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 which allow to study them 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. In our days, 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 this project have direct connections with these disciplines. The project also includes several educational activities such as training of Ph.D. students at IUPUI and co-organizing of the international research/educational program on universality and integrability in random matrix theory which will be held in the Fall of 2021 at MSRI, Berkeley. This project continuous the PI’s long term research efforts in the theory of integrable systems. The principal goal of this research program is to address various new analytical questions of the theory of integrable systems which have emerged from recent developments in random matrix theory and in the theory of exactly solvable quantum models. In this project, the PI will concentrate on three directions of research: (a) The general beta-ensembles and the Calogero-Painleve system; (b) The study of the isomonodromic tau functions, their asymptotics, Fredholm determinant representations and their relations to conformal field theory; and (c) The asymptotic analysis of Toeplitz + Hankel determinants emerging from random matrix theory 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. Success in part (a) of the project would have a notable impact in the development of the general concept of universality and integrability in both the random matrix theory and in the theory of interacting particles as well as in the modern nonlinear science at large. The part (b) of the project will further enhance the "nonlinear special function" status of Painleve transcendences, which are also sometimes called "the Special Functions of 21st century". Success in part (c) would significantly contribute to the development of the general theory of Toeplitz and Hankel determinants which, for many decades, have been playing a central role in the study of some of the fundamental models of statistical and quantum mechanics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.