Mathematical Sciences: Circle Packings and Complex Analysis

这项数学研究的来源是一种猜想 由W.瑟斯顿五年前的大意是，如果一个 平面单连通域中的填充圆，然后 用一个离散映射把这些圆联系起来， 变小，映射将收敛到解析黎曼 两个域之间的映射。 猜想的几种证明 随后不久。 该项目继续研究 圆填充和解析函数，目的是 开发了部分几何上忠实的离散模拟 经典函数论 初步发展情况 圆形填料的应用依赖于六边形填料。 为 出于美学和实用的原因，这种限制正在被 删除.深入了解圆的基本行为 预计这项工作将产生填料。 背离 平面内圆填充问题的研究 双曲几何也导致了 组合学和几何学 已经出现的，因为 例如，是施瓦茨引理和皮克的双曲类似物 引理在圆包装映射的上下文中。 最近在这些研究中引入的一个新因素是， 使用随机游走作为一个视角， 从一种包装过渡到另一种包装。 该模型显示，对于 例如，如何在边界的差异变化的影响 半径分布在填料周围。 的精度 该模型是准确的，反映了深刻的刚性循环 填料。 这项工作的第一个也是最重要的目标是 确定Mostow刚度的相应结果 定理：是无限圆填充双曲 自同构的唯一平面？ 还将开展工作， 回答哪些紧致曲面支持圆的问题 填料。

The source of this mathematical research is a conjecture made by W. Thurston five years ago to the effect that if one packed circles inside planar simply connected domains and then associated the circles by a discrete map, then as the circles became smaller, the maps would converge to the analytic Riemann map between the two domains. Several proofs of the conjecture followed shortly thereafter. This project continues the study of connections between circle packings and analytic functions, with the aim of developing a geometrically faithful discrete analogue of parts of classical function theory. Initial developments in the application of circle packing relied on hexagonal packings. For reasons both aesthetic and practical, this restriction is being removed. Deeper insights into the fundamental behavior of circle packings is expected to result from this work. A departure from studies of circle packing in the plane to the setting of hyperbolic geometry is also leading to new connections between combinatorics and geometry. What has already emerged, for example, are hyperbolic analogues for the Schwarz lemma and Pick lemma in the context of circle packing maps. A new element recently introduced into these studies is that of using a random walk as a perspective from which to view the transition from one packing to the next. This model shows, for instance, how the effects of differential changes in boundary radii distribute themselves about the packing. The precision of the model is exact, reflecting the profound rigidity in circle packings. The first and most important goal of this work will be to determine the corresponding result of the Mostow rigidity theorem: Are infinite circle packings which fill the hyperbolic plane unique up to automorphisms? Work will also be done to answer the question of which compact surfaces support circle packings.