FRG: Collaborative Research: Understanding Low Volume Hyperbolic 3-Manifolds

FRG：协作研究：了解小体积双曲 3 流形

• 批准号: 0554374
• 负责人: David Gabai
• 金额: $ 39.8万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0554374&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Understanding; Low

The goal of this Focused Research Group is to prove the following Complexity Conjecture: that the complete low-volume hyperbolic 3-manifolds can be obtained by filling cusped hyperbolic 3-manifolds of small topological complexity. In particular, our goal is to find the low-volume closed and cusped manifolds and to explain the success of the SnapPea census in determining the low-volume manifolds. Up to the mid 1990's the best lower bounds for volume of closed orientable hyperbolic 3-manifolds appeared to be approximately 1/1000 of the likely lowest volume. Then the paper "Homotopy Hyperbolic 3-Manifolds Are Hyperbolic" improved the low-volume bounds by a factor of one hundred. Subsequently, many authors have used this result to achieve further improvements in the lower bound estimate. Now, the PI's believe they have developed a fundamental new tool (the MOM technology) which will not only find the low-volume closed and cusped hyperbolic 3-manifolds, but also explain in sharp detail why the Complexity Conjecture is correct. Our method is a satisfying mix of elementary hyperbolic geometry, 3-manifold topology, Morse Theory, and rigorous computer analysis. The implementation of our approach will involve mathematicians with expertise in different core areas of math, and with a sound knowledge of the other areas utilized in our methodology. 180 years ago, W. Bolyai, C. F. Gauss, and N. Lobachevsky started a revolution in scientific thought by creating an alternative geometry to Euclidean geometry. This non Euclidean geometry, called hyperbolic geometry, has proven to be a remarkable tool in mathematics. For example, the work of W. Thurston in the 1970's and 1980's showed that the vast majority of 3-dimensional spaces (3-manifolds) possessed geometric structures modeled on hyperbolic geometry, and that this geometric structure could be used to answer fundamental questions about the underlying 3-dimensional manifold. In fact, hyperbolic 3-manifolds have been the subject of intense scrutiny these last 40 years with striking results achieved; most recently, the proofs of the Ending Lamination and Tameness Conjectures, by Y. Minsky et al. Despite these advances and the possible spectacular resolution of the Geometrization Conjecture by G. Perelman, one of the most basic elements of the theory remains to be understood. In particular, the most natural tool for analyzing a hyperbolic 3-manifold is to use the geometry to measure its size, i.e., to compute its volume, but the behavior of the volume function remains mysterious: Thurston proved that there is a least volume hyperbolic 3-manifold, and a next lowest volume, and a next lowest, and so on, but despite 25 years of effort, none of the 3-manifolds possessing these low volumes have been conclusively identified. This proposal introduces a startling new technique--the MOM Technology--that the PIs plan to develop to find all these low-volume manifolds and to explain what properties low-volume hyperbolic manifolds must have.

这个重点研究小组的目的是证明以下复杂性猜想：可以通过填充小小的倍曲线3个manifolds的小拓扑复杂性来获得完整的低量双曲线3个manifolds。特别是，我们的目标是找到小批量封闭和垂垂的歧管，并解释Snappea人口普查在确定低容量歧管方面的成功。直到1990年代中期，对于封闭的可定向双曲线3个序列的最佳下限似乎是可能最低体积的1/1000。然后，纸张“同质双曲线3个manifolds是双曲线”，将低量的边界提高了一百倍。随后，许多作者使用此结果来进一步改善较低估计值。现在，PI的人认为他们已经开发了一种基本的新工具（MOM技术），它不仅会发现小批量封闭和刺痛的双曲线3个杂质，而且还详细说明了为什么复杂性构想是正确的。我们的方法是基本双曲几何形状，3个manifold拓扑结构，摩尔斯理论和严格的计算机分析的令人满意的组合。我们的方法的实施将涉及数学家在数学的不同核心领域具有专业知识，并且对我们方法中使用的其他领域有了合理的了解。 180年前，W。Bolyai，C。F。Gauss和N. Lobachevsky通过为欧几里得几何形状创造了替代的几何形状，开始了科学思想的革命。 这种称为双曲线几何形状的非欧几里得几何形状已被证明是数学中的非凡工具。 例如，W。Thurston在1970年代和1980年代的工作表明，绝大多数的三维空间（3个manifolds）具有建立在双曲线几何形状上的几何结构，并且可以使用这种几何结构来回答有关基础3维歧管的基本问题。 实际上，在过去的40年中，双曲线3型manifolds一直是严格审查的主题，并取得了惊人的结果。最近，Y. Minsky等人的结局层压和驯服猜想的证据。尽管这些进展以及G. Perelman对几何化猜想的壮观解决，该理论的最基本要素之一仍然有待理解。尤其是，分析双曲双曲线3个字符的最自然工具是使用几何形状来衡量其尺寸，即计算其体积，但是音量功能的行为仍然是神秘的：瑟斯顿证明，瑟斯顿证明了最少的体积夸张的3个字符，尽管有一个下一个最低的数量，但虽然没有25年的劳动，但没有25年的劳动，这些效率是25年的劳动。最终确定。 该提案引入了一种令人震惊的新技术 - 妈妈的技术 - PIS计划要开发所有这些小批量的歧管，并解释哪些属性低量的双曲线歧管必须具有。