Quasiconformal methods in analysis, geometry and dynamics

分析、几何和动力学中的拟共形方法

• 批准号: 1305233
• 负责人: Christopher Bishop
• 金额: $ 17.61万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-06-15 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1305233&HistoricalAwards=false
• 关键词: Quasiconformal; methods; analysis; geometry; dynamics

The PI, Christopher Bishop, will study the geometric properties of conformal and quasiconformal maps, with an emphasis on the connections with other areas such as dynamics, computational geometry, numerical analysis and geometric measure theory. The proposal describes several areas that the PI will investigate using quasiconformal mappings as a primary tool. The first is the iteration theory of complex analytic functions on the plane. The PI has previously used quasiconformal methods to solve a number of problems in the field and plans to refine these techniques to attack several remaining open problems. The second area is to use quasiconformal maps and hyperbolic geometry in computational geometry, mostly in problems related to algorithms for meshing planar regions with optimal complexity and geometry. Obtaining similar results in three dimensions is one of the most important goals of the field, and the proposal describes some ideas for attacking this problem too. The final part of the proposal describes the dimension distortion properties of quasiconformal maps. This is one of the most interesting problems intrinsic to study of quasiconformal maps, but, as the proposal describes, can also be linked to famous open problems in computational geometry.The proposed work will investigate how combinatorial and discrete ideas yield new results in analysis and how ideas from conformal analysis and hyperbolic geometry can prove new theorems about discrete geometry and meshing. Conformal maps preserve angles; these maps have been intensively studied for over 150 years and are of fundamental importance to a wide variety of problems in analysis, geometry, probability, physics and engineering. Classically, conformal maps have been used in the study of various differential equations related to fluid flow, heat conduction and wave propagation. More recently, conformal maps have been fundamental to the study of statistical mechanics, percolation and random growth models. Quasiconformal maps allow a controlled amount of angle distortion. These maps are a flexible and extremely useful generalization of conformal maps that help us better understand the special case of conformal mappings, but also introduces many important new problems and techniques. The PI's previous work has used quasiconformal maps to give the best known algorithm for computing conformal maps onto polygons and the best known algorithms for meshing planar domains into triangles or quadrilaterals with optimal geometric properties. Such meshes play an important role in many numerical problems from computer graphs to finite element methods for PDEs and many of these methods work more effectively if the underlying mesh has good geometric properties. The proposed work will extend and sharpen the results already obtained and will use similar ideas to attack other problems that arise in dynamics and geometry.

PI克里斯托弗·毕晓普（Christopher Bishop）将研究共形图和准文化图的几何特性，并着重于与其他领域的联系，例如动力学，计算几何，数值分析和几何测量理论。 该提案描述了PI将使用准文明映射作为主要工具进行调查的几个领域。第一个是平面上复杂分析函数的迭代理论。 PI先前已经使用了准文献方法来解决该领域的许多问题，并计划完善这些技术以攻击几个剩余的开放问题。第二个领域是在计算几何形状中使用准形式图和双曲几何形状，主要是与具有最佳复杂性和几何形状的网络平面区域相关的问题。在三维中获得相似的结果是该领域最重要的目标之一，该提案也描述了攻击此问题的一些想法。该提案的最后一部分描述了准形式图的维度失真属性。这是研究准文化图的最有趣的问题之一，但是正如该提议所描述的那样，这也可以与计算几何形状的著名开放问题有关。拟议的工作将研究组合和离散的想法如何产生分析的新结果，以及从共同分析和多余几何形状中产生的想法如何证明新的几何形状以及涉及离散的几何形状和梅斯特和梅尔的新理想。 保形地图保留角度；这些地图已对150多年的深入研究，对于分析，几何，概率，物理和工程方面的各种问题至关重要。 从经典上讲，已在与流体流，热传导和波传播有关的各种微分方程的研究中使用了共形图。 最近，共形图是统计力学，渗透和随机增长模型的研究至关重要的。 准文化图允许受控的角度失真。这些地图是对共形图的灵活且极为有用的概括，可以帮助我们更好地理解保形映射的特殊情况，同时也引入了许多重要的新问题和技术。 PI的先前工作使用了准文化图，以将共形图的算法计算到多边形上，以及最著名的平面域的最著名算法，用于具有最佳几何特性的三角形或四角形。 此类网格在从计算机图到有限元方法的许多数值问题中起着重要作用，并且如果基础网格具有良好的几何特性，那么许多这些方法都可以更有效地工作。 拟议的工作将扩展并增强已经获得的结果，并将使用类似的想法来攻击动态和几何形状中出现的其他问题。