Exact and asymptotic methods in Random Matrix Theory and Integrable Systems

随机矩阵模型是由Wigner引入的，用于模拟核相互作用，并且由于意想不到的（当时）与理论物理学和物理学的广泛领域联系在一起。（弦理论和二维量子引力），纯数学和应用数学（数论，组合学，不可压缩粘性介质的流体动力学（Hele-Shaw细胞）和理论的完全可积动力系统在无限维。正在进行的研究项目的重点是随机矩阵理论（RMT）和某些动力系统的名称下已知的户田，沃尔泰拉，相对论户田，Ablowitz-拉迪克，2-户田晶格等这个项目的一个次要组成部分是专门与所谓的拉普拉斯增长问题的不可压缩流体的关系。在随机矩阵理论中，主要的兴趣是描述矩阵谱（特征值）的统计特性，特别是当矩阵的大小很大时。 这些特征值之间的概率密度和所有相关函数可以用某些多项式（正交多项式及其变体，如双正交多项式）表示。这些统计函数在核物理学的原始动机中的重要性在于，本征值代表量子系统的能量，并且知道它们的分布允许在统计基础上预测这种核的样品的发射和吸收光谱。过去三年的主要发现之一是与这些多项式自然相关的“光谱曲线”的相关性。这个项目的主要目的之一是将谱曲线技术扩展到越来越大的RMT领域，并使用代数结构来推断否则无法获得的非微扰渐近信息（例如，渐近谱曲线的最大亏格等）。

Random matrix models were introduced by Wigner to model nuclear interactions and have witnessed a renewed interest later on due to unexpected (at the time) ties with a widespread range of areas in both theoretical physics (string theory and two-dimensional quantum gravity), pure and applied mathematics (number theory, combinatorics, statistical modeling of finance) fluid dynamics of incompressible viscous media (Hele-Shaw cells)  and the theory of completely integrable dynamical systems in infinite dimensions. The ongoing project of research focuses on this latter  relationship between random matrix theory (RMT)  and certain dynamical systems known under the names of Toda, Volterra, Relativistic Toda, Ablowitz-Ladik, 2-Toda lattices etc. A minor component of this projects is devoted to the relation with the so-called Laplacian growth problems of incompressible fluids. In random matrix theory the main interest is to describe the statistical properties of the spectrum of the matrices (the eigenvalues), in particular when the size of the matrix is large.  The probability densities and all correlation functions between these eigenvalues can be expressed in terms of certain polynomials (orthogonal polynomials and variations thereof like biorthogonal polynomials). The importance of these statistical functions in the original motivation for nuclear physics was that an eigenvalue represent an energy of the quantum system and knowing their distributions allows to predict on a statistical basis the spectra in emission and absorption of samples of such nuclei. One of the main findings of the past three years has been the relevance of a "spectral curve" associated naturally to these polynomials. It is one of the main aims of this project to extend the spectral curve-technique to increasingly larger areas of RMT and to use the algebraic structure to infer otherwise unaccessible  nonperturbative asymptotic informations (such as -e.g- the maximal genus of the asymptotic spectral curve etc.)