Mathematical Sciences: Problems Relating to the Circle Packing Theorem

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The research supported by this award will investigate an approach to a conformal mapping theorem for non-Riemannian metrics. The principal investigator will try to extend the method of circle packing which was used to give a proof of the standard Riemann mapping theorem. He will also undertake a study of infinite circle packings and try to describe the relationship between the shape of an infinite circle packing with prescribed combinatorics and the rigidity of such a structure. This research is an extension of the classical theorem which says that a simply connected two dimensional domain, one without holes, is equivalent in a very precise way to a disk, a plane or a sphere. A method of proving this theorem involving a packing of the domain by circles was discovered several years ago. This research involves questions raised by this new proof. The theorem itself is used in a very crucial way in computing flow over airfoils and the recently discovered proof opens up new possibilities for more efficient computations of flows.