Mathematical Sciences: Ergodic Phenomena in Differential Dynamics

摘要西奈9706794 西奈半岛将在三个方向工作：1。定性行为的解决方案的一些准线性系统有或没有随机性-特别是，无压力方程的气体动力学，创造奇异性的类型的冲击，并集中质量的子流形上的小尺寸; 2。具有强统计特性的扭曲映射-特别是具有正李雅普诺夫指数的扭曲映射的构造;以及3.与量子混沌和随机矩阵合奏相关的问题-特别是维格纳，戴森和其他人发现的分布的普遍性。 该补助金用于资助动力系统和随机过程的长期行为研究。 许多重要的物理现象，如股票市场的行为，超市中的排队，以及从远处传输源接收的数据，都表现出局部不稳定的行为，通过观察长期平均行为可以更好地理解。 研究这样的随机过程及其相互之间的联系，以及它们与其他不那么随机的过程在精细结构方面的联系，是现代数学的核心。 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这种数学模型所能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.