Strings, Solitons and Random Matrices

摘要提案：DMS-9802077首席研究员：Mark Adler随机矩阵与可积系统的联系是什么？这个连接有用吗？这些问题的答案导致了随机矩阵理论的一个新的和统一的方法。在矩阵集合的概率分布中引入适当的t相关性，可以得到n点相关函数的顶点算子表达式（无限小区间内n个特征值的概率）和相应的Fredholm行列式（R的Borel集合中没有特征值的概率）。这种表示导致了Fredholm行列式的偏微分方程。这包括有限厄米特系综，辛和对称系综，甚至耦合矩阵系综。这与2-Toda和1-Toda的减少以及KP等级有关。我们在工程和数学物理的非线性方程的一般理论中工作。我们研究的方程类型涉及光纤中的超短传输，激光中的强脉冲，窄通道中的水波传输，慢核反应和高温下大块金属的物理性质以及重核的一般激发理论。我们还研究量子场论中与弦理论相关的方程，量子场论是现代物理学中一个令人兴奋的研究领域。

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.