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

这个项目涉及无限群在流形上的作用，这是一个连接和联合许多数学领域(拓扑学、几何群论、叶理论和拓扑动力学)的区域。首席调查员(PI)将通过研究基本的数学对象的对称性和对象在变换下的变化来研究基本的数学对象：这是对群体行为的研究。该项目将描述和区分刚性行为，其中对变换系统的微小扰动不会定性地改变长期结果，而不稳定或混乱的行为对扰动高度敏感。随着我们试图了解真实世界对象的数学模型的长期行为，从天气模式到洋流，再到机械系统的配置空间，这个问题的变化在我们周围随处可见。当感兴趣的对象高度复杂或不容易参数化时，这些问题很难解决，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计划，以及一个关于在不同研究领域有效交流数学的大型研讨会，目的是加强不同领域的数学家之间的交流。PI工作中的一个主要指导原则是，从拓扑动力学的意义上讲，群体行动的刚性往往是潜在几何结构的结果。这种现象的一个例子是PI最近关于曲面上的分叶圆丛的结果，其中证明了每个刚性的这样的丛都是几何的。这个项目建立在这个项目的成功基础上，将这个主题扩展到新的背景和应用程序。PI将继续研究平面丛和刚性单元群作用，使用叶化理论和负曲率中的粗略几何技术来研究边界作用。这个项目的另一个方面是在A.Navas的工作之后，通过圆序空间的拓扑来研究作用在圆上的无限离散群的刚性。最后，PI将把几何群论的技术应用到非局部紧致群的新环境中，并将其应用于研究曲面上的群作用的动力学，并提炼Gromov引入的变换群中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优势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.