Conformal Mapping

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。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性质如何反映在驱动函数中。这是一个经典问题，直到最近才取得进展。Loewner方程还与Marshall和K\ uhnau发现的一种数值共形映射算法有关。Marshall将分析“拉链”算法的收敛性和误差估计，并使用“分代”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一个复杂的区域转换为一个简单的区域，如圆盘。然后将复杂区域上的偏微分方程转换为圆盘上的类似方程，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。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最近发明的随机Loewner演化SLE，即Loewner微分方程和概率的融合，在数学中的共形映射和物理中的共形场论的重要领域之间架起了一座桥梁。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。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。通过我们的复杂分析“工作研讨会”指导博士后学者和研究生，支持了几位女性的工作。对我们研究的支持增加了有兴趣从事这方面职业的学生人数。

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.