"Integrable Systems, Random Matrices and Random Processes"

自20世纪60年代末在孤子物理学领域突破性地发现逆散射方法以来，经典和量子可积系统理论的应用范围迅速扩大。这些现在包括各种各样的应用，如：随机矩阵的谱理论；可积随机过程，如“不相容”过程、晶体和聚合物生长，以及最近在规范场理论中发现的显著的“可积性”特性，控制着物质的基本相互作用。本建议所处理的问题涉及所有这些领域。

The range of areas to which the theory of classical and quantum integrable systems is applicable has been rapidly growing since the break-through discovery of the inverse scattering method within the field of soliton physics in the late 1960's. These now include such diverse applications as: the spectral theory of random matrices; integrable random processes, such as "exclusion" processes, crystal and polymer growth, and the remarkable "integrability" properties discovered recently in gauge field theory, governing the fundamental interactions of matter. The problems addressed in this proposal touch upon all these fields. At the core are both new and well-developed methods of analysis, aimed at the solution of concrete problems relating to the underlying "integrable" dynamics common to all these areas. Integrability, whether quantum or classical or probabilistic, implies that, due to special structural properties, the dynamics of complicated, generally nonlinear dynamical systems, having either a finite, or infinite number of "degrees of freedom", can be reduced to the study of systems having only one degree of freedom. This appears in various guises, depending on the domain. In classical dynamical systems and classical field theory, it is related essentially to "separation of variables". In quantum systems, it appears in the form of factorization of the scattering matrix of the interacting system into two-particle amplitudes. In "integrable random processes", it appears in the fact that all correlation functions can be expressed in the form of determinants, having two-particle correlation functions as entries. These results are applicable to a wide variety of physical problems of current physical interest, including: multi-periodic and quasi-periodic solutions to systems describing coherent nonlinear phenomena in optics, fluids, polymers and superconductors; the statistical distributions of the eigenvalues of random matrices and operators; the determination of generating functions for a variety of random processes, such as crystal growth, eigenvalue dynamics of random matrices, mutually-avoiding multiple random-walkers and the computation of scattering amplitudes in superconformally invariant gauge field theory in the "planar" limit.