Some Analytical Aspects of the Theory of Integrable Systems

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AbstractItsThe research project is focused on the analytical aspects of the theoryof integrable systems related to the Riemann-Hilbert and isomonodromymethods. The problems under consideration include: (a) the investigationof the double scaling limits in the theory of random matrices andorthogonal polynomials and their applications to the two-dimensionalquantum gravity and enumerative topology; (b) the asymptotic analysisof the Toeplitz and Fredholm determinants and the multiple integralsassociated with the correlation functions of exactly solvable quantumfield and statistical mechanics models; (c) the related aspects of theglobal 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 anincreasingly important role as one of the principal sources of newanalytical and algebraic ideas for many branches of modern mathematicsand theoretical physics. At the same time, it provides an efficientanalytic tool for the study of some of the fundamental mathematicalmodels arising in modern nonlinear science and technology. Specifically, the research directions indicated above deal with themathematical models which form the theoretical basis for the followingfields: condensed matter, high energy and plasma physics, nonlinearhydrodynamics, high-bit rate telecommunication systems, and informationscience. A special attention in the project is given to the analysisof the distributions of random matrix theory which govern statisticalproperties of the large systems which do not obey the usual laws ofclassical probability. This kind of systems appear in many differentareas of applied science and technology including heavy nuclei, polymergrowth, high-dimensional data analysis, and certain percolationprocesses. Simultaneously, the random matrices are playing increasinglyimportant role, as an extremely powerful analytic apparatus, in severalpure theoretical domains such as the string theory, enumerative topologyand certain fundamental aspects of number theory.