Mathematical Sciences: Low Dimensional Topology and Infinite Group Theory

数学科学：低维拓扑和无限群论

• 批准号: 9302520
• 负责人: Peter Shalen
• 金额: $ 16.79万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-06-01 至 1997-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9302520&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Low; Dimensional; Topology

A striking feature of the development of mathematics in the past decade has been the extent to which the diverse branches of mathematics have interacted, with methods from one branch being used to solve outstanding problems in another. Culler and Shalen's ongoing program for obtaining lower bounds for volumes of hyperbolic manifolds is in large measure an application of qualitative results from topology to quantitative estimates in geometry. Their program for the Property P conjecture would be likely to exploit K-theory and arithmetic geometry in attacking a problem from the topological theory of 3-manifolds. Culler's project of implementing Manin's program by means of Thurston's theory of pleated surfaces is a way of using synthetic hyperbolic geometry to obtain insight into number theory. Finally, Shalen's project involving fundamental groups of 2-complexes lies in the domain of geometric and combinatorial group theory but is motivated in part by the prospect of applications to the study of discrete subgroups of algebraic groups and Kac-Moody groups. Three-dimensional manifolds are the mathematical models of our physical universe. While the cosmological question of whether our universe is finite may be unresolved, it is certain that the study of finite-volume 3-manifolds is of fundamental importance in understanding the physical universe. The quantitative estimates obtained by Culler and Shalen may be regarded as being mathematical analogues of measurements of fundamental physical constants, such as the speed of light. The geometry, as well as the topology, of the universe is important in physics. While the non-euclidean geometries may have been considered curiosities in the nineteenth century, the development of general relativity shows that negatively curved spaces, such as hyperbolic 3-manifolds, are the appropriate models for the universe. Recent mathematical research has shown that they are ubiquitous and mathematically rich as well. The research proposed here is aimed at expanding our understanding of these important spaces.

20世纪30年代数学发展的一个显著特点是： 在过去的十年里，各种各样的分支机构 数学相互作用，其中一个分支的方法是 用于解决另一个突出问题。 卡勒和 Shalen正在进行的获得体积下限的程序 在很大程度上，双曲流形是 从拓扑到定量估计的定性结果， 几何 他们对性质P猜想的程序是 很可能利用K理论和算术几何来攻击一个 问题的拓扑理论的3-流形。 卡勒 通过瑟斯顿的方法实施马宁计划的项目 褶曲曲面理论是一种利用合成双曲 几何学来深入了解数论。 最后，Shalen的 涉及2-复合物的基本群的项目在于 几何和组合群论的领域，但 部分原因是应用于研究的前景， 代数群和Kac-Moody群的离散子群。 三维流形是数学模型， 我们的物质宇宙 虽然宇宙学的问题 我们的宇宙是否是有限的可能还没有解决，但可以肯定的是， 有限体积三维流形的研究是一个基本的问题， 理解物理宇宙的重要性。 的 Culler和Shalen获得的定量估计可能是 被认为是测量的数学类比， 基本的物理常数，如光速。 的 宇宙的几何学和拓扑学都很重要 在物理学中。 虽然非欧几里德几何可能是 被认为是世纪的珍品， 广义相对论表明，负弯曲的空间，如 作为双曲3流形，是适当的模型， 宇宙 最近的数学研究表明， 无处不在，数学上也很丰富 研究 这里提出的目的是扩大我们的理解 这些重要的空间。