Topology and Geometry Seminar

Title: A (gentle) introduction to Poisson geometry, Part 1. 

Abstract: I will give an introduction to Poisson geometry, starting with its role in the dynamics of physical systems and its relationship to symplectic geometry. Through the use of illustrative examples, I will highlight some of the basic features of Poisson manifolds, such as their symplectic leaves. 

Title: A (gentle) introduction to Poisson geometry, Part 2. 

Abstract:  This is the second part of an introduction to Poisson geometry.  In this part, I will discuss the geometry of Poisson manifolds, such as their symplectic leaves, through the use of illustrative examples. I will also briefly discuss the quantization of Poisson manifolds. 

Title: Introduction to deformation quantization

Abstract: A deformation quantization of a manifold is a noncommutative deformation of its algebra of functions; the idea originated in physics, as a way of relating the classical and quantum descriptions of mechanical systems.  At leading order in the deformation parameter, a deformation quantization gives rise to a Poisson structure on the manifold.  A deep theorem of Kontsevich states that this gives an equivalence between Poisson structures and noncommutative deformations; it thus provides a direct link between geometry and noncommutative algebra.  In fact, he gives an explicit formula for the quantization of any Poisson bracket, via integrals of differential forms associated to graphs (a Feynman expansion).  I will give an introduction to deformation quantization, with several examples, leading to a sketch of Kontsevich's proof.

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