Some Analytical Aspects of the Theory of Integrable Systems

Indiana University, Bloomington IN

Investigators

Abstract

The term, "Integrable Systems," usually refers to mathematical objects, most often differential equations, with special symmetry properties that allow them to be studied in a very detailed way and sometimes even to solve them in a closed form. The class of integrable systems includes several fundamental equations of nature and the mathematical foundations of integrable systems go back to classical works of Liouville, Gauss, and Poincare. Currently, the theory of integrable systems has become 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. Simultaneously, it provides an efficient analytical tool for the study of some of the fundamental mathematical models arising in modern nonlinear science and technology. In addition to the traditional domain of differential equations, integrable techniques are becoming common in such diverse fields as orthogonal polynomials, string theory, enumerative topology, statistical mechanics, random processes, quantum informatics, and number theory. Many of the problems considered in the project have direct connections with these disciplines. This research project continuous the principal investigator's long term research efforts in the theory of integrable systems. The principal goal of the project is to address the following two directions which have emerged from recent developments in random matrix theory and in the theory of exactly solvable quantum models: (a) The study of the isomonodromic tau functions, their asymptotics, Fredholm determinant representations and their relations to the conformal field theory; (b) The asymptotic analysis of Toeplitz, Hankel, and Fredholm determinants arising in the study of critical phenomena in random matrices and statistical mechanics. Each of these directions is represented by a collection of concrete problems, and they will be investigated within the same analytical framework, viz., the Riemann-Hilbert method. The main focus of the first direction is on the long standing question of evaluation the asymptotic connection formulae, including the constant pre-factors, for the generic families of Painleve tau functions. The principal concern of the second direction is various types of double scaling and transitional limits - topics which are among the principal questions in both the theory of random matrices and in statistical mechanics.