Mathematical Sciences: Research in Conformal Mapping

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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. ***