Mathematical Sciences: Circle Packings and Complex Analysis

这项数学研究的来源是W. Thurston五年前提出的一个猜想，即如果一个人在平面单连通域内填充圆，然后通过一个离散映射将圆联系起来，那么当圆变小时，映射将收敛于两个域之间的解析黎曼映射。此后不久，这个猜想得到了几个证明。这个项目继续研究圆填充和解析函数之间的联系，目的是发展一个几何上忠实的经典函数理论的离散模拟。圆形填料应用的最初发展依赖于六边形填料。出于美观和实用的原因，这一限制正在被取消。预计这项工作将对圆填料的基本行为产生更深入的了解。从平面上的圆填充研究到双曲几何的设置，也导致了组合学和几何之间的新联系。例如，在圆填充图的背景下，已经出现了Schwarz引理和Pick引理的双曲类比。最近在这些研究中引入了一个新元素，即使用随机漫步作为一个视角来观察从一个包装到下一个包装的过渡。例如，这个模型显示了边界半径的差异变化对填料的影响。模型的精度是准确的，反映了圆形填料的高度刚性。这项工作的第一个也是最重要的目标将是确定Mostow刚性定理的相应结果：填充双曲平面的无限圆填充是否唯一到自同构？还将做一些工作来回答哪些致密表面支持圆形填料的问题。

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.