Geometry and Topology in Two and Three Dimensions

二维和三维几何和拓扑

• 批准号: 9803619
• 负责人: Zheng-Xu He
• 金额: $ 6.78万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803619&HistoricalAwards=false
• 关键词: Geometry; Topology; Two; Three; Dimensions

9803619He Infinitesimal deformation of disk patterns in the plane can becharacterized by harmonic functions on an electric network that isobtained by a simple geometric construction. Properties of theelectrical network may be used to derive rigidity properties of diskpatterns. The technique may also be applied to prove characterizationsof three-dimensional convex hyperbolic polyhedra, generalizing bothAndreev's Theorem and a characterization theorem due to Hodgson andRivin. The investigator will explore relationships between thedeformation of disk patterns and that of three-dimensional hyperboliccone-manifolds, and study the applications of these results to otherfields. He will also continue the research on the Koebe uniformizationconjecture. The study of geometry and topology in two and three dimensions areintimately related. Many three-dimensional geometric topology problemsmay be reduced to problems in two dimensions, and geometric structuresmay be applied to solve topological problems. This project's researchon disk patterns, three-dimensional hyperbolic polyhedra, circledomains and disk packings, three-dimensional geometric structures andtheir applications, and on topological hydrodynamics, will contributeto a better understanding of the geometry and physics of our real world.It is also an interesting fact that geometric structures on manifoldscan be applied to yield information on their topology, and in turn theinformation on their topology may be used in some problems inhydrodynamics. There is a related study of Moebius energy of knots andlinks, where several most elementary questions remain unsolved. This isa field where theory and experiment meet well.***

小行星9803619 平面内圆盘图案的无穷小变形可以用一个电网络上的调和函数来表征，该电网络由简单的几何构造得到。 电网络的性质可以用来导出圆盘图案的刚性性质。 该技术也可用于证明三维凸双曲多面体的特征，推广Andreev定理和Hodgson和Rivin的特征定理。 研究者将探索盘斑图的变形与三维双曲锥流形的变形之间的关系，并研究这些结果在其他领域的应用。 他还将继续研究Koebe单值化猜想。 二维和三维的几何学和拓扑学的研究是密切相关的。 许多三维几何拓扑问题可以归结为二维问题，几何结构可以用来解决拓扑问题。 本计画对圆盘图案、三维双曲多面体、圆域与圆盘填充、三维几何结构及其应用、以及拓扑流体力学的研究，将有助于更好地了解我们真实的世界的几何与物理。而这些拓扑结构的信息又可用于流体力学中的某些问题。 有一个关于纽结和链环的莫比乌斯能量的相关研究，其中有几个最基本的问题仍然没有解决。这伊萨一个理论与实验相结合的领域。