CAREER: Universal Circles Between Dynamics and Geometry

动力系统的数学概念允许在一个统一的框架中考虑每个时变的物理系统，作为组织可能的瞬时配置海洋的“状态空间”，以及描述状态随时间演变的“流”。从这个角度来看，捕食者-猎物系统与摆动的钟摆有着惊人的相似之处：两者都有二维状态空间，其轴对应于捕食者和猎物的总数，或者对应于钟摆点的位置和速度。这说明了数学动力学的统一力量，这是这个项目的主要焦点。PI的研究项目将把低维相空间的几何与它们所支持的系统的动力学联系起来，着眼于几何和动力学的应用。与此同时，PI的教育计划将把数学动力学的工具和观点带给STEM学科的广泛受众。这个项目的研究部分是关于3流形上的拟椭球和拟阿诺索夫流。这些流动产生了较低维度的结构——流动空间、叶片空间和通用圈——它们将流动的大规模动态与环境流形基本群的作用交织在一起，并充当底层动力学和几何之间的管道。PI的目的是表明，人们可以使用准等速流的万向圆作用来重建整个流形，连同原始流的伪anosov代表，并表征出现为万向圆的圆上的群作用。此外，PI的目标是使用拟椭球流作为工具来探测3流形的结构，并在Cannon猜想和理解Thurston范数球的面方面具有潜在的应用。在教育方面，PI提出了他所谓的国家空间项目，这是一个跨学科的工作小组（包括来自工程、化学、生物、物理和统计学的教师），包括研究生课程和研讨会。PI还计划继续开展更广泛的影响活动，包括学生指导、课程开发、参与数学圈和编辑服务。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。

The mathematical notion of a dynamical system allows for every time-varying physical system to be considered in a uniform framework, as a "state space" that organizes the sea of possible instantaneous configurations, together with a "flow" that describes the evolution of states with time. From this viewpoint a predator-prey system bears a striking resemblance to a swinging pendulum: both have 2-dimensional state spaces whose axes correspond to either the total predator and prey populations or to the position and velocity of the pendulum point. This illustrates the unifying power of mathematical dynamics, which is the primary focus of this project. The PI's research program will connect the geometry of low-dimensional phase spaces with the dynamics of the systems that they support, with an eye towards applications in both geometry and dynamics. At the same time, the PI's educational program will bring the tools and viewpoint of mathematical dynamics to a wide audience across the STEM disciplines. The research component of this project is concerned with quasigeodesic and pseudo-Anosov flows on 3-manifolds. These flows give rise to lower-dimensional structures — flowspaces, leaf spaces, and universal circles — that entwine the large-scale dynamics of the flow with an action of the ambient manifold’s fundamental group, and act as a conduit between the underlying dynamics and geometry. The PI aims to show that one can use the universal circle action of a quasigeodesic flow to reconstruct the entire manifold, together with a pseudo-Anosov representative of the original flow, and characterize the group actions on circles that appear as universal circles. In addition the PI aims to use quasigeodesic flows as tools to probe the structure of 3-manifolds with potential applications towards the Cannon Conjecture and understanding the faces of the Thurston norm ball. For the educational component the PI proposes what he calls the State Space Project, an interdisciplinary working group (which includes faculty from engineering, chemistry, biology, physics and statistics) which includes graduate courses and a seminar. The PI also plans to continue broader impact activities related to student mentoring, curriculum development, involvement in Math Circles and editorial service.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.