Groups in Geometry and Topology

AbstractAward：DMS 1604241，首席研究员：Michael Kapovich群自然地表现为几何和物理对象的对称性，如墙壁图案，矿物，雪花，最终是整个宇宙。 本计画研究群的代数性质与群呈现对称性的空间几何之间的关系。该项目的另一部分涉及机械设备的几何形状，如杆和关节机构和齿轮系。该项目将涉及此类装置的严格数学定义和装置可能的运动空间（配置空间）的分析。该项目是首席研究员前几年在几何学、拓扑学、几何群论和机械装置理论领域研究的延续。计划研究的主题围绕几何群体行动的各种空间，建筑物的几何，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系等机械设备的几何形状。 这里的目标是给这样的设备的数学定义，推广的定义作为一个有限的度量图的机械连杆和证明的普遍性定理，这些设备的配置空间，即任何代数偏微分关系可以实现为一些设备的配置空间。

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.