Mathematical Sciences: Problems Relating to the Circle Packing Theorem

该奖项支持的研究将调查 非黎曼度量的保角映射定理的一种方法。 首席研究员将尝试扩展圆的方法 一种用于证明标准黎曼定理的包装 映射定理他还将研究无限循环 包装，并试图描述形状之间的关系， 一个无限圆包装与规定的组合和 这种结构的刚性。 该研究是经典定理的推广， 说一个单连通的二维域，一个没有 孔，在一个非常精确的方式相当于一个磁盘，一个平面或 球体。证明这一定理的一种方法涉及到一个包装的 几年前就发现了圆域。本研究 涉及这个新证据提出的问题。定理本身是 在计算翼型绕流时， 最近发现的证据为更多的人提供了新的可能性 流的高效计算。

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.