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

Farb将继续与L.Mosher在理解群的渐近几何方面的联合工作。每个有限生成的群都可以被视为一个度量空间，在某种程度上，它对于有界失真的映射是唯一的。最近发现，许多有限生成的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及其与莫斯托刚性定理、格罗莫夫多项式增长定理和动力系统理论的关系。法布还将在芝加哥的芝麻课程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，并为他们提供课程工具，他们可以带回学校在课堂上使用。小组是描述对称性的正式数学方式。太空可以展示什么样的对称性？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？这些都是自然界普遍存在的基本问题。对称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现象，一个空间的粗略、大规模的几何结构通常可以决定它可能具有的对称性。

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.