Mathematical Sciences: Discrete Analytic Function Theory Via Circle Packing

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This project continues investigations into the connections between circle packing and the theory of analytic functions of a complex variable. Circle packings have long been an intriguing concept in mathematical research. In 1985 William Thurston proposed that circle packings - packings of plane domains by circles of various sizes - could lead to a completely new point of view in conformal mapping. This turned out to be the case and this project is one of several by-products of that now famous lecture. The idea has grown and prospered. Circle packings are exhibiting faithful discrete analogues to the classical results of geometric function theory such as the Schwarz-Pick lemma, Dirichlet problem, uniformization theorem, Brownian motion, etc. The current project concentrates on the discrete version of the length-area method, a powerful, but unwieldy method exercised by a few experts and not easily understood by consumers of function theory. The second part concerns infinite packings, possibly the most speculative part of the project. Most research to date considers only finite packings. The final component of the research involves packings using circle with overlap. There are fundamental results which allow for overlap, but little work has been done on this generalization. Past research led to the development of software for manipulating, displaying and animating circle packings. This is both a research tool for performing experiments and a practical method for creating the pictures needed for presentation. The software is freely available. Complex function theory is the mathematical basis for computing with and visualizing the behavior of complex functions of a complex variable. The subject has an extensive history and is normally studied concurrently as analysis and geometry. This project adds a new dimension: computation. Circle packing leads naturally to computer simulation. This in turn has led to new insights and proofs in the classical theory. To prove its worth, the new point of view will have to lead to new , unexpected discoveries.