Isomonodromic deformations and tau functions

Prerequisite

basic knowledge of function of one complex variable, algebraic topology, compact Riemann surface

Abstract

I will talk about a general theory of monodromy preserving deformation, which is developed for a system of linear ordinary differential equations dY/dx = A(x)Y, where A(x) is a rational matrix. In the first part, I will talk about the asymptotic solutions for the equation with regular and irregular singularities respectively, then the corresponding monodromy data including connection matrix, Stokes matrix and so on. In the second part, I will talk about the equations of isomonodromic deformation and assoicatied tau function, which plays a central role in the deformation theory. For example, in the special case (studied by Riemann), the tau function reduces to the theta functions. I will also talk about Schlesinger transformation and Characterisitic matrix in the theory of tau function. Finally, I will talk about its applications on the model of mathematical physics.

Lecturer Intro.

2009-2013 四川大学数学学院基础数学 本科 2013-2018 北京大学北京国际数学研究中心 博士 2018-2021 清华大学丘成桐数学科学中心 博士后 2021- 北京怀柔应用数学研究院 助理研究员 研究兴趣：1. 可积系统，特别是GW理论、LG理论中出现的无穷维可积系统，兴趣在于理解其中的无穷个对称性的代数结构和相关计算。2. 其他兴趣：mixed Hodge structure，quantum group and KZ equation, W-algebra and W-symmetry, augmentation representation