Dynamics of Group Actions on Parameter Spaces

这个项目是动力学、几何、数论、概率论和李群理论的交叉。它主要研究几何对象参数空间上群体作用的动力学。研究的重点是欧几里得空间中的格空间和高格曲面上的平面结构。这些项目的灵感来自于比较和对比这两种类型的动态和几何结构。对于PI和许多其他研究人员来说，这是一种富有成效的方法，他们发现一种情况下的结果可以转化为另一种情况，并且这个过程可以产生对两组问题的见解。另一个常见的线索是使用概率语言和机制；特别是使用随机行走来建模和研究这些空间中的各种流动，并试图理解参数空间中“随机对象”的属性。人们希望这些项目能够激发跨学科的合作，并揭示看似不同的数学领域之间更深层次的联系。其中一些问题可以用非常简单、易于理解的形式表述，并为现代数学的几个重要领域提供了一个入口，包括动力系统、几何和数论。因此，这些问题提供了几个层次的指导机会，从博士后到博士生，到本科生，到高中生和教师。平面的研究对相互作用粒子的动力学和其他重要的物理系统具有重要的具体应用。晶格及其变形的研究对于理解晶体的结构、堆积问题和材料科学都是非常重要的。

This project lies at the intersection of dynamics, geometry, number theory, probability theory, and the theory of Lie groups. It centers on the dynamics of group actions on parameter spaces of geometric objects. The particular focuses of the study are the space of lattices in Euclidean space and flat structures on higher-genus surfaces. These projects are inspired by comparing and contrasting these two types of dynamics and geometry. This has been a fruitful approach for the PI and many other researchers, who have found that results in one setting can be translated to the other, and that this process yields insights to both sets of problems. Another common thread is the use of probabilistic language and machinery; in particular using random walks to model and study various flows on these spaces, and also attempting to understand the properties of a `random object' in a parameter space.It is hoped that these projects will instigate interdisciplinary collaborations and uncover deeper connections between seemingly disparate areas of mathematics. Several of the problems can be stated in very simple, accessible form, and offer a gateway to several important areas of modern mathematics, including dynamical systems, geometry, and number theory. As such, these problems provide mentoring opportunities at several levels, from postdoctoral fellows, to PhD students, to undergraduates, to high school students and teachers. The study of flat surfaces has significant concrete applications to the dynamics of interacting particles, and other important physical systems. The study of lattices and their deformations is important to understanding the structure of crystals, packing problems, and in materials sciences.