Topics in Discrete Groups and Geometry

离散群和几何主题

• 批准号: 0604426
• 负责人: Richard Schwartz
• 金额: $ 32.76万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604426&HistoricalAwards=false
• 关键词: Topics; Discrete; Groups; Geometry

Richard Schwartz proposes to study several problems in geometry and dynamics. The first problem has to do with the deformations of complex hyperbolic discrete groups. Schwartz proved a general surgery theorem for many of these groups, along the lines of Thurston's hyperbolic Dehn surgery theorem, and he hopes to use this theorem to analyze some concrete examples. The second problem has to do with triangular billiards. Recently Schwartz proved that a triangular shaped billiard table has a periodic billiard path provided that all its angles are less than 100 degrees. This represents the first substantial progress on the 200-year old triangular billiards problem, which asks if every triangular shaped billiard table has a periodic billiard path. The third problem has to do with the iteration of simple geometric constructions, such as barycentric subdivision.The common theme to Schwartz's research is the idea of looking at the consequences of performing a simple operation infinitely often. In nature one sees great complexity in things like trees and coral reefs, and this complexity is often produced by the same process happening multiple times. For instance, some kind of basic branching operation produces the shape of a tree. The problems studied by Schwartz are idealizations of the sort of thing one might see in nature - for instance, a complex hyperbolic discrete group encodes the way light bounces of mirrors in a certain curved 4 dimensional space, and the triangular billiards problem asks what happens if one plays pool on a frictionless pool table with a billiard ball that is the size of a point. The barycentric subdivision problem is somehow an idealization of the idea of chopping up a high dimensional diamond over and over again, and studying the shapes of the facets.

理查德·施瓦茨提出要研究几何和动力学中的几个问题。 第一个问题与复双曲离散群的变形有关。 施瓦茨证明了一个一般手术定理的许多这些群体，沿着线瑟斯顿的双曲德恩手术定理，他希望用这个定理来分析一些具体的例子。 第二个问题与三角形台球有关。 最近施瓦茨证明，一个三角形的台球桌有一个周期性的台球路径提供所有的角度都小于100度。 这代表了200年历史的三角形台球问题的第一个实质性进展，该问题询问是否每个三角形台球桌都有一个周期性的台球路径。第三个问题与简单几何构造的迭代有关，例如重心细分。施瓦茨研究的共同主题是研究无限频繁执行简单操作的后果的想法。 在自然界中，人们可以看到树木和珊瑚礁等事物的巨大复杂性，这种复杂性通常是由多次发生的同一过程产生的。 例如，某种基本的分支操作产生树的形状。 施瓦茨研究的问题是理想化的排序的事情，人们可能会看到在性质-例如，一个复杂的双曲离散群编码的方式轻反弹的镜子在一定的弯曲4维空间，和三角台球问题问会发生什么，如果一个发挥池上的摩擦台球桌上的台球是一个点的大小。重心细分问题在某种程度上是一个理想化的想法，一次又一次地切碎一个高维钻石，并研究小平面的形状。