Geometry and Topology in Two and Three Dimensions

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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.***