This research project develops new methods involving completely integrable differential equations (of the Painleve type), operator theory and asymptotic analysis to study the level spacing distributions for various random matrix ensembles. The level spacing distribution for various orthogonal polynomial ensembles can be expressed in terms of a Fredholm determinant. The methods of the PI and his collaborator Harold Widom show that this determinant is also a tau-function. In the general problem that will be analyzed the tau function has two distinct types of variables. There are the ``KP type variables'' which describe a KP flow on the Sato Grassmannian and there are the ``deformation type variables.'' It is the combination and interplay between these two types of variables that give the general tau-function. Particular examples of this general tau function are important in disordered conductors, numerical analysis associated with the condition number of a matrix, in addition to the standard applications of random matrix ensembles to the analysis of eigenvalue statistics. The methods developed in this program will give detailed and explicit formulas for these important cases. %%% Complex systems that arise in nuclear physics, atomic and molecular physics, condensed matter physics dealing with media with impurities require a type of mathematics that gives results for averaged quantities, since on the macroscopic everyday world the variables that are controlled in various experiments (like concentration of an impurity, the energy of the system) do not completely specify the microscopic system. Therefore one develops statistical methods that predict average behavior or give the probabilities of deviation from this average behavior. The subject of random matrices is one such statistical theory that has been successfully applied to the above problems in physics along with a host of more theoretical problems in mathematics and practical problems in numerical analysis. This research project develops new methods using the tools of differential equations to study these statistical theories.
