Group Actions, Rigidity, and Invariant Measures

群体行动、刚性和不变措施

• 批准号: 2400191
• 负责人: Aaron Brown
• 金额: $ 35.32万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-01 至 2027-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2400191&HistoricalAwards=false
• 关键词: Group; Actions; Rigidity; Invariant; Measures

This project focuses on questions at the interface of dynamical systems and rigidity of group actions. Many mathematical objects admit large groups of symmetries. The structure of such groups may highly constrain the underlying object or properties of the action. Questions across fields of mathematics can often be reformulated as questions about the (non-)fractal nature of invariant geometric structures (particularly sets and measures) for certain group actions. The project will employ tools from the field of dynamical systems to study group actions, with broad aims of classifying actions and the objects on which groups act, classifying certain invariant geometric structures, and showing certain actions do not admit fractal invariant structures. The project will also support the training of PhD students. The project will focus on actions of groups, including higher-rank abelian groups and higher-rank lattices, with an emphasis on classifying actions with certain dynamical properties, classifying the underlying spaces on the group acts, or classifying invariant measures and orbit closures. The project will employ tools from hyperbolic dynamical systems (dynamical systems with positive Lyapunov exponents) with a common theme of studying invariant measures for the action (or certain subgroups). Classifying or ruling out fractal properties of certain invariant measures will produce further rigidity properties of the action including additional invariance of the measure, local homogeneous structures for the action, or dimension constraints on the space.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.

本课题主要研究动力系统的界面问题和群体行为的刚性问题。许多数学对象承认大群的对称性。这种组的结构可能高度约束动作的底层对象或属性。跨越数学领域的问题通常可以被重新表述为关于某些群体行动的不变几何结构(特别是集合和度量)的(非)分形性的问题。该项目将使用动力系统领域的工具来研究群体行为，广泛的目标是对行为和群体所作用的对象进行分类，对某些不变的几何结构进行分类，并表明某些行为不允许使用分形不变结构。该项目还将支持博士生的培训。该项目将侧重于群的作用，包括高阶阿贝尔群和高阶格，重点是对具有某些动力学性质的作用进行分类，对群作用上的基本空间进行分类，或对不变度量和轨道闭合进行分类。该项目将使用来自双曲动力系统(具有正Lyapunov指数的动力系统)的工具，其共同主题是研究作用(或某些子群)的不变度量。分类或排除某些不变度量的分形属性将产生动作的进一步刚性属性，包括度量的附加不变性、动作的局部均匀结构或空间的维度约束。该奖项反映了NSF的法定使命，并已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。