URegina Topology & Geometry Seminar: Brent Pym

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.