Strings, Solitons and Random Matrices

摘要 提案：DMS-9802077主要研究者：Mark Adler 随机矩阵与可积系统有什么联系？ 这种连接有用吗？这些问题的答案导致一个新的和统一的方法来理论的随机矩阵。 在概率中引入适当的t依赖 矩阵系综的分布，导致顶点算子 表达式的n点相关函数（概率的n个本征值在无穷小的间隔）和相应的Fredholm行列式（概率没有本征值的Borel集的R）。 这种表示导致Fredholm行列式的偏微分方程。 这涵盖了有限厄米系综，辛和对称系综，甚至耦合矩阵系综。这些是 与2-户田和1-户田的减少和KP等级有关。 我们的工作在一般理论的非线性方程的工程 和数学物理。我们研究的方程类型 光纤中的超短传输、激光中的强脉冲、狭窄通道中的水波传输、缓慢的核反应以及 大块金属在高温下的物理性能以及 重核的一般激发理论。 我们也研究方程 与量子场论中的弦理论有关， 现代物理学的研究。

Abstract Proposal: DMS-9802077 Principal Investigator: Mark Adler What is the connection of random matrices with integrable systems? Is this connection useful?The answer to these questions leads to a new and unifying approach to the theory of random matrices. Introducing an appropiate t-dependence in the probability distribution of the matrix ensemble, leads to vertex operator expressions for the n-point correlation functions (probabilities of n eigenvalues in infinitesmal intervals) and the corresponding Fredholm determinants (probabilities of no eigenvalues in a Borel set of R). This representation leads to PDE's for the Fredholm determinant. This covers the finite Hermitean ensemble,the symplectic and symmetric ensemble and even a coupled matrix ensemble. These are related to reductions of 2-Toda and 1-Toda and the KP hierarchy. We work in the general theory of nonlinear equations of engineering and mathematical physics.The type of equations we study figure into ultra-short transmissions in fiber optics,intense pulses in lasers, water wave transmission in narrow channels,slow nuclear reactions and the physical properties of bulk metals at high temperatures as well as the general excitation theory of heavy nuclei. We also study equations connected with string theory in quantum field theory, an exciting area of research in modern physics.