# University of Toronto's Symplectic Geometry Seminar

Thursday May 19, 2005

SS 5017A

## Rui Fernandes

###
IST-Lisbon

##
Linearization of Poisson brackets

** Abstract: **
If one fixes a symplectic leaf of a Poisson manifold, there is a
linear model for the Poisson bracket in a neighborhood of the leaf. The
linearization problem is to decide if there is a Poisson diffeomorphism
from a tubular neighborhood of the leaf to this linear model. An old
result of J. Conn states that Poisson brackets can be linearized around
fixed points (i.e., zero dimensional leafs) with compact semisimple linear
part. I will give a new geometric proof of Conn's result, using Moser's
path method, and formulate a conjecture for higher dimensional leafs.