Problems in Low Dimensional Geometry and Topology

低维几何和拓扑问题

• 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

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.

这一建议解决了低维拓扑和几何中的中心问题。PI计划调查末端分层空间的拓扑结构，以及与Dehn手术有关的问题。此外，还研究了低体积双曲三维流形以及超大型双曲流形的拓扑结构。后者具有大体积、Cheeger常数下界和强不可约极小亏格Heegaard分裂。PI还计划研究光滑的勋蝇4球。三维流形拓扑学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。二流形拓扑学是研究球面、环面和高亏格曲面等的拓扑学。虽然像曲面和标准3-space这样的物体非常简单，但存在于其中的物体的类别是令人难以置信的丰富和迷人的。例如，在三维空间中有纽结理论，在曲面上有曲线和分层理论。这些理论反过来又从根本上与许多其他数学结构相联系，例如非欧几里德几何。这项建议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，平面上的任何光滑闭曲线，如圆，都有光滑圆的边界。事实上，这样的定理在第三维是正确的，在第四维是未知的。圆周率计划调查这个非常神秘的问题，它是光滑四维拓扑的核心。