Conformal Mapping

共形映射

• 批准号: 0602509
• 负责人: Donald Marshall
• 金额: $ 12.13万
• 依托单位: University of Washington
• 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 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0602509&HistoricalAwards=false
• 关键词: Conformal; Mapping

The Loewner differential equation was introduced in 1923 to study extremal problems for conformal maps in the unit disc. Schramm's recent discovery of the stochastic Loewner evolution SLE, the Loewner equation driven by one-dimensional Brownian motion, has opened up a new area of investigations involving conformal mappings, probability theory and mathematical physics. It has led to the discovery of new results in percolation and random walks, for example, as well as the discovery mathematical proofs of results known to the theoretical physics community. The Loewner equation is also related to an algorithm for numerical conformal mapping discovered by Marshall and K\"uhnau. The Loewner equation has as input an arbitrary continuous function and produces a continuous family of conformal mappings. Marshall plans to investigate properties of the solutions of Loewner's equation under various assumptions on the smoothness of the driving function, and conversely to investigate how smoothness of the boundaries of the associated regions implies smoothness of the driving function. This is a classical problem where progress has been made only recently. Marshall will analyze convergence and error-estimates for the numerical mapping method called ``zipper'', closely related to Loewner's equation, and improve the speed of convergence using ``generational'' techniques.The main theme of Marshall's research program is to study conformal mappings generated by the Loewner differential equation, and related topics. Conformal mappings have been used as a tool in science and engineering for many years. They are often used to change coordinates from a complicated region to a simpler region like a disc. Physical processes are modeled by partial differential equations. A partial differential equation on the complicated two dimensional region can be changed by a conformal map to a similar equation on the disc, a setting where it is easier to solve. Classically, this method was used for problems related to Laplace's equation, such as electrostatics and two dimensional fluid flow. Numerous non-classical applications have been developed in the last three decades such as electro-magnetics, vibrating membranes and acoustics, transverse vibrations and buckling of plates, elasticity, and heat transfer. Fundamental research in conformal mapping and conformal field theory are important directions in mathematics and physics respectively. The fusion of Loewner's differential equation and probability forms a bridge between these two areas. This project is likely to increase the understanding of solutions to Loewner's equation, as foundational work, which should increase its usefulness in understanding stochastic processes. Broader impacts include the continued improvement and dissemination of the conformal mapping computer codes, which have been used by a number of investigators not in mathematics, as well as mathematicians. Greater speed and new knowledge of convergence should lead to wider applicability and use of this algorithm. Broader impacts also include helping to cement a stronger relationship between modern complex analysis and mathematical physics, which should benefit both.

1923年引入了Loewner微分方程来研究单位圆盘上共形映射的极值问题。Schramm最近发现的随机Loewner演化SLE，即一维布朗运动驱动的Loewner方程，开辟了一个涉及保角映射、概率论和数学物理的新研究领域。例如，它导致了在渗透和随机漫步方面的新结果的发现，以及理论物理界已知的结果的数学证明的发现。Loewner方程还与Marshall和K\ uhnau发现的一种数值共形映射算法有关。洛厄纳方程以任意连续函数作为输入，产生连续的保角映射族。Marshall计划研究在驱动函数光滑性的各种假设下Loewner方程解的性质，反过来研究相关区域边界的光滑性如何意味着驱动函数的光滑性。这是一个经典问题，直到最近才取得进展。马歇尔将分析与洛厄纳方程密切相关的称为“拉链”的数值映射方法的收敛性和误差估计，并使用“分代”技术提高收敛速度。Marshall的主要研究课题是研究由Loewner微分方程生成的保角映射，以及相关课题。保角映射作为一种工具在科学和工程中已经使用了很多年。它们通常用于将坐标从一个复杂的区域转换为一个简单的区域，如圆盘。物理过程是用偏微分方程来模拟的。复杂的二维区域上的偏微分方程可以通过向圆盘上的类似方程的保角映射来改变，这是一种更容易求解的设置。经典地，这种方法被用于与拉普拉斯方程相关的问题，如静电学和二维流体流动。在过去的三十年中，许多非经典应用得到了发展，如电磁学、振动膜和声学、横向振动和板的屈曲、弹性和传热。保角映射和保角场论的基础研究分别是数学和物理学的重要方向。洛厄纳微分方程和概率的融合在这两个领域之间架起了一座桥梁。这个项目可能会增加对Loewner方程解的理解，作为基础工作，这应该会增加它在理解随机过程中的有用性。更广泛的影响包括保角映射计算机代码的不断改进和传播，这些代码已被许多非数学领域的研究人员以及数学家使用。更快的速度和新的收敛知识将导致该算法更广泛的适用性和使用。更广泛的影响还包括帮助巩固现代复杂分析和数学物理之间更牢固的关系，这对双方都有利。