3-manifolds and geometry

3-流形和几何形状

• 批准号: 0905770
• 负责人: Walter Neumann
• 金额: $ 22.69万
• 依托单位: Barnard College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905770&HistoricalAwards=false
• 关键词: manifolds; geometry

The PI will work with Behrstock to complete the quasi-isometric classification of 3-manifold groups (or equivalently, the bi-Lipschitz classification of universal covers of 3-manifolds), which has involved a large number of researchers for over a decade. He will also use bi-Lipschitz geometry and knot-theoretic tools in the search for more flexible approaches to classification questions for complex surface singularities. For hyperbolic manifolds there are postulated connections between geometric, representation-theoretic, and quantum based invariants of manifolds, and the PI will work towards specific conjectures relating these invariants. The PI will also continue his investigation of commensurability properties of 3-manifolds.This proposal addresses questions concerning 3-dimensional space forms that arise from disparate areas of mathematics. These questions create interactions of low dimensional topology with algebra, algebraic geometry, number theory, and theoretical physics. Links between disparate areas provide much of the power of mathematics, and strengthening these links increases the power. Specific tools that will be used are the study of the geometry up to bounded distortion ("bi-Lipschitz geometry") and the use of algebraic and number-theoretic invariants.

PI 将与 Behrstock 合作完成 3 流形组的准等距分类（或等效的 3 流形通用覆盖物的双 Lipschitz 分类），十多年来，该工作已吸引了大量研究人员。他还将使用双利普希茨几何和纽结理论工具来寻找更灵活的方法来解决复杂表面奇点的分类问题。 对于双曲流形，流形的几何、表示理论和基于量子的不变量之间存在假设的联系，PI 将致力于与这些不变量相关的特定猜想。 PI 还将继续研究 3 流形的可通约性性质。该提案解决了来自不同数学领域的有关 3 维空间形式的问题。这些问题创造了低维拓扑与代数、代数几何、数论和理论物理学的相互作用。 不同领域之间的联系提供了数学的大部分力量，而加强这些联系则可以增强数学的力量。将使用的具体工具是对有界变形几何（“双利普希茨几何”）的研究以及代数和数论不变量的使用。