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共形映射定理的一种方法。主要研究人员将尝试推广圆填充方法，该方法用于给出标准黎曼映射定理的证明。他还将对无限圆填料进行研究，并试图用规定的组合学描述无限圆填料的形状与这种结构的刚性之间的关系。这项研究是经典定理的推广，该定理说一个简单连通的二维区域，一个没有洞的区域，以非常精确的方式等价于一个圆盘、一个平面或一个球体。几年前，发现了一种证明这一定理的方法，该方法涉及域的圆填充。这项研究涉及到这一新证据提出的问题。在翼型绕流的计算中，该定理本身被用于非常关键的方式，最近发现的证明为更有效地计算流动开辟了新的可能性。

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.