$\hbar$-Riemann-Hilbert correspondence

复切触流形的微局部黎曼-希尔伯特对应

The microlocal Riemann-Hilbert correspondence for complex contact manifolds

本文在复切触流形上建立了微局部化D-模与反常微层之间的黎曼-希尔伯特对应，统一了Kashiwara的范畴层构造与作者新近的反常微层理论。

仿射Poisson括号下直到7阶的Kontsevich星积：Riemann zeta值在哪里？

Kontsevich's star-product up to order 7 for affine Poisson brackets: where are the Riemann zeta values?

该研究针对仿射Poisson括号下的Kontsevich星积，推导了到7阶的调和权重公式，验证其满足循环关系与结合律，发现Riemann zeta值因Jacobi恒等式的微分推论消失，得到仅含有理系数的简洁约化公式。

Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painlevé equation of type VI, Part I

In this paper, we will give a complete geometric background for the geometry of Painlevé $VI$ and Garnier equations. By geometric invariant theory, we will construct a smooth coarse moduli space $M_n^{\balpha}(\bt, \blambda, L) $ of stable parabolic connection on $\BP^1$ with logarithmic poles at $D(\bt) = t_1 + ... + t_n$ as well as its natural compactification. Moreover the moduli space $\cR(\cP_{n, \bt})_{\ba}$ of Jordan equivalence classes of $SL_2(\C)$-representations of the fundamental group $π_1(\BP^1 \setminus D(\bt),\ast)$ are defined as the categorical quotient. We define the Riemann…

Irregular Riemann-Hilbert correspondence, Alekseev-Meinrenken dynamical r-matrices and Drinfeld twists

In 2004, Enriquez-Etingof-Marshall suggested a new approach to the Ginzburg-Weinstein linearization theorem for a quasitriangular Lie bialgebra $(\g,r)$. This approach is based on solving a system of PDEs for a gauge transformation between the classical r-matrix $r$ and the Alekseev-Meinrenken dynamical r-matrix. They proved that the semiclassical limit of an admissible Drinfeld twist gives rise to a solution of the PDEs. In this paper, we explain that preferred gauge transformations can be constructed as connection maps for a certain irregular Riemann-Hilbert problem (provided $r$ is the stan…

非阿贝尔 Hodge 对应的温和导引

A Gentle Introduction to the Non-Abelian Hodge Correspondence

本文以教学方式介绍非阿贝尔 Hodge 对应，阐明基本群表示与全纯丛之间的联系，并梳理其核心定理、几何结构与相关应用。