Measured Group Theory and Dynamics

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起着核心作用。这个NSF奖支持研究和教育，其中包含几个方向，研究动态系统中存在的大型团体。这些包括具有许多显式或隐式对称性的情况下的动力学性质。这样的系统往往获得额外的特殊性能。相反，某些动力学性质常常可以作为丰富的基本对称性的指示。 该项目的教育影响是通过博士学位。本研究计画将涵盖动力系统、群论、几何学与算子代数等交叉领域的许多主题。研究方向的共同主题在于这些领域之间的综合和相互作用。这些相互作用的许多方面都涉及刚性，即结构在扰动下的鲁棒性现象。具体的项目涉及到动力系统中的特殊现象，这些现象发生在大群体的存在下，例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。正在考虑的三个主要议题是：措施等价刚性，李雅普诺夫谱随机游动和推广，和Gromov双曲群的长度谱。本研究的一个重点是在群论和几何，特别是刚性现象的研究中应用动力学方法。另一个重点将是应用群论阐明动力系统和几何学中的问题。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.