Aspects of Unipotent Dynamics on Homogenous Spaces

数论和几何中的许多问题在其结构中都涉及多个无限对称群。使用动力系统的思想探索这些对称组之间的相互作用，并将其应用于适当创建的代数模型，是该项目涉及的数学工作的核心。该项目的主要目标是进行相关的新观察，并开发适用于解决各种动力学、数论和几何问题的新技术。这项工作结合了几个不同数学领域的概念，预计该项目将在这些不同学科之间建立概念桥梁。研究生和博士后研究人员将参与该研究项目，为他们在数学的一些前沿领域的工作做好准备。研究者和他的研究生将研究由数论或几何问题引起的均匀空间单能流的广泛动力学问题。该项目研究了四类问题：（1）描述无限体积几何有限双曲流形中单能轨道的闭包和完全测地线浸没； (2) 在适当的丢番图条件下，控制基点上单能轨迹的不发散率或短单能曲线的扩展平移； (3) 求齐次空间中曲线的展开平移在极限内均匀分布的几何条件； (4) 计算非紧齐次空间上的流的具有指定偏移率的点集的豪斯多夫维数。这项研究将结合数学各个领域的技术，包括动力系统、李群、表示论、代数几何、概率论、数论和组合学。该研究旨在获得新成果并开发动力学、数论和组合性质的新技术。

Many questions in number theory and geometry involve multiple infinite groups of symmetries in their structures. Exploring the interactions between these groups of symmetries using ideas from dynamical systems, applied to suitably created algebraic models, is at the core of mathematical work involved in this project. The primary goal of the project is to make new observations that relate, and to develop new techniques applicable for solving, a variety of dynamical, number theoretical, and geometrical problems. The work combines concepts from several different areas of mathematics, and it is anticipated that the project will build conceptual bridges between these different disciplines. Graduate students and post-doctoral researchers will participate in the research project, preparing them to work on some of the forefront areas of mathematics.The investigator and his graduate students will study a wide range of dynamical questions on unipotent flows on homogeneous spaces that arise from number theoretic or geometrical questions. The project investigates four types of problems: (1) Describe closures of unipotent orbits and totally geodesic immersions in infinite-volume geometrically-finite hyperbolic manifolds; (2) Control the non-divergence rate of unipotent trajectories or expanding translates of short unipotent curves under suitable Diophantine conditions on the base point; (3) Find geometric conditions under which expanding translates of a curve in a homogeneous space are uniformly distributed in the limit; (4) Compute Hausdorff dimensions of the sets of points with specified excursion rates for flows on non-compact homogeneous spaces. This study will combine techniques from various areas of mathematics, including dynamical systems, Lie groups, representation theory, algebraic geometry, probability theory, number theory, and combinatorics. The research aims to obtain new results and develop new techniques of dynamical, number theoretical, and combinatorial nature.