Riemann-Hilbert Problems, Integrable Systems and Random Matrix Theory

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The PI plans to work on a variety of problems from mathematics, applied mathematics and physics. All the problems under consideration are asymptotic in nature in the sense that the problems depend on a large parameter, such as time or space, or a small parameter, such as perturbation strength. The main issue is to determine the behavior of the systems when the parameter(s) go to infinity, or to zero, respectively. It turns out thatthe problems under consideration have a Riemann-Hilbert representation which provides a non-commutative analog, for these problems, of the integral representations of the classical special functions, such as the Bessel functions or the Airy function, etc. And just as the classical special functions can be analyzed asymptotically by the steepest-descent/stationary phase method, so too the Riemann-Hilbert problems can be analyzed by the non-linear steepest-descent method introduced by the PI and X.Zhou in 1993. Amongst the problems to be considered by the PI and his collaborators are: spiral asymptotics for the modes of lasers with rectangular plane-parallel reflecting surfaces, as the Fresnel number goes to infinity; asymptotics for the Emptiness Formation Probability of the XY spin-1/2 chain, as the anisotropy and field strength vary; perturbation theory of infinite dimensional integrable systems such as the perturbed Nonlinear Schroedinger Equation, in the focusing case when solitons are present. In addition the PI will consider problems in random matrix theory and in the asymptotics of Toeplitz and Hankel determinants. It is a remarkable, and unanticipated, fact that a great variety of problems in mathematics, applied mathematics and physics can be rephrased as Riemann-Hilbert problems. This makes it possible to analyze their behavior with the same efficiency and accuracy as the classical problems, such as electricity and magnetism, of the 19th century. In particular, various random matrix ensembles can be analyzed by Riemann-Hilbert methods. Random matrices in themselves provide models for an extraordinary range of problems, from the scattering of neutrons off heavy nuclei, to the zeros of the Riemann-zeta function on the critical line. In transportation theory, for example, the PI and his collaborators recently showed how the bus system in Cuernevaca, Mexico, could be described by random matrix theory: this bus system has special features and is used in many parts of Latin America. The list of problems that can be modeled by random matrix theory includes combinatorics, multivariate statistics, condition numbers in numerical analysis, tiling problems, interacting particle systems , quantum transport problems and wireless communication, amongst many others. The PI and his collaborators are also involved in writing various texts on Riemann-Hilbert methods and also on random matrix theory that should be accessible to researchers across the scientific spectrum.