


Holomorphic Curves in Symplectic Topology


Professor Yakov Eliashberg

Stanford University, USA

As it was discovered in the early fifties symplectic structures and symplectic transformations have amazing rigidity properties. For example, the group of symplectic transformations is C0-closed in the group of volume preserving transformations. The group of symplectomorphisms also admits a bi-invariant metric, which is highly unusual for non-compact infinite-dimensional groups. The theory of holomorphic curves in symplectic manifolds which was introduced by M. Gromov serves as a major tool for handling these questions, as well as other problems in Symplectic Topology. In the course we will discuss the main notions and problems of Symplectic topology and will explain the role played by the theory of holomorpic curves. We will also discuss the relevant algebraic formalism, called Symplectic Field Theory.

o        Basic notions of symplectic and contact topology. Flexible and rigid questions in symplectic topology. Basic problems of symplectic topology.

o        Holomorphic curves in symplectic manifolds. Gromov compactness theorem. Moduli spaces of holomorphic curves.

o        Applications to symplectic topology: C0-closedness of the group of symplectomorphisms, non-squeezing results.

o        Further applications: Lagrangian intersection theory.

o        Gromov-Witten invariants and quantum cohomology.

o        Holomorphic curves in symplectization of contact manifolds and symplectic cobordisms. Applications to contact topology.

o        Symplectic field theory.



o            D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford Scientific Publications, 1995.

o            M. Gromov, Pseudo-holomorphic curves in Symplectic manifolds, Invent. Math., 82(1985), 307-347.

o            Y. Eliashberg, A. Givental and H. Hofer, Introduction to Symplectic Field Theory, GAFA, Special volume "Vision 2000".


Dates / Time:

A series of 7 lectures on Wednesdays and Fridays to be held 10:30am - 12:00noon, June 1 - 22, 2001


Room 517, Meng Wah Complex