CAREER: Topics at the Intersection of Geometry, Topology and Group Theory

职业：几何、拓扑和群论交叉的主题

• 批准号: 9984815
• 负责人: Benson Farb
• 金额: $ 20万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9984815&HistoricalAwards=false
• 关键词: CAREER; Topics; Intersection; Geometry; Topology

Farb will continue his joint work with L. Mosher on understanding the asymptotic geometry of groups. Every finitely-generated group can be viewed as a metric space in a way which is unique up to maps of bounded distortion. It was recently discovered that many finitely-generated groups are actually determined uniquely by their asymptotic geometry. Farb plans to continue his work on thisphenomenon and how it relates to the Mostow Rigidity Theorem, Gromov's Polynomial Growth Theorem, and the theory of dynamical systems. Farb will also develop and teach a geometry component in Chicago's SESAME program. This program teaches mathematics and science to elementaryteachers from the Chicago public school system, and provides them with curriculum tools they can take back to their schools for use in their classrooms.Groups are the formal mathematical way to describe symmetry. What symmetries can a space exhibit? How do the symmetries of a space relate to its geometry, and to the way that geodesics (light-rays) travel in that space? These are fundamental issues that occur throughout nature. The mathematical underpinnings of the theory of symmetry are important not only in understanding known phenomena, but in discovering new ones. One example is the recently discovered phenomenon that the coarse, large-scale geometric structure of a space can often determine the symmetries it can have.

Farb将继续与L.关于群的渐近几何的理解。每一个有限生成群都可以被看作是度量空间，在某种程度上，直到有界畸变的映射都是唯一的。 最近发现，许多有限生成群实际上是由它们的渐近几何唯一确定的。 法布计划继续他的工作，这一现象，以及它如何涉及到莫斯托刚性定理，格罗莫夫的多项式增长定理，和理论的动力系统。 法布还将开发和教授芝加哥的SEERAL计划的几何组成部分。 这个项目向来自芝加哥公立学校系统的小学教师教授数学和科学，并为他们提供课程工具，他们可以带回学校在课堂上使用。 一个空间可以表现出什么样的对称性？ 空间的对称性如何与其几何形状以及测地线（光线）在该空间中的传播方式相关联？ 这些都是大自然中普遍存在的基本问题。 对称性理论的数学基础不仅在理解已知现象方面很重要，而且在发现新现象方面也很重要。 一个例子是最近发现的一种现象，即空间的粗糙的、大尺度的几何结构通常可以决定它可能具有的对称性。