Some Analytical Aspects of the Theory of Integrable Systems

Indiana University, Bloomington IN

Investigators

Abstract

Abstract Its The research project is focused on the analytical aspects of the theory of integrable systems related to the Riemann-Hilbert and isomonodromy methods. The problems under consideration include: (a) the investigation of the double scaling limits in the theory of random matrices and orthogonal polynomials and their applications to the two-dimensional quantum gravity and enumerative topology; (b) the asymptotic analysis of the Toeplitz and Fredholm determinants and the multiple integrals associated with the correlation functions of exactly solvable quantum field and statistical mechanics models; (c) the related aspects of the global asymptotic analysis of the solutions of integrable differential equations of the Painleve' and KdV types. The theory of integrable systems is an expanding area which plays an increasingly important role as one of the principal sources of new analytical and algebraic ideas for many branches of modern mathematics and theoretical physics. At the same time, it provides an efficient analytic tool for the study of some of the fundamental mathematical models arising in modern nonlinear science and technology. Specifically, the research directions indicated above deal with the mathematical models which form the theoretical basis for the following fields: condensed matter, high energy and plasma physics, nonlinear hydrodynamics, high-bit rate telecommunication systems, and information science. A special attention in the project is given to the analysis of the distributions of random matrix theory which govern statistical properties of the large systems which do not obey the usual laws of classical probability. This kind of systems appear in many different areas of applied science and technology including heavy nuclei, polymer growth, high-dimensional data analysis, and certain percolation processes. Simultaneously, the random matrices are playing increasingly important role, as an extremely powerful analytic apparatus, in several pure theoretical domains such as the string theory, enumerative topology and certain fundamental aspects of number theory.