Computational Discrete Conformal Geometry and Applications

9972769研究者进一步开发了软件工具CirclePack，并用它来探索在几个应用领域出现的数学问题。圆填充是由Thurston首先在轨道构造的背景下引入的，然后在保角映射的近似中引入。这个主题已经发展到提供了一个令人惊讶的全面的离散理论，它既平行又近似于解析函数的经典理论，即二维的保角映射。发展在很大程度上是由经典联系推动的——离散分析、极值长度、谐波测度、黎曼映射等等——而这些联系反过来又解释了圆填充在各种其他主题中的效用。在函数理论中，研究的重点是圆填充技术在研究一个猜想的小，共形瓷砖，和格罗登迪克的设计的结构。在应用中，圆填充为图嵌入提供了一种独特的新工具，并特别强调了脑平坦化和2-dim淬火的协作。圆封装的可计算性是一个核心优势，但是不断增长的需求给软件包“CirclePack”带来了相当大的压力。该项目解决了移植和模块化问题，并研究了实现并行打包算法的策略。科学、技术和数学的许多领域从根本上依赖于组合信息，即具体的和抽象的模式——DNA扭曲链中的交叉模式、晶体中的原子位置、视觉皮层中的激活位点。在二维环境中，这种模式通常被表示为平面图形，然后需要以视觉方式呈现。抽象拓扑学中出现了一种令人愉悦的新方法。“圆填充”是具有特定切线模式的圆的配置。它具有双重性质：在规定的模式下是“组合的”，但在实际圆的半径上是“几何的”。人们应该把圆看作是把“自发的”几何带到最初纯粹抽象的组合情况中。幸运的是，这种几何继承了一个重要的经典数学主题，保形几何的关键特征，因此它带来了超过150年的理论发展的应用。然而，与经典理论不同的是，圆填充既具体又可计算。该项目涉及圆填充算法和显示软件的开发和应用，特别强调与美国国立卫生研究院关于“大脑映射”的项目合作，关于二维淬火的物理，以及数学分析中的离散解析函数理论。

9972769The investigator further develops the software tool CirclePack and uses it to explore mathematical questions that arise in several areas of application. Circle packing was introduced by Thurston, first in the context of orbifold constructions and then in the approximation of conformal mappings. The topic has been developed to the point that it provides a surprisingly comprehensive discrete theory that both parallels and approximates the classical theory of analytic functions, i.e., conformal mappings in two dimensions. Developments have been driven largely by classical connections --- discrete analyticity, extremal length, harmonic measure, Riemann mappings, and so forth --- and in turn those same connections account for circle packing's utility in a variety of other topics. In function theory, the research focuses on circle packing techniques in studies of a conjecture of Smale, conformal tilings, and construction of Grothendieck's dessins d'enfants. In applications, circle packing provides a unique new tool for graph embedding, and particular emphasis is placed on collaborations in brain-flattening and 2-dim quenching. Computability of circle packings is a central strength, but growing demands have placed considerable stress on the software package "CirclePack." The project addresses porting and modularization issues, and investigates strategies for implementing parallel packing algorithms.Many areas of science, technology, and mathematics rely in fundamental ways on combinatorial information, that is, patterns, both concrete and abstract --- crossing patterns in a twisted strand of DNA, atomic locations in a crystal, activation sites in the visual cortex. In 2-dimensional settings such patterns are often represented as planar graphs, which then need to be presented visually. A pleasing new approach has emerged from abstract topology. A "circle packing" is a configuration of circles having a specified pattern of tangencies. It has dual natures: "combinatoric" in the prescribed pattern, but "geometric" in the radii of the actual circles. One should think of the circles as bringing a "spontaneous" geometry to what began as a purely abstract combinatorial situation. Fortuitously, this geometry inherits key features of an important classical mathematical topic, conformal geometry, so it brings to applications over 150 years of theoretical development. However, unlike the classical theory, circle packings are both concrete and computable. This project involves the development and application of circle packing algorithms and display software, with particular emphasis on collaboration with an NIH project on "brain mapping," on the physics of 2-dimensional quenching, and on discrete analytic function theory in mathematical analysis.