Isomonodromic deformations and conformal field theory

Study of the relation between isomonodromic deformations and conformal field theory starts from the paper by Gamayun, Iorgov and Lisovyy where the general solution of the Painlevé VI equation was given as a Fourier transformation of c=1 conformal block. We will discuss this statement and then one of the approaches to its proof and generalization. In particular, I will tell about free-fermionic construction which allows us to express isomonodromic tau functions as averages of some group elements. This construction also gives Fredholm determinant representations for these isomonodromic tau functions.