Mathematical Sciences: Research in Conformal Mapping

9400733罗丹，这项由该奖项支持的工作开始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。如果用固定半径的圆填充单连通区域，则存在单位圆盘的组合等价填充，并且当半径趋于零时，这种圆填充同构收敛到区域到圆盘上的黎曼映射。这一思想在最近几年导致了一种全新的共形映射思想，现在将被应用于研究多重连通区域到其补完全由从原点辐射的狭缝组成的区域上的映射。这一结果在许多特殊情况下都是已知的--甚至某些具有无限连通性的域也是如此。一般情况下，人们可以做到这一点，这是Koebe在1908年首次猜测的。他对有限连通域证明了这一点。结合圆填充的新工具和保角映射的其他进展，将集中力量解决Koebe猜想的全部问题。保角映射是通过保持无限小角度和方向的变换来研究平面(和高维)区域之间的映射。映射在解析函数几何理论和位势理论中起着核心作用，它将定义在任意区域上的许多函数问题归结为限制在一类高度对称区域上的相同问题。***

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. ***