Mathematical Sciences: Integrable Models in Mathematics and Physics

该研究项目开发了新的方法， 完全可积微分方程 型），算子理论和渐近分析，研究水平 各种随机矩阵集合的间距分布。 的 各种正交多项式水平间距分布 集合可以用Fredholm行列式来表示。 PI和他的合作者Harold Widom的方法表明， 这个行列式也是一个τ函数。 在一般问题中 tau函数有两种不同的类型， 变量有“KP类型变量”，它描述了 KP流在Sato Grassmannian上存在"变形 类型变量。这是一种结合和相互作用， 这两种类型的变量给出了一般的τ函数。 这种一般tau函数的特定示例在以下方面是重要的： 无序导体，数值分析与 矩阵的条件数，除了标准的 随机矩阵系综在分析 特征值统计 本程序中开发的方法 我将为这些重要的问题给出详细而明确的公式 例 %%% 核物理、原子物理和核物理中出现的复杂系统， 分子物理学、凝聚态物理学与介质 需要一种数学来给出结果 对于平均量，因为在宏观的日常世界中 在各种实验中控制的变量（如 杂质的浓度，系统的能量）不 详细描述了微观系统。 因此你 开发预测平均行为的统计方法， 给出偏离这个平均行为的概率。 随机矩阵的主题就是这样一种统计理论 已经成功地应用于上述问题， 物理学沿着着大量的理论问题， 数学和数值分析中的实际问题。 这 研究项目开发了新的方法， 微分方程来研究这些统计理论。

This research project develops new methods involving completely integrable differential equations (of the Painleve type), operator theory and asymptotic analysis to study the level spacing distributions for various random matrix ensembles. The level spacing distribution for various orthogonal polynomial ensembles can be expressed in terms of a Fredholm determinant. The methods of the PI and his collaborator Harold Widom show that this determinant is also a tau-function. In the general problem that will be analyzed the tau function has two distinct types of variables. There are the ``KP type variables'' which describe a KP flow on the Sato Grassmannian and there are the ``deformation type variables.'' It is the combination and interplay between these two types of variables that give the general tau-function. Particular examples of this general tau function are important in disordered conductors, numerical analysis associated with the condition number of a matrix, in addition to the standard applications of random matrix ensembles to the analysis of eigenvalue statistics. The methods developed in this program will give detailed and explicit formulas for these important cases. %%% Complex systems that arise in nuclear physics, atomic and molecular physics, condensed matter physics dealing with media with impurities require a type of mathematics that gives results for averaged quantities, since on the macroscopic everyday world the variables that are controlled in various experiments (like concentration of an impurity, the energy of the system) do not completely specify the microscopic system. Therefore one develops statistical methods that predict average behavior or give the probabilities of deviation from this average behavior. The subject of random matrices is one such statistical theory that has been successfully applied to the above problems in physics along with a host of more theoretical problems in mathematics and practical problems in numerical analysis. This research project develops new methods using the tools of differential equations to study these statistical theories.