"Integrable Systems, Random Matrices and Random Processes"

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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.