Topics in Discrete Groups and Geometry

理查德·施瓦茨提出研究几何学和动力学中的几个问题。第一个问题与复杂双曲离散群的变形有关。Schwartz沿着Thurston的双曲Dehn手术定理的思路，为许多这样的群体证明了一个一般的外科定理，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Schwartz证明了一个三角形台球桌在其所有角度都小于100度的情况下具有周期台球路径。这代表了有200年历史的三角形台球问题的第一个实质性进展，该问题要求每个三角形台球桌是否都有一个周期的台球路径。第三个问题与简单几何结构的迭代有关，比如质心细分。施瓦茨的研究有一个共同的主题，那就是研究无穷次地执行一个简单操作的后果。在自然界中，我们可以在树木和珊瑚礁等事物中看到巨大的复杂性，而这种复杂性通常是由多次发生的相同过程产生的。例如，某种基本的分支操作产生了树的形状。施瓦茨研究的问题是人们在自然界中可能看到的事物的理想化——例如，一个复杂的双曲离散群编码了在某个弯曲的四维空间中镜子反射光线的方式，三角形台球问题问的是，如果一个人在无摩擦的台球桌上用一个点大小的台球打台球会发生什么。以质量为中心的细分问题在某种程度上是一种理想化的想法，即一遍又一遍地切割高维钻石，并研究切面的形状。

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.