Aspects of Polish group dynamics

该项目将主要关注某些数学对象的对称群。这类对象的两个例子是：具有度量的空间和一般广义曲线。对这些物体及其对称性的兴趣分别起源于经典力学和几何拓扑，但它们将使用来自各种数学领域的方法进行研究。这些领域的相互作用具有独立的数学意义。该项目涉及研究生培训。该项目将解决描述性集合理论的边界问题，或者更广泛地说，逻辑，遍历理论和拓扑。本文将对保测度变换产生的闭群进行精确分析。本文将研究一类正则连续体的同胚群的拓扑动力学。与一类n维单纯形映射的Ramsey理论相关的组合性质将被探讨。这些不同的主题将由一个来自数理逻辑的观点联系在一起。该奖项反映了美国国家科学基金会的法定使命，并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。

The project will mostly focus on symmetry groups of certain mathematical objects. Two examples of such objects are: a space with a measure and a generic generalized curve. The interest in these objects and their symmetries originates in classical mechanics and in geometric topology, respectively, but they will be investigated using methods coming from a variety of mathematical fields. The interaction of these fields is of independent mathematical interest. The project involves graduate student training.The project will tackle problems on the boundary of Descriptive Set Theory, or more broadly, Logic, Ergodic Theory, and Topology. A precise analysis will be made of closed groups generated by measure preserving transformations. Topological dynamics of the homeomorphism group of a certain canonical continuum, called the pseudoarc, will be investigated. Combinatorial properties related to Ramsey theory of a class of maps of $n$-dimensional simplex will be explored. These diverse themes will be tied together by a point of view coming from Mathematical Logic.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.