Loewner Evolutions and Random Maps

Loewner 演化和随机地图

• 批准号: 1068105
• 负责人: Steffen Rohde
• 金额: $ 34.5万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1068105&HistoricalAwards=false
• 关键词: Loewner; Evolutions; Random; Maps

Rohde will continue his investigations of conformal mappings generated by the Loewner differential equation, and will investigate the conformal geometry of random surfaces, particularly random planar maps such as the Benjamini-Schramm Uniformly Infinite Planar Triangulation. One aspect of Rohde's research is to understand similarities and differences between the deterministic and the stochastic Loewner equation, such as trace properties and their dependence on the regularity of the driving term. The Loewner equation is closely related to the "Zipper-algorithm" of Kuhnau and Marshall, and to conformal welding. Another aspect of Rohde's research is to prove convergence of this welding algorithm, and to study generalized weldings corresponding to non-simple curves. Rohde's research also explores the analytic foundations of the emerging theory of random maps and random Riemann surfaces, such as the type problem.What is the shape of a large molecule such as DNA? How does water percolate through soil? What are the patterns of spin-alignments in ferro-magnets? These problems are too complex to allow for a precise and simple mathematical answer. Even much simpler mathematical models, proposed by chemists and physicists, were out of reach of mathematicians for decades. This changed in 1999 with the invention of the Schramm-Loewner Evolution SLE, the Loewner equation driven by one-dimensional Brownian motion. The last decade has witnessed tremendous progress on old mathematical problems and even provided physicists with new and unexpected insights, as well as furthering interactions between probabilists, mathematical physicists, and complex analysts. Rohde's research aims at foundational questions of this emerging theory. Specifically, he is working on path properties of solutions of the Loewner equation in both the deterministic and probabilistic setting, and he is working towards an understanding of the large-scale behavior of random sphere-triangulations.

Rohde将继续对Loewner微分方程产生的共形映射进行调查，并将研究随机表面的共形几何形状，尤其是随机平面图，例如Benjamini-Schramm统一的无限平面三角剖分。 Rohde研究的一个方面是了解确定性和随机Loewner方程之间的相似性和差异，例如痕量特性及其对驾驶术语规律性的依赖。 Loewner方程与Kuhnau和Marshall的“拉链 - 算法”和保形焊接密切相关。 Rohde研究的另一个方面是证明这种焊接算法的融合，并研究与非简单曲线相对应的广义焊接。 Rohde的研究还探讨了随机图和随机Riemann表面的新兴理论的分析基础，例如类型问题。大分子（例如DNA）的形状是什么？水如何通过土壤渗透？铁磁铁中的自旋对齐的模式是什么？这些问题太复杂了，无法提供精确而简单的数学答案。几十年来，化学家和物理学家提出的更简单的数学模型已经无法触及数学家。随着Schramm-loewner Evolution Sle的发明，这是1999年发生的，这是由一维布朗尼运动驱动的Loewner方程。在过去的十年中，在旧数学问题上取得了巨大的进步，甚至为物理学家提供了新的和意外的见解，以及概率主义者，数学物理学家和复杂分析师之间的相互作用。 Rohde的研究旨在探讨这一新兴理论的基本问题。具体而言，他正在确定性和概率环境中介绍Loewner方程解的路径属性，并且正在努力理解随机球体 - 三角形的大规模行为。