Poisson geometry and normal forms: a guided tour through examples

Abstract

There are no special prerequisites to follow this minicourse except for basic differential geometry. Poisson manifolds constitute a natural generalization of Symplectic manifolds. Many problems in mechanics turn out to be naturally formulated in the Poisson language. In contrast to symplectic manifolds where there are no local invariants (Darboux), Poisson manifolds do present local invariants. The aim of the minicourse is to present an introduction to Poisson Geometry and the study of normal forms (how the structures look like locally). We also plan to explain some of the problems considered in their semilocal and global study and introduce the study of symmetries (group actions) in these manifolds. We will end up the minicourse presenting an application of the study of group actions to integrable systems (stressing the particular case of b-Poisson manifolds).