Mathematical Sciences: Ergodic Phenomena in Differential Dynamics

9404437西奈半岛拟议的研究将涉及三个不同的主题。第一个是研究类标准映射的遍历类性质。主要的困难是，在这些情况下，稳定的和不稳定的叶理永远不会相互横切。新的思路和方法来自于双曲台球理论。在第二个主题中，我们将研究具有随机初值和随机系数的拟线性方程。这些方程出现在一些宇宙学问题中。统计性质的分析引出了概率论中新的非常有趣的问题。第三个主题涉及量子混沌理论中的一些数学问题。这是数学物理学的一个新方向，非常有前途。总的来说，我们将讨论混沌理论中的几个基本问题。第一个问题可以被认为是哈密顿动力学中最重要的公开问题，它涉及到纯确定性运动定律中混沌行为的出现。第二种方法处理连续介质的方程，其中通过初始数据将随机性纳入其中。它在宇宙学中有重要的应用。最后一个问题涉及一个新的、困难的话题，称为量子混沌，这是物理学和数学的重大兴趣。***

9404437 Sinai The proposed research will concern three different topics. The first is the study of ergodic like properties of standard-like maps. The main difficulty is due to the fact that in these cases, stable and unstable foliations are never transversal to each other. The new ideas and methods are taken from the theory of hyperbolic billiards. In the second topic, we shall study quasi-linear equations with random initial data and random coefficients. These equations appear in some cosmological problems. The analysis of statistical properties lead to new very interesting problems in probability theory. The third topic deals with some mathematical problems in the theory of quantum chaos. This is a new direction in mathematical physics which can be very promising. In general, we shall deal with several basic problems in the theory of chaos. The first one can be considered the most important open problem in Hamiltonian dynamics and concerns the appearance of chaotic behavior from pure deterministic laws of motion. The second one deals with equations of continuous media where the randomness is incorporated through initial data. It has important applications in cosmology. The last one concerns a new and difficult topic called quantum chaos and is of big interest for physics and mathematics. ***