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

基本信息

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

项目摘要

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将继续他与Mosher L. Mosher的联合工作，以了解群体的渐近几何形状。每个有限生成的群体都可以将其视为一个唯一的方式，可以将其视为一个度量空间，直到有限失真的地图。最近发现，许多有限生成的群体实际上是由它们的渐近几何形状唯一决定的。 Farb计划继续他在ThisPhenomenon上的工作及其与最大的刚性定理，Gromov的多项式生长定理以及动力学系统理论的关系。 Farb还将在芝加哥的芝加哥计划中开发和教授几何部分。该计划向芝加哥公立学校系统的小学老师讲授数学和科学，并为他们提供可以将其带回学校使用的课程工具。集团是描述对称性的正式数学方法。空间可以展示哪些对称性？空间的对称性与其几何形状以及在该空间中传播的方式相关联？这些是整个自然界中发生的基本问题。对称理论的数学基础不仅在理解已知现象，而且在发现新现象方面很重要。一个例子是最近发现的现象，即空间的粗大几何结构通常可以确定其对称性。