Dynamics, Topology, and Combinatorics

动力学起源于物理学，用于模拟粒子在连续或离散时间内在空间中的运动。抽象拓扑动力学将时间的概念从真实的数和整数的群扩展到任意的、可能是无限维的拓扑群，并将空间限制为有界的并包含其所有极限点。抽象拓扑动力学包含了许多数学领域的方法和应用。本项目主要研究它与组合数学，特别是与Ramsey理论的密切关系。每一个拓扑群都有一个自然表示，它是某种数学结构的对称群。事实证明，群的狂野动力学行为对应于相应结构的混沌行为，即缺乏拉姆齐现象，而更驯服的动力学行为对应于近似undelying结构的有限数量的模式。这一关系在一般理论和具体实例两个层面上都是问题的丰富来源。该项目的目的是解决这两个手牵手，同时隔离适合本科生和研究生研究的情况。 在很大程度上，拓扑群的动力学行为可以从它的最小流，特别是从射影最大流，泛最小流中读出。它以前曾被研究的功能分析手段，而PI开发了一个双重的方法，通过超滤子，为群体的自同构离散结构，更普遍的近超滤子一般拓扑群。相应的拉姆齐现象也可以直接翻译成超滤语言，并澄清了动力学和组合学之间的联系。我们的目标是通过（近）超滤子继续研究泛极小流，例如什么拓扑空间可以是泛极小流，以及引导理论方向的群和底层结构的具体例子，例如，离散结构的自同构群是否表现出所有可能的动力学行为，该奖项反映了NSF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。

Dynamics originated in physics to model movement of particles in a space in continuous or discrete time. Abstract topological dynamics extends the notion of time from the groups of real numbers and integers to arbitrary, possibly infinite-dimensional, topological groups, and restricts the space to be bounded and to contain all its limit points. Abstract topological dynamics encompasses methods from and applications into numerous areas of mathematics. This project mainly investigates its intimate relationship with combinatorics, in particular, Ramsey theory. Every topological group admits a natural representation as a group of symmetries of some mathematical structure. It turns out that wild dynamical behaviour of the group corresponds to chaotic behaviour of the corresponding structure, that is, lack of Ramsey phenomena, and tamer dynamical behaviour corresponds to a bounded number of patterns approximating the undelying structure. This relationship is a rich source of questions both on the level of general theory and concrete examples. The intent of the project is to address both hand in hand while isolating instances suitable for undergraduate and graduate research. To a great extent, dynamical behaviour of a topological group can be read from its minimal flows, in particular from the projectively largest one, the universal minimal flow. It has previously been studied by functional analytical means, whereas the PI developed a dual approach via ultrafilters, for groups of automorphisms of discrete structure, and more generally near ultrafilters for general topological groups. The corresponding Ramsey phenomena have a direct translation into the ultrafilter language as well and clarify the connection between dynamics and combinatorics. The goal is to continue studying universal minimal flows via (near) ultrafilters in general, such as what topological spaces can be the universal minimal flows, as well as concrete examples of groups and underlying structures that lead the direction of the theory, for example, whether groups of automorphisms of discrete structures exhibit all possible dynamical behaviour, or whether metric structure (allowing for small error) bring in new phenomena.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.