Sol-流形的谱：算术和量子单值性

具有拟置换单值矩阵的矩阵Riemann-Hilbert问题的解

Solution of matrix Riemann-Hilbert problems with quasi-permutation monodromy matrices

在本文中，我们解决了一个任意矩阵Riemann-Hilbert（逆monodromy）问题与准置换monodromy表示外的一个因子的monodromy数据空间。这个因子的特征在于θ-因子的雅可比流形上的辅助紧黎曼曲面实现为适当的分支覆盖的$\CP1$。解是根据黎曼曲面上的Szegö核的推广给出的。特别是，我们的建设提供了一类新的解决方案的Schlesinger系统。这些解决方案的isomonodromytau-函数的计算无处消失的因素独立的monodromymatrix的元素。

引力孤立子和单值变换方法求解Einstein方程的可积约化

Gravitational Solitons and Monodromy Transform Approach to Solution of Integrable Reductions of Einstein Equations

本文在所谓的“单值变换方法”的背景下，研究了具有二维阿贝尔等距群的真空Einstein方程解空间的Belinskii和Zakharov孤子生成变换，为研究Einstein方程的各种可积时空对称约化提供了一般的基础。类似于在已知的谱变换中使用的散射数据，在该方法中，用于相关线性系统的解的单值数据完全表征约化的爱因斯坦方程的任何解，并且解的许多物理和几何性质可以直接表示为对应的单值数据函数的谱平面上的分析结构。Belinskii和Zakharov真空孤子生成变换可以显式表示为相应的单值数据函数的简单（线性-分数）变换，其系数为谱参数的多项式。这允许确定许多物理参数的生成孤子的解决方案，而无需（或之前）的所有组件的解决方案的计算。对电真空孤子产生变换也给出了类似的刻画。

On the monodromy manifold of $q$-Painlevé VI and its Riemann-Hilbert problem

We study the sixth $q$-difference Painlevé equation ($q{\textrm{P}_{\textrm{VI}}}$) through its associated Riemann-Hilbert problem (RHP) and show that the RHP is always solvable for irreducible monodromy data. This enables us to identify the solution space of $q{\textrm{P}_{\textrm{VI}}}$ with a monodromy manifold for generic parameter values. We deduce this manifold explicitly and show it is a smooth and affine algebraic surface when it does not contain reducible monodromy. Furthermore, we describe the RHP for reducible monodromy data and show that, when solvable, its solution is given explic…

Tau-functions and monodromy symplectomorphisms

We derive a new Hamiltonian formulation of Schlesinger equations in terms of the dynamical $r$-matrix structure. The corresponding symplectic form is shown to be the pullback, under the monodromy map, of a natural symplectic form on the extended monodromy manifold. We show that Fock-Goncharov coordinates are log-canonical for the symplectic form on the extended monodromy manifold. Using these coordinates we define the symplectic potential on the monodromy manifold and interpret the isomonodromic tau-function as the generating function of the monodromy map. This, in particular, solves a recent …

常微分方程的拟精确解、共振和平凡单值性

Quasi-exact solvability, resonances and trivial monodromy in ordinary differential equations

利用六次非谐振子与一对三阶常微分方程之间的对应关系，研究了阶数大于2的微分算子本征值问题的拟精确可解性.特别是，与Bender-Dunne多项式和独立的解决方案之间的共振的链接被观察到某些二阶的情况下，并扩展到高阶问题。

Monodromy of the equivariant quantum differential equation of the cotangent bundle of a Grassmannian

We describe the monodromy of the equivariant quantum differential equation of the cotangent bundle of a Grassmannian in terms of the equivariant K-theory algebra of the cotangent bundle. This description is based on the hypergeometric integral representations for solutions of the equivariant quantum differential equation. We identify the space of solutions with the space of the equivariant K-theory algebra of the cotangent bundle. In particular, we show that for any element of the monodromy group, all entries of its matrix in the standard basis of the equivariant K-theory algebra of the cotang…