Computational Uniformization

研究人员应用圈填充中的新兴计算方法来研究表面上的共形结构，特别是复杂的非平面表面。本质上是局部的共形结构的全局实现被经典地称为非均匀化。圆填充方法导致了离散均匀化的概念，它既模拟又近似于经典概念。然而，离散的几何性质在传统数值数学之外提出了计算问题。特别是，同质化是一个在实践中极具挑战性的自组装过程-例如，包含数百万个圆圈的圆圈填充。在中心计算工作中，调查者实现了一个递归框架来管理这种自组装，该框架避免了全局扭曲，同时适应了高效的并行实现。计算问题并不是孤立地考虑的，而是与研究人员和他的合作者正在进行的应用相关的，这些主题包括大脑成像、保形瓷砖、乳剂和保形焊接。在应用和实验中提出的许多理论问题中，研究人员特别关注“流均匀”的概念及其在离散保形焊接和形状分析中的潜在应用。表面-光滑的肥皂膜、复杂的大脑灰质、小平面晶格-在自然科学、物理科学、工程、计算机可视化和许多其他领域中普遍存在。用于数学研究和描述表面的工具出现在19世纪，与角度测量相关的特别丰富的几何纹理被称为“保形结构”。数学上最著名的结果之一是1851年的黎曼映射定理，它证明了每个具有保角结构的曲面，无论多么复杂，都可以用三个非常简单的熟悉曲面之一-球、平面或圆盘-来识别，其方式是保持保角结构。尽管这个理论很棒，而且尽管有了巨大的计算资源，但直到最近十年，以圆填充形式出现的新数学才为实际实现黎曼定理提供了一种实用的方法。研究人员在几个应用的背景下发展了圆填充的数学和计算方面。其中包括将人脑皮质表面扁平化以帮助神经科学家进行分析，通过计算机视觉中的保形焊接过程来研究平面形状，以及在各种主题中构建数学黎曼表面，这些都是第一次可以进行实验。

The investigator applies emerging computational methods in circlepacking to study conformal structures on surfaces, with particularemphasis on complex, nonplanar surfaces. The global realization ofconformal structures which are local in nature is known classically asuniformization. Circle packing methods have led to notions of discreteuniformization which both mimic and approximate the classical notion.However, the geometric nature of the discretization raisescomputational issues outside traditional numerical mathematics. Inparticular, uniformization is a self-assembly process which isextremely challenging in practice --- for instance, with circlepackings containing millions of circles. In the central computationalwork, the investigator implements a recursive framework for managingthis self-assembly which avoids global distortions while accommodatingefficient parallel implementation. The computational issues are notconsidered in isolation, but rather in relation to ongoingapplications by the investigator and his collaborators to topicsincluding brain imaging, conformal tiling, dessins d'enfants, andconformal welding. Of the many theoretical issues raised inapplications and experiments, the investigator pays special attentionto the notion of ``flow uniformization'' and to its potential use indiscrete conformal welding and shape analysis.Surfaces --- a smooth soap film, the convoluted gray matter of thebrain, a faceted crystal lattice --- are ubiquitous in the natural andphysical sciences, engineering, computer visualization, and scores ofother areas. The tools for studying and describing surfacesmathematically came out of work in the nineteenth century, with aparticularly rich geometric vein associated with angular measure goingby the name "conformal structure". Among the most celebrated results inmathematics is the Riemann mapping theorem of 1851 which proved thatevery surface with a conformal structure, no matter how complicated,can be identified with one of three very simple familiar surfaces--- a ball, a plane, or a disc --- in a way that preserves conformalstructure, that is, that preserves angles. Wonderful as this theoryis, and despite the availability of huge computational resources, ithas been only in the last decade that new mathematics in the form ofcircle packing has provided a practical means for actually realizingRiemann's theorem. The investigator develops the mathematical andcomputational aspects of circle packing in the context of severalapplications. Among these are the flattening of human brain corticalsurfaces to aid analysis by neuroscientists, the study of planeshapes through a process known as conformal welding for use incomputer vision, and the construction of mathematical Riemann surfacesin various topics which are now amenable to experimentationfor the first time.