Mathematical Sciences: Research in Conformal Mapping

小行星9400733 该奖项支持的工作旨在利用最近的数学发展，将圆填充的概念应用于保角映射的经典问题。 如果一个单连通区域用一个固定半径的圆填充，那么单位圆盘就有一个组合等价的填充，并且当半径趋于零时，这个圆填充同构收敛于区域到圆盘上的黎曼映射。 这一想法，这导致了一个全新的方法，思想的保形映射在最近几年，现在将被应用于研究映射的多连通域的补充组成的唯一缝辐射从原点。 这个结果在许多特殊情况下是已知的-甚至是一些具有无限连通性的域。 一个人一般可以做到这一点，是由Koebe在1908年首次提出的。 他证明了它的连通域。 结合新的工具，从圆包装和其他进展，共形映射将集中精力解决Koebe猜想的整体。 保角映射是研究平面（和高维）域通过保持无穷小角度和方向的变换的映射。 地图发挥核心作用的几何理论的解析函数和潜在的理论减少许多问题有关的功能定义在任意域相同的问题限制在一类高度对称的域。 ***

9400733 Rodin The work supported by this award sets out to exploit recent mathematical developments applying concepts of circle packing to classical problems of conformal mapping. If one packs a simply connected region with circles of a fixed radius then there is a combinatorially equivalent packing of the unit disk, and this circle packing isomorphism converges to the Riemann map of the domain onto the disk as the radius tends to zero. This idea, which has led to an entirely new approach to ideas of conformal mapping in recent years, will now be applied to study mappings of multiply connected domains onto domains whose complement consists solely of slits radiating from the origin. The result is known for many special cases - even some domains with infinite connectivity. That one can do it in general, was first conjectured by Koebe in 1908. He proved it for finitely connected domains. The combination of new tools from circle packing and other advances in conformal mapping will be applied in a concentrated effort to settle the Koebe conjecture in its entirety. Conformal mapping is the study of mappings of plane (and higher dimensional) domains by transformations which preserve infinitesimal angles and orientation. The maps play central roles in the geometric theory of analytic functions and potential theory by reducing many questions concerning functions defined on arbitrary domains to the same questions restricted to a class of highly symmetric domains. ***