Oporto Meeting on Geometry, Topology and Physics   Oporto Meeting on Geometry, Topology and PhysicsOporto Meeting on Geometry, Topology and PhysicsOporto Meeting on Geometry, Topology and Physics
FCUP - Isntituto Superior Técnico Faculdade de Ciências da Universidade do Porto Instituto Superior Técnico  
XVIIIth Oporto Meeting on
Geometry, Topology and Physics
    9th to 12th July 2009
 





Invited Talk

  • A geometric approach to Conn's linearization theorem (Rui Loja Fernandes, Dep. Matematica, IST Lisbon)
Abstract: I will describe joint work with Marius Crainic (Utrecht) on a soft geometric proof of the classical result due to Conn stating that a Poisson structure is linearizable around a singular point (zero) at which the isotropy Lie algebra is compact and semisimple. Although most of this proof was known to us by 2004, only recently we were able to complete a crucial step. In this lecture, I will attempt to describe the complete proof with emphasis on the above mentioned step.

Introductory courses:

  • A short  introduction to Symplectic geometry (David Martinez, IST, UTL, Lisbon)
Abstract: 1.Linear symplectic geometry: Symplectic linear forms. Dimension. Quotients. Standard symplectic form. Symplectic linear group. Subspaces. Darboux/symplectic basis. Volume forms. Symplectic linear group and compatible almost complex structures. Symplectic vector bundles and Chern classes.
 2.Symplectic manifolds: Symplectic forms. Examples. Symplectic transformations. Reduction (without actions around).
3.Moser's technique: Moser's theorem. Stability. Darboux/symplectic coordinates. Neighborhood theorems.
  • A short  introduction to Poisson geometry (Marco Zambon, CMUP/FCUP, Porto)
Abstract: A short introduction to Poisson geometry: I will introduce Poisson manifolds both from the algebraic and the geometric point of view, and discuss the classes of examples given by symplectic manifolds and duals of Lie algebras. I will also describe the role played by coisotropic submanifolds. If time permits I will mention two important problems  arising from Poisson geometry: deformation quantization and the integration of Lie algebroids.