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

本项目关注无穷群在流形上的作用，这是一个连接和统一许多数学领域（拓扑学、几何群论、叶理理论和拓扑动力学）的领域。首席研究员（PI）将研究基本的数学对象，通过研究它们的对称性，并通过研究对象在变换下如何变化：这是对群体行为的研究。该项目将描述和区分刚性行为，其中转换系统的小扰动不会定性地改变长期结果，而不稳定或混沌行为，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，从天气模式到洋流，再到机械系统的构型空间，这个问题的变体在我们周围随处可见。当感兴趣的对象高度复杂或不易参数化时，这些问题很难接近，PI的程序集中在几种新技术上，以使刚性问题易于处理。该项目还包括为数学本科生引入主动学习课程，包括一个培训研究生教学方法的项目，以及一个关于在不同研究领域有效交流数学的主要研讨会，目的是增加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，在拓扑动力学的意义上，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最近关于表面上的叶状圆束的结果，其中表明每个刚性的这样的束都是几何的。本项目以该项目的成功为基础，将这一主题扩展到新的环境和应用中。PI将继续研究平面束和刚性单群作用，使用叶理理论和负曲率粗糙几何的技术来研究边界作用。该项目的另一个部分涉及通过圆阶空间的拓扑研究作用在圆上的无限离散群的刚性，这是A. Navas的工作成果。最后，PI将把几何群论中的技术应用于非局部紧群的新背景，将其应用于研究曲面上群作用的动力学，并完善Gromov引入的变换群中的畸变和增长的概念。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。

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.