Computational Uniformization

计算统一化

• 批准号: 0609715
• 负责人: Kenneth Stephenson
• 金额: $ 20.16万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-15 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0609715&HistoricalAwards=false
• 关键词: Computational; Uniformization

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.

调查人员适用于新兴的计算方法在circlepacking研究表面上的保形结构，particularembranes在复杂的，非平面的表面。 局部共形结构的全局实现在经典上称为一致化。圆填充方法导致了离散均匀化的概念，它既模仿又近似于经典概念。然而，离散化的几何性质提出了传统数值数学之外的计算问题。特别是，均匀化是一个自组装过程，在实践中极具挑战性-例如，包含数百万个圆的圆包装。在中心计算工作中，研究者实现了一个递归框架来管理这种自组装，从而避免了全局扭曲，同时提高了并行实现的效率。计算问题不是孤立地考虑的，而是与研究者及其合作者正在进行的应用有关，包括脑成像、适形铺瓦、儿童死亡和适形焊接。在应用和实验中提出的许多理论问题中，研究者特别注意“流动均匀化”的概念及其在离散保形焊接和形状分析中的潜在用途。表面--光滑的肥皂膜，大脑的回旋灰质，多面晶格--在自然和物理科学，工程学，计算机可视化和许多其他领域中无处不在。数学上研究和描述曲面的工具出现在世纪，与角测量相关的非常丰富的几何脉络被称为“共形结构”。其中最著名的结果在数学是黎曼映射定理的1851年证明thatevery表面的共形结构，无论多么复杂，可以确定的一个非常简单的熟悉的表面--一个球，一个平面，或一个磁盘--在某种程度上保持conformalstructure，即保持角度。尽管这个理论很奇妙，尽管有巨大的计算资源，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。调查员开发的数学和计算方面的圆包装的上下文中的几个应用程序。其中包括人类大脑皮层表面的扁平化，以帮助神经科学家进行分析，通过一种被称为保形焊接的过程来研究平面形状，用于计算机视觉，以及在各种主题中构建数学黎曼曲面，现在第一次可以进行实验。