Poisson Geometry

Henrique Bursztyn and Marius Crainic , U. Utrecht (Feb - May, 2007) - Masterclass, "Symplectic geometry and beyond"

Textbooks:
- Cannas da Silva, A., Weinstein, A.: Geometric models for noncommutative algebras , Berkeley Mathematics Lecture Notes 10, AMS, 1999.
- Dufour, J.-P., Zung, N.-T.: Poisson Structures and their Normal Forms , Progress in Mathematics, Vol. 242, Birkhauser, 2005.
Lectures: Wednesdays, 10am-1pm..
Course outline: This is an introductory course on Poisson geometry. Some basic prior knowledge of symplectic geometry will be assumed.
- Poisson structures on manifolds, hamiltonian vector fields; first examples.
- Multivector fields and the Schouten bracket; the Poisson bivector field.
- Weinstein's splitting theorem, symplectic leaves; reduction and symplectic foliation.
- Dirac structures, gauge transformations; transversal structures.
- Symplectic realizations of Poisson manifolds, contravariant geometry.
- The integrability problem for Poisson structures; symplectic groupoids. Obstruction (variation of symplectic area).
- Symplectic gruopoids, moment maps and reduction.
- Poisson cohomology, the linearization problem.
- Poisson-Lie groups; Lie bialgebras, Manin triples, dual Poisson groups, doubles.
- The Poisson structure on the moduli space of flat G-bundles over a surface.
Exams: There will be a final oral exam.
Grades : based on the oral exam as well as problem sets and work on lecture notes.