Hyperbolic 3-Manifold Invariants and Applications

G. Perleman和W. Thurston的不朽著作表明了双曲型3-流形在3-流形研究中的根本重要性。双曲型3-流形是一类复杂而神秘的3-流形，对其进行全面的认识是十分重要的。分析双曲型3流形最简单的工具是体积不变量。拟人计划在双曲3-流形理论中研究一个宏大的开放问题，并利用从这项工作中获得的知识来更好地理解双曲3-流形的拓扑和几何之间的联系。宏大的开放问题是寻找第一个1尖双曲3-流形的无限体积串；即求出体积小于最小体积的2角双曲3流形的所有1角双曲3流形。提议者与D. Gabai合作，并与N. Thurston协商，提出了一个基于计算机的方案来解决这个问题。该方法的计算方面本身就很有趣。大约200年前，J.博利亚、C.高斯和N.罗巴切夫斯基革新了数学，他们声称可以通过采用欧几里得的五个经典公设并否定第五个公设（平行公设）来构造一个合法的几何。此外，他们还从理论上推断，这种新的神秘的非欧几里得几何（现在被称为“双曲几何”）将有重要的应用。他们的理论已经被证实：双曲几何在现代几何研究中是至关重要的。例如，在“三维流形”的研究中，双曲几何比欧几里德几何重要得多（我们的三维宇宙就是三维流形的一个例子）。另一个例子是，我们的宇宙很可能遵循非欧几里得几何定律，而不是欧几里得几何定律。这位提议者与D. Gabai合作，并与N. Thurston协商，计划通过研究双曲三维流形的体积或大小，研究一种基于计算机的方法来理解它们。这种对双曲3-流形体积的理解将导致对双曲3-流形的各种见解，包括它们的拓扑（形状）和几何（测量）之间的联系。这些看似无关的领域之间的联系往往对数学有着深远的影响。

The monumental work of G. Perleman and W. Thurston has shown the fundamental importance of hyperbolic 3-manifolds within the study of 3-manifolds. Hyperbolic 3-manifolds constitute a complicated and mysterious class of 3-manifolds, and it is important to understand them to the fullest extent possible. The simplest tool for analyzing hyperbolic 3-manifolds is the volume invariant. The proposer plans to work on a grandiose open problem in the theory of hyperbolic 3-manifolds, and to use the knowledge gained from this work to better understand the connections between the topology and the geometry of hyperbolic 3-manifolds. The grandiose open problem is to find the first infinite string of volumes of 1-cusped hyperbolic 3-manifolds; that is, to find all 1-cusped hyperbolic 3-manifolds with volume less than that of the minimum volume 2-cusped hyperbolic 3-manifold(s). The proposer, working jointly with D. Gabai, and in consultation with N. Thurston, has a computer-based scheme to attack this problem. The computational aspects of the approach are interesting in their own right. Almost 200 years ago, J. Bolyai, C. Gauss, and N. Lobachevsky revolutionized mathematics by claiming that a legitimate geometry could be constructed by taking the five classical postulates of Euclid and negating the fifth postulate (the parallel postulate). Further, they theorized that this new and mysterious non-Euclidean geometry (now called "hyperbolic geometry") would have important applications. Their theorizing has been borne out: hyperbolic geometry is vitally important in the modern study of geometry. For example, hyperbolic geometry turns out to be much more important than Euclidean geometry in the study of "3-dimensional manifolds" (our 3-dimensional Universe is an example of a 3-dimensional manifold). As another example, it is quite possible that our Universe adheres to the laws of non-Euclidean geometry rather than the laws of Euclidean geometry.The proposer, working jointly with D. Gabai, and in consultation with N. Thurston, plans to work on a computer-based approach to understanding hyperbolic 3-dimensional manifolds by studying their volume, or size. This understanding of volumes of hyperbolic 3-manifolds should lead to a variety of insights about hyperbolic 3-manifolds, including connections between their topology (shape) and geometry (measurement). Such connections between seemingly unrelated areas often have profound implications for mathematics.