Random matrix theory has had remarkably wide applicability. The spacing distributions arising in random matrix theory have over the past few years been shown to have deep applications in number theory; there are applications in numerical analysis and computational complexity where condition numbers of random matrices are important; random matrix theory has motivated developments in the Riemann-Hilbert method which in turn finds applications to a variety of problems in integrable systems and inverse scattering. In physics the applications range from many-body systems (both atomic and nuclear), to quantum chaos to quantum transport in mesoscopic systems. Four areas for research are specified. The first is related to the fact that in certain random matrix ensembles the measure describing the eigenvalue distribution is the Gibbs measure for charges interacting via a potential at inverse temperature beta equal to one, two or four (corresponding to orthogonal, unitary and symplectic ensembles, respectively). The limiting spacing distributions for these ensembles are now quite well understood but the methods are applicable to these values of beta only. The question for general beta, while quite difficult, is mathematically interesting and quite important in statistical physics. A new approach looks promising and we intend to pursue it. The second area of research is the question of universality of the limiting distribution of the largest eigenvalue in matrix ensembles. This would be analogous to the universality of the Gaussian distribution for sums of independent random variables, the famous Central Limit Theorem. Thirdly, we propose to study the order statistics of the spacings between eigenvalues (which is different from the spacing distributions between consecutive eigenvalues mentioned above). For example, what is the probability distribution for the largest or smallest spacing? There are known results for independent random variables but none yet for for random matrices, whose eigenvalues are far from independent. Finally, we expect to complete earlier work on the asymptotics of solutions to the periodic Toda equations by determining the asymptotics on the so-called critical curves, where the asymptotics will take a very different form. The theory of Wiener-Hopf operators and operator determinants should play a decisive role in this investigation. No doubt the pursuit of these four questions will lead to others.
In the 1950s Eugene Wigner, in his now classic study of highly excited states of large nuclei of atoms, introduced a mathematical model to describe the spacing between these states. This model goes under the name of random matrix theory. Since Wigner's work in nuclear physics, it has been shown that the mathematics of random matrix theory has far-reaching applications to condensed matter physics, atomic physics and the new area of quantum chaos. In mathematics itself, random matrix theory has begun to appear in such diverse areas as number theory, combinatorics and numerical analysis. It is natural to ask why there is such wide applicability of random matrix theory. In probability theory the bell-shaped curve is widely applicable because of a theorem which says roughly that when one adds quantities which are random and independent, the sum follows the bell shaped curve regardless of the distribution of the random objects being added. The distribution functions of random matrix theory appear to have a similar universality for a class of problems where there is a high degree of dependence in the underlying processes. In the present project the mathematics of random matrix theory will be further developed with an eye kept on possible applications. In earlier work a general mathematical framework was developed that related the distribution functions of random matrix theory with solutions to certain equations which are said to be integrable. This mathematical theory gives exact formulas for distribution functions in random matrix theory and provides efficient numerical methods for their computation. Computing these distribution functions will allow one to compare them with experimental data.
