New Developments in Liouville Theory
Classical Liouville theory is a field theory having its origins in the uniformization problem for Riemann surfaces. Its probabilistic version is one of the principle models for conformal field theory. It offers the natural framework for studying spin systems on random lattices. The mathematical problems center on constructing the probability measure for the theory. We will explain all this and describe some recent developments in solving this problem on the sphere due to David, Kupiainen, Rhodes and Vargas and discuss possible relations to work with Patrick Waters for the extension of this to higher genus.