Loewner Evolutions and Random Maps

罗德将继续他的调查共形映射所产生的Loewner微分方程，并将调查共形几何的随机表面，特别是随机平面地图，如Benjamini-施拉姆均匀无限平面三角剖分。罗德的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，如跟踪属性及其对驱动项的规律性的依赖。Loewner方程与Kuhnau和马歇尔的“拉链算法”以及保形焊接密切相关。Rohde研究的另一个方面是证明这种焊接算法的收敛性，并研究对应于非简单曲线的广义焊接。罗德的研究还探索了随机映射和随机黎曼曲面的新兴理论的分析基础，例如类型问题。DNA这样的大分子的形状是什么？水是如何渗透土壤的？铁磁体中自旋排列的模式是什么？这些问题太复杂了，无法用精确而简单的数学来回答。即使是由化学家和物理学家提出的更简单的数学模型，几十年来也是数学家们无法企及的。这在1999年随着Schramm-Loewner Evolution SLE的发明而改变，Loewner方程由一维布朗运动驱动。在过去的十年里，老的数学问题取得了巨大的进展，甚至为物理学家提供了新的和意想不到的见解，以及促进概率学家，数学物理学家和复杂分析师之间的互动。罗德的研究旨在解决这一新兴理论的基础问题。具体而言，他正在研究路径属性的解决方案的Loewner方程在确定性和概率设置，他正在努力了解大规模的行为随机球三角。

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.