Conformal Mapping

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最近发现的随机Loewner演化SLE，Loewner方程驱动的一维布朗运动，开辟了一个新的领域的调查涉及共形映射，概率论和数学物理。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。Loewner方程也与马歇尔和Kuhnau发现的数值保角映射算法有关。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平滑度的驱动功能，并反过来调查如何平滑的边界的相关地区意味着平滑的驱动功能。这是一个经典的问题，最近才取得进展。马歇尔将分析收敛性和误差估计的数值映射方法称为"拉链“，密切相关的Loewner方程，并提高收敛速度使用"生成”技术。马歇尔的研究计划的主题是研究共形映射产生的Loewner微分方程，以及相关的主题。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复杂区域更改为更简单的区域，如圆盘。物理过程由偏微分方程模拟。在复杂的二维区域上的偏微分方程可以通过保角映射变成圆盘上的类似方程，这是一种更容易求解的设置。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。共形映射和共形场论的基础研究分别是数学和物理学的重要方向。Loewner微分方程和概率的融合形成了这两个领域之间的桥梁。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，这些代码已被一些数学以外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。更广泛的影响还包括帮助巩固现代复分析和数学物理之间更强大的关系，这对双方都有利。

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.