Arithmetic Groups and Tessellations of Homogeneous Spaces

算术群和齐次空间的镶嵌

• 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

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.

该项目的AbstractWitteone焦点是研究均匀空间的细节。也就是说，如果g/h是连接的lie组的非紧缩的，简单地连接的同质空间，则问题是G/h是否存在g的不连续的亚组d，因此轨道空间d \ g/h是紧凑的。 L. Auslander，Y。Benoist，G。A。Margulis，R。J。Zimmer等研究了一些特殊案例。 在与H. Oh和A. Iozzi合作的情况下，PI最近在理解G是一个半级别等级第二的案例方面取得了进展，其中包括对G = So（2,2n）或SU（2,2N）的详细研究。 PI将继续进行这项研究，无论是实际等级和更高的实际等级。 他还将继续研究算术群在圆圈和相关问题上的作用。该项目研究我们所生活的三维宇宙以外的数学空间中的晶体。（晶体是一种材料，其原子结构非常对称。）在该学科中，最根本的问题是决定哪些空间包含晶体，以及哪些空间。 （对于这个问题，最有趣的空间是均匀的，这意味着空间的每个点看起来都与其他所有要点完全一样。）近年来，数学家在这个问题上取得了重大进展，该项目将继续进行工作。 如果确实存在晶体，则该项目将研究由晶体对称性形成的组的代数特性。