Mathematical Sciences: Invariants for Hyperbolic 3-Manifolds

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、最复杂的一类。这些流形上的双曲结构可以用来描述诸如陈-西蒙斯不变量、体积和η-不变量等不变量。根据莫斯托定理，这些几何不变量实际上是拓扑不变量，至少在封闭的情况下是这样。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。这个项目对理解这些不变量进行了多方面的攻击。

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 useable 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. The hyperbolic structures on these manifolds can be used to describe such invariants as the Chern-Simons invariant, the volume, and the eta-invariant. By Mostow's theorem, these geometric invariants are actually topological invariants, at least in the closed case. Recent research indicates that these invariants may be very useful in helping us to understand 3-manifolds. This project mounts a multi-sided attack on understanding these invariants.