Aspects of Unipotent Dynamics on Homogenous Spaces

齐次空间上的单势动力学的各个方面

• 批准号: 1700394
• 负责人: Nimish Shah
• 金额: $ 18.4万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700394&HistoricalAwards=false
• 关键词: Aspects; Unipotent; Dynamics; Homogenous; Spaces

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.

数论和几何中的许多问题都涉及到多个无限对称群的结构。利用动力学系统的思想探索这些对称群之间的相互作用，并将其应用于适当创建的代数模型，是该项目所涉及的数学工作的核心。该项目的主要目标是进行新的观察，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工作做好准备。研究员和他的研究生将研究一系列关于齐次空间上的幂幺流的动力学问题，这些问题来自数论或几何问题。本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幂幺轨道和全测地浸入的闭包;（2）在基点上适当的Diophantine条件下，控制幂幺轨线或幂幺短曲线的扩张平移的不发散率;（3）求齐次空间中曲线的展开平移在极限上均匀分布的几何条件;（4）计算非紧齐性空间上流的具有特定漂移率的点集的Hausdorff维数。这项研究将结合联合收割机技术从各个领域的数学，包括动力系统，李群，表示论，代数几何，概率论，数论和组合数学。该研究旨在获得新的结果，并开发动力学，数论和组合性质的新技术。

项目成果

Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices

• DOI: 10.3934/dcds.2020227
• 发表时间: 2016-06
• 期刊: Discrete & Continuous Dynamical Systems - A
• 作者: N. Shah;Lei Yang

Primitive rational points on expanding horocycles in products of the modular surface with the torus

模曲面与环面乘积中展开四环的本原有理点

• DOI: 10.1017/etds.2020.15
• 发表时间: 2020
• 期刊: Ergodic theory and dynamical systems
• 影响因子: 0.9
• 作者: EINSIEDLER, MANFRED;LUETHI, MANUEL;SHAH, NIMISH A.
• 通讯作者: SHAH, NIMISH A.

Equidistribution of dilated curves on nilmanifolds: EQUIDISTRIBUTION OF DILATED CURVES

尼尔流形上扩张曲线的等分布：EQUIDISTRIBUTION OF DILATED Curves

• DOI: 10.1112/jlms.12156
• 发表时间: 2018
• 期刊: Journal of the London Mathematical Society
• 作者: Kra, Bryna;Shah, Nimish A.;Sun, Wenbo
• 通讯作者: Sun, Wenbo