Arithmetic Groups and Tessellations of Homogeneous Spaces

算术群和齐次空间的镶嵌

基本信息

批准号：
0100438
负责人：
Dave Witte
金额：
$ 8.74万
依托单位：
Oklahoma State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2001
资助国家：
美国
起止时间：
2001-08-01 至 2004-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100438&HistoricalAwards=false
关键词：
Arithmetic Groups Tessellations Homogeneous Spaces

项目摘要

AbstractWitteOne focus of this project is the study of tessellations of homogeneous spaces. Namely, if G/H is a non-compact, simply connected homogeneous space of a connected Lie group G, the question is whether there is a properly discontinuous subgroup D of G, such that the orbit space D\G/H is compact. Some special cases were studied by L. Auslander, Y. Benoist, G. A. Margulis, R. J. Zimmer, and others. In collaboration with H. Oh and A. Iozzi, the PI has recently made progress in understanding the case where G is a semisimple Lie group of real rank two, including a detailed study of the case where G = SO(2,2n) or SU(2,2n). The PI will continue this research, both for real rank two and higher real rank. He will also continue his study of actions of arithmetic groups on the circle, and related questions.This project studies crystals in mathematical spaces other than the 3-dimensional universe that we live in. (A crystal is a material whose atomic structure is very symmetric.) The most fundamental problem in this subject is to decide which spaces contain crystals, and which do not. (For this question, the most interesting spaces are homogeneous, which means that every point of the space looks exactly like all of the other points.) Mathematicians have made substantial progress on this problem in recent years, and this project will continue the work. In cases where crystals do exist, the project will investigate the algebraic properties of the group formed by the symmetries of a crystal.

AbstractWitte 该项目的一个重点是研究同质空间的镶嵌。也就是说，如果 G/H 是连通李群 G 的非紧单连通齐次空间，问题是 G 是否存在适当不连续的子群 D，使得轨道空间 D\G/H 是紧的。 L. Auslander、Y. Benoist、G. A. Margulis、R. J. Zimmer 等人研究了一些特殊情况。最近，PI 与 H. Oh 和 A. Iozzi 合作，在理解 G 是实二阶半单李群的情况方面取得了进展，包括对 G = SO(2,2n) 或 G = SO(2,2n) 的情况的详细研究SU(2,2n)。 PI 将继续这项研究，无论是对于真实排名第二还是更高的真实排名。他还将继续研究圆上算术群的作用以及相关问题。这个项目研究我们生活的3维宇宙以外的数学空间中的晶体。（晶体是一种原子结构非常对称的材料.）本主题中最基本的问题是确定哪些空间包含晶体，哪些空间不包含晶体。（对于这个问题，最有趣的空间是同质的，这意味着空间的每个点看起来都与所有其他点完全相同。）近年来，数学家在这个问题上取得了实质性进展，这个项目将继续这项工作。如果晶体确实存在，该项目将研究由晶体对称性形成的群的代数性质。