Dynamics, Topology, and Combinatorics

动力学起源于物理学，用来模拟粒子在连续或离散时间内的空间运动。抽象拓扑动力学将时间的概念从实数和整数群扩展到任意的，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。本项目主要研究其与组合学的密切关系，特别是与拉姆齐理论的关系。每一个拓扑群都有一种自然的表示，即某种数学结构的对称群。结果表明，群的野蛮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，而驯服动力学行为对应于有限数量的接近底层结构的模式。这种关系在一般理论和具体例子层面上都是一个丰富的问题来源。该项目的目的是同时解决这两个问题，同时分离出适合本科生和研究生研究的实例。在很大程度上，拓扑群的动力学行为可以从它的最小流中读出，特别是从射影最大的最小流，即通用最小流中读出。它以前已经通过泛函分析手段进行了研究，而PI则通过超滤波器开发了一种对偶方法，用于离散结构的自同构群，更一般地用于一般拓扑群的近超滤波器。相应的拉姆齐现象也可以直接转化为超滤语言，阐明了动力学与组合学之间的联系。目标是继续研究一般情况下通过（近）超滤波器的普遍最小流，例如什么拓扑空间可以是普遍最小流，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态行为，或者度量结构（允许小误差）是否带来新的现象。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。

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.