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. Mosher 共同研究群的渐近几何。每个有限生成的群都可以被视为度量空间，其方式对于有界畸变映射来说是唯一的。 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。 法布计划继续研究这一现象以及它与莫斯托刚性定理、格罗莫夫多项式增长定理和动力系统理论的关系。 Farb 还将在芝加哥的 SESAME 项目中开发和教授几何组件。 该计划向芝加哥公立学校系统的小学教师教授数学和科学，并为他们提供可以带回学校在课堂上使用的课程工具。群是描述对称性的正式数学方式。 空间可以表现出什么对称性？ 空间的对称性与其几何形状以及测地线（光线）在该空间中的传播方式有何关系？ 这些是自然界中普遍存在的基本问题。 对称理论的数学基础不仅对于理解已知现象很重要，而且对于发现新现象也很重要。 一个例子是最近发现的一种现象，即空间的粗糙、大规模几何结构通常可以决定它可以具有的对称性。