CAREER: Geometric and Topological Approaches to Group Actions in Low Dimensions

该项目涉及流形上的无限群的作用，流形是一个连接和统一许多数学领域（拓扑、几何群论、叶状理论和拓扑动力学）的领域。 首席研究员（PI）将通过研究基本数学对象的对称性以及研究对象在变换下如何变化来研究基本数学对象：这是对群体行为的研究。 该项目将描述和区分刚性行为，即对转换系统的小扰动不会从质上改变长期结果，而不稳定或混乱的行为对扰动高度敏感。 当我们试图理解现实世界物体的数学模型的长期行为时，这个问题在我们周围出现，从天气模式到洋流，再到机械系统的配置空间。 当感兴趣的对象高度复杂或不容易参数化时，这些问题就很难解决，PI 的项目集中在几种新技术上，以解决刚性问题。 该项目还涉及为数学本科生引入主动学习课程，包括研究生教学方法培训计划，以及跨研究领域有效交流数学的大型研讨会，旨在加强不同领域数学家之间的交流。 PI 工作的一个主要指导原则是，在拓扑动力学意义上，群体行为的刚性通常是底层几何结构的结果。这种现象的一个例子是 PI 最近对表面上的叶状圆束的研究结果，其中表明每个刚性的此类束都是几何的。该项目建立在该计划的成功基础上，将该主题扩展到新的环境和应用程序。 PI 将继续研究扁平束和刚性单向群作用，使用叶状理论和负曲率中的粗几何技术来研究边界作用。 该项目的另一部分涉及通过圆阶空间拓扑研究作用于圆的无限离散群的刚性，这是继 A. Navas 的工作之后。 最后，PI 将把几何群论的技术应用到非局部紧群的新背景中，应用该技术来研究曲面上群作用的动态，并完善 Gromov 引入的变换群的扭曲和增长的概念。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project concerns actions of infinite groups on manifolds, an area bridging and uniting many areas of mathematics (topology, geometric group theory, foliation theory, and topological dynamics). The Principal Investigator (PI) will study basic mathematical objects by studying their symmetries, and by studying how the objects change under transformations: this is the study of group actions. The project will describe and distinguish rigid behavior, where small perturbations to a system of transformations do not qualitatively change the long-term outcome, versus unstable or chaotic behavior, which is highly sensitive to perturbation. Variations on this problem arise all around us, as we seek to understand the long-term behavior of mathematical models of real world objects ranging from weather patterns, to ocean currents, to the configuration space of a mechanical system. When the objects of interest are highly complex or not easily parametrized, these problems are difficult to approach, and the PI's program centers on several new techniques to render rigidity problems tractable. The project also involves the introduction of active-learning courses for undergraduate mathematics students, including a program to train graduate students in teaching methods, and a major workshop on effectively communicating mathematics across research areas with the aim of increasing communication between mathematicians in disparate fields. A major guiding principle in the PI's work is that rigidity of group actions, in the sense of topological dynamics, is often the result of an underlying geometric structure. One example of this phenomenon is the PI's recent results on foliated circle bundles over surfaces, where it is shown that every rigid such bundle is geometric. This project builds on this success of this program, extending this theme to new contexts and applications. The PI will continue work on flat bundles and rigid monodromy group actions, using techniques from foliation theory and coarse geometry in negative curvature to study boundary actions. Another strand of this project involves an investigation of rigidity of infinite discrete groups acting on the circle through the topology of spaces of circular orders, following work of A. Navas. Finally, the PI will adapt techniques from geometric group theory to the new context of non-locally compact groups, applying this this to study the dynamics of group actions on surfaces, and refining the notion of distortion and growth in transformation groups introduced by Gromov.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.