Loewner Evolutions and Random Maps

Rohde将继续他对由Loewner微分方程生成的共形映射的研究，并将研究随机曲面的共形几何，特别是随机平面图，如Benjamini-Schramm均匀无限平面三角剖分。Rohde研究的一个方面是了解确定性和随机Loewner方程之间的异同，例如迹属性及其对驱动项的规律性的依赖。Loewner方程与Kuhnau和Marshall的“zipper算法”密切相关，与保形焊接密切相关。Rohde的另一个研究方面是证明该焊接算法的收敛性，并研究非简单曲线对应的广义焊接。Rohde的研究还探讨了新兴的随机映射和随机黎曼曲面理论的分析基础，如类型问题。像DNA这样的大分子是什么形状的？水是如何渗透土壤的？铁磁体的自旋排列模式是什么？这些问题太复杂了，不可能有一个精确而简单的数学答案。即使是化学家和物理学家提出的更简单的数学模型，几十年来数学家也无法触及。这种情况在1999年随着Schramm-Loewner演化SLE（由一维布朗运动驱动的Loewner方程）的发明而改变。在过去的十年里，在古老的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想不到的见解，并进一步促进了概率论家、数学物理学家和复杂分析学家之间的互动。罗德的研究旨在解决这一新兴理论的基本问题。具体地说，他正在研究Loewner方程在确定性和概率设置下的解的路径性质，他正在努力理解随机球三角测量的大规模行为。

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.