Mathematical Sciences: New Invariants for Hyperbolic 3-Manifolds

三维流形在非常基本的层面上吸引着我们，但它们非常难以理解。有许多方法可以构造三维流形的例子，但关于所构造的流形的最简单的问题往往无法回答。例如，很难区分两种构造方案产生的流形是相同的还是不同的。所需要的是一个可计算的不变量集合，它将三维流形中包含的大量信息提炼成一种可用的形式。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，要么很难处理。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。双曲流形具有与Bolyai、Gauss和Lobachevsky在1820年S发现的非欧几里得几何相同的度量性质。这种度量可以用来进行测量。特别是，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3-流形。也可以定义更微妙的几何不变量。最近的研究表明，这些不变量在帮助我们理解三维流形方面可能非常有用。这个项目需要大量的机器计算来理解这些不变量。

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.