Dynamics of Group Actions on Parameter Spaces

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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.