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。每一个有限生成的群都可以被看作是一个度量空间，这种度量空间对于有界畸变的映射是唯一的。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地决定的。Farb计划继续研究这一现象，以及它与Mostow刚性定理、Gromov多项式增长定理和动力系统理论的关系。法布还将在芝加哥SESAME项目中开发和教授几何部分。这个项目为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。群是描述对称性的正式数学方法。空间可以展示什么样的对称性？一个空间的对称性与它的几何形状，以及测地线（光线）在空间中传播的方式有什么关系？这些都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理解已知现象很重要，而且对发现新现象也很重要。一个例子是最近发现的一个现象，即空间的粗糙、大规模的几何结构往往可以决定它的对称性。