Poisson geometry and Lie groupoids

The study of Poisson geometry, which generalizes the Poisson bracket of symplectic geometry, has a rich history going back to work of Lie. More recently, researchers have become aware that much of Poisson geometry is most naturally discussed in the framework of Lie groupoids. In this talk, I will draw out the relationship between Poisson geometry and Lie groupoid theory by exploring three questions one might ask about a given Poisson manifold. The first asks whether that Poisson manifold has a complete symplectic realization. The second asks whether there is a symplectic groupoid whose base space is the given Poisson manifold. Finally, the third asks whether the Poisson manifold's cotangent bundle Lie algebroid can be integrated to a Lie groupoid. These three questions were proven to be equivalent by Crainic and Fernandes, building on work of Weinstein. I will give an overview of the concepts and terminology needed to understand this result and sketch their proof of it.