Programme on Lie Groups 2001

Introduction to Symplectic and Poisson Geometry

Professor Jiang-Hua Lu
University of Arizona, USA

Symplectic geometry is the geometry of a certain 2-form on a manifold, and Poisson geometry is the geometry of a 2-vector field. They are related by the fact that a symplectic manifold is a special example of a Poisson manifold, and a Poisson manifold is decomposed as a disjoint union of symplectic manifolds.

Both symplectic geometry and Poisson geometry are rapidly expanding and evolving fields, marked by current research activities in symplectic topology, deformation quantization, and applications to mathematical physics and Lie theory.A set of fundamental concepts and constructions in symplectic and Poisson geometry is essential to all the more advanced topics mentioned above. The purpose of this course is to introduce some of these basic materials and point out further applications. Among the topics we will discuss are symplectic vector spaces and Lagrangian subspaces, Darboux's Theorem and Moser's theorem, symplectic reduction, the geometry of moment maps, Duistermaat-Heckman theorem, and symplectic leaves and cohomology of Poisson structures. We plan to work out many concrete examples with emphases on those coming from Lie theory. A set of excercise problems will be given as an integral part of the course, and we plan to hold regular discussion sessions to work on these problems.

[G]    V. Guillemin, Moment maps and combinatorial invariants of Hamiltonian Tn-spaces, Progress in mathematics 122, Birkhauser (1994).

[M-S] D. McDuff and D. Salamon, Introduction to symplectic topology, Oxford Mathematical Monographs, Oxford, 1995.

[V] I. Vaisman, Lectures on the geometry of Poisson manifolds, Birkhauser, Basel, 1994.

Dates / Time:
2:00 - 3:30pm Wednesdays and Fridays, March 7 - May 18, 2001
Room 517, Meng Wah Complex