I will explain some results on the localization and deformation theory of vertex algebras, algebraic objects encoding a class of topological associative algebras generalizing the enveloping algebra of an affine Kac-Moody Lie algebra. I will also explain how these results can be used to give geometric constructions of free field realizations, embeddings of these algebras into infinite dimensional Weyl algebras, motivated by the physics of 4d N=2 superconformal field theories. All new results that will be presented are in joint work with Sujay Nair.

I will give an introductory talk about the  ergodicity of geodesic flows, Anosov diffeomorphisms, the Hopf argument, and the Howe-Moore Theorem. Everybody is welcome, we will go for pizza after the talk.