CAREER: Rigidity of Group Actions in Ergodic Theory and Geometry

【摘要】furman该项目的研究内容涉及遍历理论、李群和双曲几何中刚性的各个方面。提出的研究项目的共同主题是分析具有可测量，拓扑或几何结构的空间上的群体行为。更具体地说，在遍历理论的背景下，提出的问题解决了李群中格的可测量作用产生的概率空间上的遍历等价关系的结构和不变量。在双曲几何的背景下，目标是通过基本群上的粗几何观点研究紧流形上的负弯曲黎曼结构。一般来说，该项目的研究部分集中在具有丰富对称性的各种系统的动力学和几何上，以寻找由于这些对称性而发生的特殊现象。该项目的教育部分旨在通过特别设计的数学研讨会和高级主题课程为研究生学习计划做出贡献。目标是通过一系列具有挑战性的问题，让学生参与小规模的“自我发现”项目，鼓励独立思考，并让学生接触到广泛的当前研究。

AbstractFurmanThe research component of the project concerns with various aspects ofrigidity in Ergodic theory, Lie groups and Hyperbolic Geometry. Thecommon theme of the proposed research projects is the analysis of group actions on spaces endowed with a measurable, topological or geometrical structure. More specifically, in the context of ErgodicTheory the proposed problems address the structure and invariants of ergodic equivalence relations on probability spaces generated bymeasurable actions of lattices in Lie groups. In the context ofHyperbolic geometry the goal is to study negatively curved Riemannian structures on compact manifolds through coarse-geometric point ofview on the fundamental group.Generally, the research component of the project is focused on dynamics and geometry of various systems which have a rich groupof symmetries in the search of special phenomena occurring due tothese symmetries. The educational component of the project is aimed to contribute to the graduate studies program via special designed Workshop in Mathematics Seminar and advanced Topic courses. The goal is to engage students in small scale "self discovery" projects via sets of challenging problems, to encourage independent thinking and to expose students to a wide spectrum of current research.