Geometry and Topology in Two and Three Dimensions

圆盘花纹在平面上的微小变形可以用简单几何构造得到的电网上的调和函数来刻画。电网络的性质可以用来推导盘状图案的刚性性质。该方法还可用于证明三维凸双曲多面体的特征，推广了Andreev定理和Hodgson和Rivin的一个特征定理。研究人员将探索圆盘图案的变形与三维双曲圆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统一化猜想。研究二维和三维的几何和拓扑学是密切相关的。许多三维几何拓扑问题可以归结为二维问题，而几何结构可以用来解决拓扑问题。这个项目对圆盘图案、三维双曲多面体、圆域和圆盘填充、三维几何结构及其应用以及拓扑流体力学的研究，将有助于更好地理解我们现实世界的几何和物理。有趣的是，流形上的几何结构可以用来产生关于它们的拓扑信息，反过来，关于它们的拓扑信息可以用来解决一些流体力学问题。有一项关于纽结和链环的Moebius能量的相关研究，其中几个最基本的问题仍然没有解决。这是一个理论和实验很好结合的领域。

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.***