Mathematical Sciences: Topics in Ergodic Theory and Analysis

贝娄教授和琼斯教授将继续研究遍历理论和更广泛的分析中出现的几乎处处收敛问题。他们将专注于一系列问题。其中最核心的是布尔加恩的熵准则和强扫除性质。他们还将研究涉及薄集合平均的问题，如个别遍历定理中的Banach密度为零的集合，单参数流情况下的离散平均集，两参数流情况下的环空等。他们计划花费一些时间的一个项目涉及可积函数的几乎处处收敛。这是一个困难的领域，但主要调查人员希望利用Calderon-Zygmund分解取得一些进展。这个项目涉及对遍历理论的研究。一般说来，遍历理论涉及理解系统的平均行为，这些系统的动力学太复杂或太混乱，以至于无法在微观细节上进行跟踪。在“动力学”的标题下，可以把一组抽象的变换如何作用于光滑空间的现代理论放在下面。通过这种方式，遍历理论在寻求对均匀空间上的流动进行分类的过程中与几何相联系。

Professor Bellow and Jones will continue their study of almost everywhere convergence questions which arise in ergodic theory and more generally in analysis. They will focus on a number of issues. Central among them are Bourgain's entropy criterion and the strong sweeping out property. They will also study problems involving averaging over thin sets such as sets of Banach density zero in the individual ergodic theorem, discrete averages in the case of a one parameter flow, annuli in the case of a two parameter flow, etc. One project on which they plan to spend some time involves almost everywhere convergence for integrable functions. This is a difficult area but the principal investigators hope to make some progress using Calderon-Zygmund decompositions. This project involves research in ergodic theory. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics" can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In this way ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces.