Riemann-Hilbert Problems, Integrable Systems and Random Matrix Theory

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The PI plans to work on a variety of problems, particularly asymptotic problems, in the general area of integrable systems. Two key tools will play a prominent role in the PI's work: Riemann-Hilbert Problems (RHP) and the non-linear, non-commutative steepest descent method, on the one hand, and Random Matrix Theory (RMT) on the other. RHPs can be viewed as the natural, non-commutative generalization of the scalar integral representations of the classical special functions, and just as the classical steepest descent method allows one to evaluate the asymptotic behavior of the classical functions, so too the non-linear, non-commutative steepest descent method allows one to evaluate the asymptotic behavior of modern integrable systems such as the KdV equation and the Painleve equations, amongst many others. RMT, and in particular, the eigenvalues of random matrices, provide an extremely versatile tool to describe the behavior of a wide variety of stochastic phenomena when the underlying variables are no longer independent. Amongst the problems that the PI will address with these tools are: the asymptotic behavior of the modes of a laser as the Fresnel number goes to infinity; the behavior of random n-band matrices of size N as n,N go to infinity; universality for general orthogonal and symplectic matrix ensembles; the perturbation theory of the focusing Non-Linear Schroedinger Equation on the line.It is an extraordinary fact that the eigenvalues of random matrices provide a quantitative model for a wide variety of statistical phenomena from physics, pure and applied mathematics, and engineering. Applications of RMT range from the analysis of resonances in nuclear scattering theory, to the analysis of the zeroes of the Riemann zeta function on the critical line, in analytic number theory.Quite spectacularly, and surprisingly, even the arrival times between buses in the Mexican city of Cuernavaca, have been shown to obey random matrix theory statistics! RMT as a discipline has two components: the development of random matrix theory per se, and the application of the theory to new physical and mathematical problems. The work of the PI will impact both components. In the theory of integrable systems, two of the problems in particular proposed by the PI, arise directly from technology, viz., laser mode asymptotics, and the effect of perturbations on fiber optical transmission systems. The PI and his collaborators are also writing basic texts on RHPs and RMT aimed at the researchers as well as advanced graduate students entering the field.