Analysis of conformal and quasiconformal maps

共形和拟共形映射的分析

• 批准号: 1006309
• 负责人: Christopher Bishop
• 金额: $ 20.04万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-15 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1006309&HistoricalAwards=false
• 关键词: Analysis; conformal; quasiconformal; maps

AbstractAward: DMS-1006309Principal Investigator: Christopher BishopThe principal investigator will study the geometric properties of conformal and quasiconformal maps, with an emphasis on the connections with other areas such as numerical analysis, computational geometry, complex dynamics and geometric measure theory. He will continue his work on fast conformal mapping algorithms, including the class of Schwarz-Christoffel iterations. These include Davis' method and the CRDT algorithm of Driscoll and Vavasis as special cases, and other methods involving the medial axis and quasiconformal mappings. One of the basic themes is to start with certain quasiconformal maps that approximate the desired conformal map, but are easier to compute and have better properties. These maps have natural connections to several well known problems such as Brennan's conjecture, and the PI will investigate these connections. The PI will also continue his work on various related problems involving optimal meshing algorithms, diffusion limited aggregation, the geometry of Brownian motion, the quasiconformal Jacobian problem and conformal welding.A fundamental problem of mathematics and engineering is to quickly and accurately solve various differential equations related to fluid flow, heat conduction and wave propagation on complex regions. For 2-dimensional surfaces, a standard method is to use a conformal mapping (i.e., angle preserving on small scales) to replace the region with a simpler one, such as a disk or rectangle. This approach has been studied for over 150 years, but not until the 1980's did computers made conformal mapping practical for highly complex regions. The PI will investigate how to improve existing methods and develop new ones that are faster and more reliable. He will also investigate the theoretical behavior of conformal maps on extremely complex domains such as fractals and the connections between conformal maps and probability theory. The PI has already applied the insights gained from these problems to meshing. This is the process of dividing a complex region into simple pieces such as triangles. Meshing is a basic part of most numerical methods and the usefulness of a mesh in applications depends on the number of pieces (fewer is better) and their shapes (better to avoid small and large angles). It is difficult to construct a mesh which is good in both respects, but PI has used non-Euclidean geometry to construct 2-dimensional quadrilateral meshes with optimal size and shapes for simple polygons and is working to extend this to more general domains. Greater generality is needed in various problems involving crack formation, interfaces between materials and computer learning. Efficient meshing also has numerous applications in high-performance computing such as modeling surfaces for engineering and computer graphics. This award is jointly funded by the programs in Analysis and Geometric Analysis.

Abstractaward：DMS-1006309原理研究者：Christopher Bishopthe主要研究者将研究保形和准形式图的几何特性，并强调与其他领域的联系，例如数值分析，计算几何，复杂的动力学和几何学测量理论。他将继续在快速的保形映射算法上进行工作，包括Schwarz-Christoffel迭代类。其中包括Davis的方法和Driscoll和Vavasis的CRDT算法，以及其他涉及内侧轴和准文献映射的方法。 基本主题之一是从某些近似所需的共形映射近似的准形式图开始，但易于计算和具有更好的属性。这些地图与几个众所周知的问题（例如Brennan的猜想）有自然的联系，PI将研究这些联系。 PI还将继续他在各种相关问题上的工作，这些问题涉及最佳的网格算法，扩散限制聚集，布朗运动的几何形状，准形式的Jacobian问题和保形焊接。数学的基本问题和工程的基本问题将迅速，准确地求解各种差分方程与流体流动相关的抑制，热量和波动率在热量上，潮汐和波浪反应。 对于二维表面，标准方法是使用共形映射（即，在小尺度上保留角度）用更简单的区域（例如磁盘或矩形）代替该区域。这种方法已经研究了150多年，但直到1980年代才为高度复杂的区域进行了保形映射。 PI将研究如何改善现有方法并开发更快，更可靠的方法。他还将研究在极其复杂的领域（例如分形）以及保形图和概率理论之间的联系的理论行为。 PI已经将这些问题从这些问题中获得的见解应用于网格划分。这是将复杂区域分为简单片段（例如三角形）的过程。 网格是大多数数值方法的基本组成部分，网格在应用中的有用性取决于碎片的数量（较少的碎片）及其形状（最好避免较小和大角度）。 在这两个方面都很难构造一个良好的网格，但是PI已使用非欧几里得几何形状来构建具有最佳尺寸和形状的二维四边形网格，用于简单多边形，并正在努力将其扩展到更一般的域。 在涉及裂纹形成，材料和计算机学习之间的界面的各种问题中，需要更大的通用性。 有效的网格划分在高性能计算中还具有许多应用，例如用于工程和计算机图形的建模表面。 该奖项由分析和几何分析计划共同资助。