Mathematical Sciences: The Large-Scale Geometry of Groups, and Spaces with Nonpositive Curvature

提出的研究是群论、微分几何、动力学和拓扑学之间的界面，源于对负曲面的研究。更具体地说，研究者对离散群与其Cayley图的大规模结构之间的关系感兴趣，并将这些关系应用于涉及负曲面的几何和动力学问题。在数学、物理和化学中，群经常被用来描述各种对称性质。离散群是有限的，因此，对象可以相当明确地写下来。弯曲空间的基本群是承载空间大尺度对称性信息的群；这个群体在很多情况下是离散的。

9626911 Kleiner The proposed research lies in the interface amongst group theory, differential geometry, dynamics and topology, stemming from the study of negatively curved surfaces. More specifically, the investigator is interested in relations between a discrete group and the large scale structure of its Cayley graphs, and applications of these to geometric and dynamical problems involving negatively curved surfaces. In mathematics as well as in physics and chemistry, groups are often used to describe various symmetry properties. Discrete groups are finite, hence, objects that can be written down rather explicitly. The fundamental group of a curved space is a group carrying information about large scale symmetries of the space; this group turns out to be discrete in many cases.