Topology and Geometry of 3-Dimensional Manifolds

3 维流形的拓扑和几何

• 批准号: 0204142
• 负责人: Peter Shalen
• 金额: $ 33.44万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204142&HistoricalAwards=false
• 关键词: Topology; Geometry; Dimensional; Manifolds

Culler and Shalen are continuing to investigate consequences of theirwork on the character variety of a knot group. This involves studyingthe relationship between the boundary slopes of a knot and thetopological properties of essential surfaces that realize theslopes. It also includes their ongoing project with Dunfield and Jacoabout smallish knots in non-Haken manifolds, which is part of aprogram to prove the Poincare Conjecture. A joint project by Agol,Culler and Shalen, concerning the construction of covering spaces of atriangulated 3-manifold by an inductive process, is also relevant tothis program. Agol is also continuing his work on geometric finitenessof geometrically defined subgroups of knot groups, volume estimatesfor non-fibered Haken manifolds and hyperbolic orbifolds, and thecomplexity of algorithms in 3-manifold theory.A fundamental problem in many areas of mathematics is to classify allexamples of a certain type of mathematical object. The objects ofstudy in this proposal are 3-manifolds, which are mathematical modelsof 3-dimensional spaces. Since our universe is a 3-dimensional space,the classification of 3-manifolds is directly related to ourunderstanding of nature itself. The classification problem for3-manifolds is far from solved, but the work of many mathematiciansover the last 25 years has at least produced a conjecturalanswer. Remarkably, the conjectures, if true, will provide aunification of the most classical, rigid kind of geometry---both theEuclidean version first studied by the ancient Greeks and thenon-Euclidean kind that constituted an exciting discovery in the 19thcentury---and topology, a subject devoted to studying much moreflexible geometric structures, which until recently had developedquite independently of the more classical theories. The worksupported by this grant forms part of the effort to verify theconjectured geometric classification of 3-manifolds.

Culler和Shalen正在继续研究他们的工作对结群性格变化的影响。这包括研究结的边界斜率与实现斜率的基本表面的拓扑性质之间的关系。它还包括他们正在进行的关于非哈肯流形中的小结的项目，这是证明庞加莱猜想计划的一部分。Agol，Culler和Shalen的一个合作项目，关于用归纳法构造三角化3流形的覆盖空间，也与这个方案有关。Agol也在继续研究结群的几何定义子群的几何有限性，非纤维Haken流形和双曲轨道的体积估计，以及3流形理论中算法的复杂性。数学许多领域的一个基本问题是对某一类数学对象的样本进行分类。本文研究的对象是三维流形，即三维空间的数学模型。由于我们的宇宙是一个三维空间，三维流形的分类直接关系到我们对自然本身的理解。3流形的分类问题还远未解决，但在过去25年里，许多数学家的工作至少产生了一个推测性的答案。值得注意的是，如果这些猜想是正确的，它们将把最经典、最刚性的几何——古希腊人首先研究的欧几里得几何，以及19世纪令人兴奋的非欧几里得几何——和拓扑学统一起来。拓扑学致力于研究更灵活的几何结构，直到最近才完全独立于更经典的理论发展起来。这项资助的工作是验证3流形的推测几何分类的一部分。