Conformal Mapping

共形映射

• 批准号: 0900814
• 负责人: Donald Marshall
• 金额: $ 28.53万
• 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: 2009
• 资助国家: 美国
• 起止时间: 2009-06-01 至 2012-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0900814&HistoricalAwards=false
• 关键词: Conformal; Mapping

The main theme of Marshall's research program is to study conformal mappings generated by the Loewner differential equation, and related topics. 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 driving function, and conversely to investigate how properties of the boundaries of the associated regions are reflected in the driving function. This is a classical problem where progress has been made only recently. The Loewner equation is also related to an algorithm for numerical conformal mapping discovered by Marshall and K\"uhnau. Marshall will analyze convergence and error-estimates for the "zipper"' algorithm and improve the speed of convergence using "generational"' techniques. 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. A partial differential equation on the complicated region is then changed 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. The Loewner differential equation was introduced in 1923 to study extremal problems for conformal maps in the unit disc. Schramm's recently invention of stochastic Loewner evolution SLE, the fusion of Loewner's differential equation and probability, has formed a bridge between the important areas of conformal mapping in mathematics and conformal field theory in 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. 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 by mathematicians. Greater speed and new knowledge of convergence should lead to wider applicability and use of this algorithm. The mentoring of postdoctoral scholars and graduate students through our complex analysis "working seminar", has supported the work of several women. Support of our research increases the number of students interested in pursuing a career in this direction.

马歇尔的主要研究课题是研究由Loewner微分方程产生的共形映射以及相关的课题。 Loewner方程具有任意连续函数作为输入，并产生连续的共形映射族。马歇尔计划调查性质的解决方案Loewner方程的各种假设下的驱动功能，并反过来调查如何属性的边界的相关地区反映在驱动功能。这是一个经典的问题，最近才取得进展。Loewner方程也与马歇尔和Kuhnau发现的数值保角映射算法有关。 马歇尔将分析"拉链"算法的收敛性和误差估计，并使用"代"技术提高收敛速度。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复杂区域更改为更简单的区域，如圆盘。然后将复杂区域上的偏微分方程变为圆盘上的类似方程，这是一个更容易求解的设置。经典上，这种方法用于与拉普拉斯方程有关的问题，如静电和二维流体流动。 在过去的三十年中，已经开发了许多非经典应用，例如电磁学，振动膜和声学，板的横向振动和屈曲，弹性和传热。 Loewner微分方程是在1923年推出的研究极值问题的共形映射的单位圆盘。Schramm最近提出的随机Loewner演化SLE，是Loewner微分方程与概率的融合，它在数学中的共形映射和物理学中的共形场论这两个重要领域之间架起了一座桥梁。例如，它导致了在渗流和随机游走中发现新的结果，以及发现理论物理界已知结果的数学证明。 该项目可能会增加对Loewner方程解的理解，作为基础工作，这将增加其在理解随机过程中的有用性。更广泛的影响包括保形映射计算机代码的持续改进和传播，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。更快的速度和新的知识收敛应导致更广泛的适用性和使用这种算法。通过我们的复杂分析"工作研讨会"，对博士后学者和研究生进行辅导，支持了几名妇女的工作。我们的研究的支持增加了有兴趣在这个方向追求职业生涯的学生人数。