Internal symmetry of soliton equations and its applications

An algebro-geometric interpretation and construction （a la Krichever theory） of a general form of string equation， and some "genus zero" property it possesses in the situation which is closer to the bispectral proManagement， are shown. Calogero-Moser type KP solutions， related to bispectral proManagement and Krichever theory， is studed。结果包括证明，什么不符合双重特征等。关于具有理性协同作用的任何一级普通差异运算符的光谱曲线的事实，是统一的。与矩阵集成和组合相关的各种主题已经与Adler和van Moerbeke一起研究。一个公式计算一些奇怪的步态的数目是给定的，BKP的高计量类型的解决方案与Schur Q函数所有严格部分的产品之和有关。（λ_1 > λ_2 >··）与λ_1 <小于或等于h，在组合学问题的背景下研究Tracy and Widom（result of Sasha Orlov）。基于Krichever theory and yet without using complex analysis on principally polarized abelian varies的基本方法，我们在KP层次结构中研究了Jacobian变体的特征化的详细情况，其中包括了许多不同的等式。

An algebro-geometric interpretation and construction (a la Krichever theory) of a general form of string equation, and some "genus zero" property it possesses in the situation which is closer to the bispectral problem, are shown.Calogero-Moser type KP solutions, related to bispectral problem and Krichever theory, is studied. Results include a proof, which does not assume bispectrality etc., of the fact that the spectral curve of any rank one ordinary differential operator with rational coefficients is unicursal.Various topics related to matrix integrals and combinatorics has been studied jointly with Adler and van Moerbeke. A formula counting the number of some vicious walks is given.BKP solutions of hypergeometric type is observed to be related to the sum of products of Schur Q-functions over all the strict partitions λ = (λ_1 > λ_2 > ・・・) with λ_1 【less than or equal】 h, studied by Tracy and Widom in the context of combinatorics problems (result of Sasha Orlov) .Elementary approach to the Schottky problem - Based on the Krichever theory and yet without using complex analysis on principally polarized abelian varieties, we worked out details of characterization of Jacobian varieties in terms of finitely many differential equations in the KP hierarchy.