北京雁栖湖应用数学研究院

In this semester, we will continue discussing the isomonodromic deformations. We will focus on the two topics:
1. Riemann-Hilbert problems with quasi-permutation monodromies and isomonodromic tau-function
In general, the RH problem cannot be solved in terms of special functions. But, for an arbitrary quasi-permutation monodromy group, the RH problem was solved outside of the so-called Malgrange divisor. This divisor is characterized in terms of the theta-divisor on the Jacobi manifold of an auxiliary compact Riemann surface realized as an appropriate branched covering of CP^1. The solution is given in terms of a generalization of Szego kernel on the Riemann surface. Correspondingly, the solutions of Schlesinger equation and tau function can be computed. We will explain the details as well as discussing the relation between this tau function and exact conformal block.

2. Isospectral deformation and Widom constant
Tau functions play a central role in the theory of integrable equations, both in fields of isospectral and isomonodromic deformations. On the side of isospectral deformations, Sato defined the tau function starting from his interpretation of the KP hierarchy in terms of the geometry of Grassmannian manifolds. Even through the definitions of tau functions (isomonodromic tau functions and Sato-Segal-Wilson tau functions) look so different in both the worlds of isomonodromic and isospectral deformations, tau functions coincide with a simple object: Widom constant.