Classical conformal blocks as generating functions

In this talk, I will consider a CFT on a four punctured sphere. I will first gather three known results in the literature about the role classical (c-> infinity) conformal blocks play as generating functions for: accessory parameters, monodromy coordinates, and the connection constant of Heun equations.  Secondly, I will outline analogous results for the one-point torus and provide a road-map to proving these results rigorously using probability techniques. Finally, I will discuss potential challenges in rigorous proofs for conformal blocks on any other geometry.