Mathematical Sciences: Ergodic Phenomena in Differential Dynamics

摘要西奈9706794西奈将在三个方向上工作：一些具有或不具有随机性的拟线性系统解的定性行为——特别是气体动力学的无压力方程，各种冲击奇点的产生，以及小维子流形上质量的集中2. 具有强统计性质的扭转映射——特别是具有正Lyapunov指数的扭转映射的构造和3。与量子混沌和随机矩阵集合有关的问题——特别是维格纳、戴森和其他人早先发现的分布的普适性。该基金用于研究动力系统和随机过程的长期行为。许多重要的物理现象，如股票市场的行为、超市里的排队、从远距离传输源接收的数据等，都表现出局部的不稳定行为，而通过观察长期平均行为可以更好地理解这种不稳定行为。研究这些随机过程及其相互之间的联系，以及它们的精细结构与其他不那么随机的过程之间的联系，是现代数学的一个核心问题。只有通过这样的研究，我们才能更好地理解，从而能够控制或预测数学可以模拟的物理现象的演变。

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.