In modern geometry, classical mechanics is understood through symplectic manifolds. An important question for physicists is the quantization process i.e., going from the commutative setup of a symplectic manifold to a quantum system where the observables do not commute anymore. An important tool for the quantization process on symplectic manifolds is the star-product but it is not clear how to construct such a product, especially when one generalises the question to Poisson manifolds. Kontsevich proved that every Poisson manifolds posseses a star-product. However his methods are very algebraic and ignore every convergence property. Analytical constructions of star-products have since been done in the case of symplectic manifolds by various authors. I will talk about an ongoing project with Jean-Marie Lescure and Omar Mohsen to generalize these ideas to (integrable) Poisson manifolds. This talk will also be an introduction to symplectic and Poisson geometry.
