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.

马歇尔研究计划的主要主题是研究由Loewner微分方程产生的保形映射以及相关主题。 Loewner方程具有任意连续函数的输入，并产生连续的保形映射系列。马歇尔计划在各种假设下对驾驶功能的各种假设进行研究的属性，相反，研究如何在驾驶功能中反映相关区域边界的属性。这是一个古典问题，直到最近才取得进展。 Loewner方程还与Marshall和K \“ Uhnau。Marshall发现的数值共形映射算法有关，Marshall将分析“ Zipper”算法的收敛性和错误估计值，并提高了使用“世代”技术的收敛速度。多年来，它们被用作科学和工程的工具。在光盘上，可以更易于解决的设置。 1923年，引入了电磁性，振动膜和声学，横向振动和板的屈曲，弹性和热传递。施拉姆（Schramm）最近对Loewner微分方程和概率融合的随机Loewner Evolution SLE发明的发明已经形成了数学中相结合映射的重要领域与物理学中的保形场理论之间的桥梁。它导致发现了渗透和随机步行的新结果，例如，理论物理界已知的结果的发现数学证明。 该项目可能会增加对Loewner方程解决方案的理解，因为基础工作应该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，这些计算机代码已被许多研究人员在数学中以及数学家使用。更高的速度和新知识应导致更广泛的适用性和使用该算法。通过我们复杂的分析“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。对我们的研究的支持增加了有兴趣从事这一方向的职业的学生人数。