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

这个项目关注流形上无限群的作用，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。 主要研究者（PI）将通过研究它们的对称性来研究基本的数学对象，并通过研究对象在变换下如何变化：这是对群作用的研究。 该项目将描述和区分刚性行为，其中对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。 当我们试图理解真实的世界物体的数学模型的长期行为时，从天气模式到洋流，再到机械系统的配置空间，这个问题的变化就出现在我们周围。 当感兴趣的对象是高度复杂的或不容易参数化，这些问题是很难接近，和PI的程序集中在几个新的技术，使刚性问题易于处理。 该项目还涉及为本科数学学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，以及一个关于跨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工作中的一个主要指导原则是，在拓扑动力学的意义上，群体作用的刚性通常是潜在几何结构的结果。这种现象的一个例子是PI最近的结果在表面上的叶状圆丛，它表明，每一个刚性这样的丛是几何的。该项目建立在该计划的成功基础上，将这一主题扩展到新的背景和应用。PI将继续研究扁平束和刚性单值群作用，使用叶理理论和负曲率下的粗糙几何来研究边界作用。 另一股这个项目涉及刚性的调查无限离散群体作用于圆通过拓扑空间的循环秩序，以下工作的A。纳瓦斯 最后，PI将调整技术从几何群论到非局部紧群的新背景下，应用这一点来研究表面上的群作用的动力学，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进行评估，被认为值得支持的搜索.

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.