Some Analytical Aspects of the Theory of Integrable Systems

Indiana University, Bloomington IN

Investigators

Abstract

The principal goal of the research project is to address some of the new analytical questions of the theory of integrable systems which emerged from the recent developments in random matrix theory and in the related areas of exactly solvable quantum models. The problems under consideration include the analytical description of the Painlev\'e - type universalities in random matrix theory and the related study of special families of Painlev\'e transcendents and generalized Painlev\'e transcendents arising in the critical asymptotics in random matrices, statistical mechanics and in the enumerative topology. The third direction of the project is concerned with the analytical investigation of the crossover phenomena in the classical theory of Toeplitz and Hankel determinants, i.e. with the study of the Painlev\'e-type transition behavior between different non-critical asymptotic regimes exhibited by large size Toeplitz and Hankel determinants. Each of the above mentioned directions is represented by a collection of concrete problems, and they are proposed to be investigated within the same analytical framework, which is the Riemann-Hilbert method. 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. Simultaneously, it provides an efficient analytical tool for study of some of the fundamental mathematical models arising in modern nonlinear science and technology. The new areas where the analytic techniques of integrable systems become more and more common include random matrices, quantum field and string theories, enumerative topology, stochastic processes, number theory, and, most recently, entanglement in quantum chain systems which is expected to play prominent role in future quantum computing technology. The problems considered in the proposal have direct connections with the mentioned disciplines. In particular, part of the proposal related to the Toeplitz and Hankel determinants is primary motivated by the needs of the analytical theory of quantum entanglement and by the challenges of analytical description of critical behavior in important stochastic and quantum field models. This part of the proposal is also essential for testing the random matrix predictions in number theory. Proposed study of the Painlev\'e - type universalities and the families of special Painlev\'e functions has direct relation to enumerative topology and string theory. Success in achievement of the proposal's goals will have a notable impact on research in all these areas.