Groups in Geometry and Topology

摘要奖：DMS 1604241，首席研究员：Michael Kapovich群自然地以几何和物理对象的对称性出现，如墙壁图案、矿物、雪花，最终是整个宇宙。本课题研究群的代数性质与群表现为对称的空间几何之间的关系。该项目的另一部分涉及机械装置的几何形状，如杆和关节机构和轮系。该项目将涉及对这类设备的严格数学定义和对设备可能的运动空间(构型空间)的分析。本项目是主要研究人员在几何、拓扑、几何群论和机械设备理论领域的研究的继续。计划研究的主题围绕着各种空间上的群作用几何、建筑物几何、代数几何、双曲几何和几何群论的相互作用。该项目的另一部分涉及齿轮传动等机械设备的几何形状。本文的目的是给出这类装置的数学定义，将机械连杆机构的定义推广为有限度量图，并证明了这类装置的位形空间的普适性定理，即任何代数偏微分关系都可以实现为某一装置的位形空间。

AbstractAward: DMS 1604241, Principal Investigator: Michael KapovichGroups appear naturally as symmetries of geometric and physical objects, like wall-patterns, minerals, snowflakes and, ultimately, the entire universe. This project studies relation between algebraic properties of groups and geometry of spaces for which groups appear as symmetries. Another part of the project deals with geometry of mechanical devices such as bar-and-joint mechanisms and gear trains. The project will involve rigorous mathematical definitions of such devices and analysis of possible motion spaces (configuration spaces) of the devices.This project is a continuation of the principal investigator's research of previous years in the areas of geometry, topology, geometric group theory and theory of mechanical devices. The subjects of the planned research revolve around geometry of group actions on various spaces, geometry of buildings, interactions of algebraic geometry, hyperbolic geometry and geometric group theory. Another part of the project deals with geometry of mechanical devices such as gear trains. The goal here is to give mathematical definition of such devices, generalizing the definition of a mechanical linkage as a finite metric graph and proving a universality theorem for configuration spaces of these devices, namely that any algebraic partial differential relation can be realized as the configuration space of some device.