Dynamics on Translation Surfaces and on Spaces of Translation Surfaces

动力系统领域研究闭合系统中的点，如行星运动，随着时间的推移在该系统内移动的方式。遍历理论使用度量的概念来研究动力系统，度量是平面上集合的面积或空间中集合的体积的推广。随着时间的推移，遍历理论已经发展出与概率论、数论、物理学和其他领域的联系。使用遍历理论的工具，这个项目计划研究连接到由在无摩擦多边形中运动的点质量组成的系统的动力学系统，其中假设点质量在撞击侧面时遵循弹性碰撞定律。它们还与数学的其他领域相联系，包括几何拓扑学和代数几何。该项目还将涉及这一领域的研究生培训。该项目旨在提高我们对地层平移面上的角环流和“真正的REL”流以及平移面上的平移流本身的理解。这些系统是相互关联的，对一个系统的研究为另一个系统的研究提供了工具和问题。主要研究人员将研究这些系统的基本性质，包括点的行为、不变测度集的结构和混合性质。此外，还将考虑一些由几何拓扑学和几何群论引发的问题。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。

The field of dynamical systems studies the way points in a closed system, such as planetary motion, travel within that system over time. Ergodic theory uses the notion of measure, which is a generalization of area for sets in the plane or volume for sets in space, to study dynamical systems. Over time, ergodic theory has developed connections to probability theory, number theory, physics and other areas. Using tools from ergodic theory, this project plans to study dynamical systems that are connected to systems consisting of a point mass traveling in a frictionless polygon, where the point mass is assumed to obey the laws of elastic collision when hitting the side. They are also connected to other areas of mathematics including geometric topology and algebraic geometry. This project will also involve the training of graduate students in this area. This project seeks to improve our understanding of the horocycle flows and "Real rel" flows on strata translation surfaces and translation flows on translation surfaces themselves. These systems are related, and the study of one informs the study of the other with tools and questions. The principal investigator will study the basic properties of these systems, including the behavior of points, the structure of the set of invariant measures and mixing properties. Additionally, some questions motivated by geometric topology and geometric group theory will be considered.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.