Some Analytical Aspects of the Theory of Integrable Systems

Indiana University, Bloomington IN

Investigators

Abstract

The class of equations under study, called integrable systems, includes several fundamental equations of nature. This is an old and venerable area of study which goes 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 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 string theory, statistical mechanics, random processes, quantum informatics, and number theory. Many of the problems considered in the project have direct connections with these disciplines. Indeed, the concrete analytical questions addressed in the project are primarily motivated by the needs of the analytical theory of quantum entanglement - a phenomenon which is expected to play a key role in a practical realization of quantum computing, and by the challenges of analytical description of fundamental issue of phase transition in the quantum and statistical mechanics. The principal goal of this research project 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 proposes to focus on three directions of research: the asymptotic analysis of Toeplitz and Hankel determinants and the applications of these determinants in random matrices and statistical mechanics, the study of Fredholm determinants arising in random matrix theory, and the asymptotics of the correlation functions of non-free fermionic quantum spin models. 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.