Mathematical Sciences: New Invariants for Hyperbolic 3-Manifolds

数学科学：双曲 3 流形的新不变量

• 批准号: 9008592
• 负责人: G. Robert Meyerhoff
• 金额: --
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-08-15 至 1991-09-23
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9008592&HistoricalAwards=false
• 关键词: Mathematical; Sciences; New; Invariants; Hyperbolic

Three-dimensional manifolds appeal to us at a very basic level, but they are very difficult to understand. There are many ways to construct examples of 3-manifolds, but the simplest sorts of questions about the constructed manifolds often cannot be answered. For example, it is very difficult to tell whether two construction schemes yield the same or different manifolds. What is needed is a computable collection of invariants which distill the vast amount of information contained in the three-manifold into a usable form. The standard invariants - Euler characteristic, homology, fundamental group - either contain almost no information, or are very difficult to work with. Hyperbolic 3-manifolds are the most important - and most complicated - class of geometric 3-manifolds. Hyperbolic manifolds have the same metric properties as the non-Euclidean geometry discovered by Bolyai, Gauss, and Lobachevsky in the 1820's. The metric can be used to carry out measurements. In particular, such invariants as the volume (the size of the manifold) and the length spectrum (the set of all lengths of non- trivial closed geodesics in the manifold) can be used to study these 3-manifolds. Subtler geometric invariants can be defined as well. Recent research indicates that these invariants may be very useful in helping us to understand 3-manifolds. This project involves extensive machine computation in an approach to understanding these invariants.

三维流形在非常基础的层面上吸引了我们，但它们非常难以理解。有很多方法可以构造三维流形的例子，但是关于所构造的流形的最简单的问题往往无法回答。例如，很难判断两个构造方案产生的流形是相同还是不同。所需要的是一个可计算的不变量集合，它将包含在三流形中的大量信息提炼成可用的形式。标准不变量-欧拉特征、同调、基本群-要么几乎不包含任何信息，要么非常难以处理。双曲三维流形是几何三维流形中最重要也是最复杂的一类。双曲流形与波尔约、高斯和罗巴切夫斯基在19世纪20年代发现的非欧几何具有相同的度量性质。该度量可用于执行测量。特别地，诸如体积（流形的大小）和长度谱（流形中所有非平凡闭测地线的长度的集合）等不变量可以用来研究这些三维流形。更微妙的几何不变量也可以被定义。最近的研究表明，这些不变量可能是非常有用的，在帮助我们理解三维流形。这个项目涉及广泛的机器计算的方法来理解这些不变量。