Measured Group Theory and Dynamics

测量群论和动力学

• 批准号: 2005493
• 负责人: Alexander Furman
• 金额: $ 45.39万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005493&HistoricalAwards=false
• 关键词: Measured; Group; Theory; Dynamics

Groups are algebraic objects that formally describe the concepts of symmetry. They play a central role in all branches of mathematics. This NSF award supports research and education that contain several directions to study Dynamical Systems in the presence of large groups. These include dynamical properties in situations with many explicit or implicit symmetries. Such systems tend to attain additional special properties. Conversely, some dynamical properties can often serve as indications of rich underlying symmetries. Educational impact of the project is through Ph.D. supervision, dissemination efforts, and high school outreach.This research project will encompass a number of topics at the intersection of dynamical systems, group theory, geometry and operator algebras. The common theme of the research directions lies in the synthesis and interactions between these areas. Many of the aspects of these interactions involve rigidity, a phenomenon of robustness of a structure under perturbations. Specific projects are related to special phenomena in dynamical systems that occur in the presence of large groups, such as explicit or implicit symmetries, rigidity of certain exotic geometries, and applications of dynamical methods to geometry and analysis. The three main topics under consideration are: measure equivalence rigidity, the Lyapunov spectrum for random walks and generalizations, and the length spectrum for Gromov hyperbolic groups. One focus of this research is the application of dynamical methods in the study of group theory and geometry, and especially rigidity phenomena. The other focus will be the application of group theory to shed light on problems in dynamical systems and geometry.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.

群是形式上描述对称概念的代数对象。它们在数学的所有分支中都发挥着核心作用。该NSF奖支持包含在大型团队中学习动态系统的几个方向的研究和教育。这包括具有许多显式或隐式对称性的情况下的动力学性质。这样的系统往往会获得额外的特殊性质。相反，一些动力学性质通常可以作为丰富的潜在对称性的指示。该项目的教育影响是通过博士指导、传播努力和高中出游。本研究项目将涵盖动力系统、群论、几何和算子代数的许多主题。这些研究方向的共同主题在于这些领域之间的综合和互动。这些相互作用的许多方面都涉及刚性，这是一种结构在扰动下的稳健性现象。具体项目涉及动力系统中存在大群时的特殊现象，如显式或隐式对称性、某些奇异几何的刚性以及动力学方法在几何和分析中的应用。所考虑的三个主要问题是：度量等价刚性，随机游动和推广的Lyapunov谱，以及Gromov双曲群的长度谱。本研究的重点之一是动力学方法在群论和几何研究中的应用，特别是刚性现象的研究。另一个重点将是应用群论来阐明动力系统和几何中的问题。这一奖项反映了NSF的法定使命，并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。