Applications of Descriptive Set Theory in Ergodic Theory and Smooth Dynamical Systems

物理现象的模型通常被表述为动力系统，即指定数量如何随时间演变的规则。有许多不同类型的动力系统，一个重要的数学问题是对这些系统的统计或定性行为进行分类。这个项目使用描述集合论来展示什么样的分类是可能的，并建立一般分类的数学限制。该项目涉及研究生。动力系统的分类可以通过对紧致流形的同构的统计和定性行为进行分类，并建立判定时间向前与时间向后同构的标准（在适当的类别中）来实现。 这些问题与哪些抽象的保测变换同构于紧致流形的保体积同构密切相关。 该项目研究在什么情况下可以使用固有可数方法进行分类，并探索将抽象变换实现为同构的方法。该奖项反映了NSF的法定使命，通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Models of physical phenomena are often formulated as dynamical systems, rules that specify how quantities evolve in time. There are many different types of dynamical systems, and one important mathematical question is to classify the statistical or qualitative behavior of these systems. This project uses descriptive set theory to show what kinds of classification are possible and to establish the mathematical limitations of general classifications. The project involves graduate students.Classification of dynamical systems can be approached by classifying the statistical and qualitative behavior of diffeomorphisms of compact manifolds and establishing criteria for deciding when time forwards is isomorphic to time backwards (in the appropriate category). These questions are closely related to which abstract measure-preserving transformations are isomorphic to volume-preserving diffeomorphisms of compact manifolds. This project studies when classifications are possible with inherently countable methods and explores methods for realizing abstract transformations as diffeomorphisms.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.