Problems in Low Dimensional Geometry and Topology

低维几何和拓扑问题

基本信息

批准号：
1006553
负责人：
David Gabai
金额：
$ 79.1万
依托单位：
Princeton University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2010
资助国家：
美国
起止时间：
2010-07-01 至 2017-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1006553&HistoricalAwards=false
关键词：
Problems Low Dimensional Geometry Topology

项目摘要

This proposal addresses central problems in low dimensional topology and geometry. The PI plans to investigate the topology of ending laminations space as well as problems related to Dehn surgery on links and taut foliations. In addition to study the topology of low volume hyperbolic 3-manifolds as well as those which are ultra large. The latter have large volume, Cheeger constant bounded below and have strongly irreducible minimal genus Heegaard splittings. The PI also plans to study smooth Schoenflies 4-balls. Three-manifold topology is the study of objects that locally look like the standard three-dimensional space in which we live. Two-manifold topology is the study of surfaces such as the sphere, torus and the surface of higher genus. While objects such as surfaces and standard 3-space are very simple, the class of objects that exist in them is incredibly rich and fascinating. For example, there is knot theory in 3-space and the theory of curves and laminations in surfaces. These theories in turn are fundamentally connected with many other mathematical structures such as non Euclidean geometry. This proposal addresses basic mathematical questions in these areas. A classical fundamental theorem in topology is that any smooth closed curve in the plane, such as the circle, bounds a smooth disc. In fact such a theorem is true in dimension three and unknown in dimension four. The PI plans to investigate this very mysterious question at the heart of smooth 4-dimensional topology.

该提案解决了低维拓扑和几何中的核心问题。 PI 计划研究最终层压空间的拓扑结构以及与链接和拉紧叶状结构的 Dehn 手术相关的问题。此外还研究小体积双曲 3 流形以及超大体积的拓扑。后者具有很大的体积，奇格常数在下方有界，并且具有强不可约的最小赫加德分裂。 PI 还计划研究光滑的 Schoenflies 4 球。三流形拓扑是对局部看起来像我们生活的标准三维空间的对象的研究。二流形拓扑是对球面、圆环面和高等曲面等曲面的研究。虽然曲面和标准 3 空间等对象非常简单，但其中存在的对象类别却极其丰富且令人着迷。例如，3-空间中的纽结理论以及曲面中的曲线和叠层理论。这些理论又与许多其他数学结构（例如非欧几里得几何）有着根本的联系。该提案解决了这些领域的基本数学问题。拓扑学中的一个经典基本定理是，平面上任何光滑的闭合曲线（例如圆）都限制着光滑的圆盘。事实上，这样的定理在第三维中是正确的，但在第四维中是未知的。 PI 计划研究平滑 4 维拓扑核心的这个非常神秘的问题。