A story of isomonodromic deformations on the torus

In the first half of this talk, I will provide a brief introduction to Isomonodromic deformations with the one-point torus as my main example, and show the relation to the elliptic form of Painlevé VI equation as well as the Lamé equation. In the second half of this talk, I will present an overview of my results in the past few years concerning the associated tau-functions, conformal blocks, and accessory parameters. Finally, I will motivate how probabilistic methods in conformal field theory help us understand the data within Lamé type equations.