Hyperbolic 3-Manifold Invariants and Applications

G·珀尔曼和W·瑟斯顿的不朽工作表明了双曲3-流形在3-流形研究中的基本重要性。双曲三维流形构成了一类复杂而神秘的三维流形，最大限度地了解它们是很重要的。分析双曲三维流形最简单的工具是体积不变量。作者计划研究双曲三维流形理论中一个宏大的公开问题，并利用这项工作所获得的知识更好地理解双曲三维流形的拓扑和几何之间的联系。一个宏大的公开问题是寻找1-尖点双曲3-流形的第一个无穷体积串，即找到所有体积小于最小体积的1-尖点双曲3-流形(S)。提出者与D.Gabai共同工作，并与N.瑟斯顿协商，提出了一个基于计算机的方案来解决这个问题。这种方法的计算方面本身就很有趣。大约200年前，J.Bolyai、C.Gauss和N.Lobachevsky声称，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)可以构造合法的几何，从而使数学发生了革命性的变化。此外，他们还推测，这种新的、神秘的非欧几里德几何(现在称为“双曲几何”)将会有重要的应用。他们的理论已经得到证实：双曲几何在现代几何研究中至关重要。例如，在研究“3维流形”(我们的3维宇宙就是3维流形的一个例子)时，双曲几何被证明比欧几里得几何重要得多。作为另一个例子，我们的宇宙很可能遵循非欧几里德几何定律，而不是欧几里得几何定律。提出者与D.Gabai合作，并与N.瑟斯顿协商，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双曲三维流形。这种对双曲三维流形体积的理解应该导致对双曲三维流形的各种见解，包括它们的拓扑(形状)和几何(度量)之间的联系。这种看似互不相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.