Mathematical Sciences: Topology and Geometry of 3-Manifolds

数学科学：3-流形的拓扑和几何

• 批准号: 9626676
• 负责人: Peter Shalen
• 金额: $ 12.06万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 1999-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626676&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topology; Geometry; Manifolds

9626676 Shalen One objective of this research project is to understand as much as possible about the incompressible surfaces in the exterior of a knot in a homotopy 3-sphere. This is an important aspect of classical knot theory and has potential applications to the Poincare conjecture. A second goal is to improve the known bounds for the distance between various types of exceptional Dehn fillings of a 3-manifold with torus boundary. This amounts to achieving a quantitative understanding of the phenomenon that a generic Dehn surgery on a hyperbolic knot produces a hyperbolic manifold. A third objective is to improve some of the known lower bounds for volumes of hyperbolic manifolds under various topological restrictions. Such bounds give a quantitative understanding of the Mostow Rigidity Theorem, which implies that the volume of a hyperbolic manifold is a topological invariant. Topology is the branch of mathematics that deals with properties of geometric objects that are so universal that they are unaffected by any distortion of the shape of the object. A simple example of such a property is the winding number of a closed circuit in the plane. If the circuit does not pass through the origin, then it winds around the origin a certain number of times, and that number is not affected by any distortion of the circuit (just so long as it never passes through the origin during its deformation). This was the main idea in Gauss's first proof that every polynomial equation has a solution in the complex numbers and was the origin of the subject of topology. The geometric objects being studied by Culler and Shalen are 3-dimensional spaces, one example of which is our physical universe. (The geometric and topological properties of the physical universe are not yet understood, but modern physics has shown that it is definitely not the 3-dimensional Euclidean space that we study in high school.) This research focuses on various numerical quantities that are determined by the geometric properties of the space. The goal is to understand how the fundamental topological properties of the space are reflected in these quantities. In very general terms, this is analogous to the process by which the chemical composition of a star can be deduced by studying its spectrogram. ***

9626676 Shalen 该研究项目的目标之一是尽可能多地了解同伦 3 球体中结外部的不可压缩表面。 这是经典纽结理论的一个重要方面，并且对庞加莱猜想具有潜在的应用。 第二个目标是改善具有环面边界的 3 流形的各种类型的特殊 Dehn 填充之间的距离的已知界限。 这相当于对双曲结上的一般 Dehn 手术产生双曲流形的现象进行了定量理解。 第三个目标是改进各种拓扑限制下双曲流形体积的一些已知下界。 这样的界限给出了对莫斯托刚性定理的定量理解，该定理意味着双曲流形的体积是拓扑不变量。 拓扑学是数学的一个分支，它研究几何对象的属性，这些属性是如此普遍，以至于它们不受对象形状的任何扭曲的影响。 这种属性的一个简单例子是平面中闭合电路的匝数。 如果电路不经过原点，那么它会绕原点缠绕一定的次数，并且该次数不受电路任何变形的影响（只要它在变形期间不经过原点）。 这是高斯第一个证明每个多项式方程都有复数解的主要思想，也是拓扑学学科的起源。 卡勒和沙伦正在研究的几何对象是三维空间，其中一个例子就是我们的物理宇宙。 （物理宇宙的几何和拓扑性质尚不清楚，但现代物理学已经表明，它绝对不是我们在高中学习的3维欧几里得空间。）这项研究重点关注由空间几何性质决定的各种数值量。 目标是了解空间的基本拓扑特性如何反映在这些量中。 一般来说，这类似于通过研究恒星的光谱图来推断恒星化学成分的过程。 ***