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 that the 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.
Percy A. Deift, Principal Investigator For certain key models in statistical mechanics, important physical quantities can be expressed in terms of Toeplitz determinants. This is true in particular for spontaneous magnetization in the celebrated Ising model in two dimensions. For temperatures below a critical temperature T, spontaneous magnetization is present, but for temperatures above T, magnetization is absent. Precisely at temperature T, the system displays anomalous behavior. These physical properties are reflected mathematically in the properties of the symbol for the Toeplitz determinant: Below the critical temperture, the symbol is regular, and above, and at, the critical temperture, the symbol develops certain singularities. There was a long-standing conjecture for the behavior of Toeplitz determinants whose symbols have very general singularities, of the kind that arise in the Ising model above or at the critical temperture T. The PI et al have verified this conjecture, making it possible to analyse the asymptotic behavior of a wide variety of problems in mathematics and physics, including problems in analytic number theory, statisical mechanics and the asymptotic behavior of the eigenvalues of Toeplitz matrices. In another direction, the PI et al have developed techniques to analyse precisely the long-time behavior of solutions of the initial-boundary value problem for integrable systems such as the Nonlinear Schroedinger Equation. In other work , the PI et al considered an optimization problem that arises in the analysis of Sigma-Delta modulation theory. Sigma-Delta modulation theory is a popular method for analog-to-digital conversion of band-limited signals. By the sampling theorm, signals of bandwidth 2K can be reconstructed from the (precise) values of the signal at sampling times {n/R}, where n runs over the integers and R, the sampling rate, is greater than or equal to 2K. Sigma-Delta modulation theory rests on the observation that if L=R/2K, the oversampling ratio, is very large, then the signal does not need to be sampled so accurately, and in fact very coarse sampling is sufficient to reconstruct the signal: In particular, if L is sufficiently large, it is enough to assign a measurement +1 or -1 to the signal at the times {n/R} (this analog-to-digital "quantization" is known as a one-bit Sigma-Delta modulation scheme). Such schemes were known to reconstruct the signal with errors decaying as 2^(-rL) as the oversampling ratio L becomes large, for some r <1. By solving an associated optimization problem the PI et al where able to devise a one-bit Sigma-Delta scheme which recovers a signal with best known accuracy: In particular they were able to effect a significant increase in the previously best known value of r. Finally, in a set of numerical experiments the PI et al considered the basic problem of the computation of the eigenvalues of a random symmetric matrix. They considered a variety of standard algorihms, and for each algorithm they considered matrices chosen from different random matrix ensembles. The main discovery was that, quite remarkably, for each given algorithm, the fluctuations in the number of steps to compute the eigenvalues of a matrix to any given accuracy, turned out to be universal for all ensembles. This universality is consonant with the universality that Eugene Wigner invoked in the 1950's, when he famously modeled the scattering resonances of nuclei off a heavy atoms by random matrix theory. The computations of the PI et al suggest that universality features should be present far more generally in numerical computation; subsequent computations on various other numerical problems confirm this suggestion.
