Abstract Sinai 9706794 Sinai will work in three directions: 1. Qualitative behavior of solutions of some quasi-linear systems with or without randomness - in particular, pressure-less equations of gas dynamics, the creation of singularities of types of shocks, and concentration of masses on submanifolds of small dimensions; 2. Twist maps with strong statistical properties - in particular, the construction of twist maps with positive Lyapunov exponents; and 3. Problems related to quantum chaos and ensembles of random matrices - in particular, the universality of distributions found earlier by Wigner, Dyson and others. This grant is to fund the study of the long term behavior of dynamical systems and stochastic processes. Many important physical phenomena, as varied as the behavior of the stock market, lines in the supermarket, and data being received from a distant transmission source, exhibit locally erratic behavior which can be understood better by looking at the long term average behavior instead. The study of such stochastic processes and their connections with one another, as well as their connections in terms of their fine structure with other less random processes, is a central one in modern mathematics. It is only through such studies that we will be able to understand better and thus be able to control or predict the evolution of the physical phenomena which this mathematics can model.
