Geometric Function Theory and Loewner Evolutions

摘要几何函数论和Loewner演化Rohde将继续研究由Loewner微分方程生成的共形映射及其相关主题。Loewner微分方程描述了在连续递增的单连通平面域序列上保形映射的流。它将这样的一系列域编码为一个实值函数，即方程的驱动项。Oded Schramm最近发现的随机Loewner演化SLE（驱动项是一维布朗运动），以及Smirnov关于临界渗流簇收敛于SLE的证明（6），开辟了一个新的研究领域，涉及共形映射、概率论和数学物理。统计物理学中被证明收敛于SLE的临界晶格过程的列表不断增长，并且正在研究不同于布朗运动的驱动项。自相似集（所谓的“分形”），如货车科赫雪花，在数学中发挥了重要作用，因为它们既为物理现象提供了天文模型，又可用于数学分析（动力学，遍历理论，共形映射）。最近在数学、概率论和统计物理学的各个分支中经常出现的“随机分形”（类似于分形但仅在统计上自相似的集合）需要一个允许严格分析的数学基础。施拉姆的系统性红斑狼疮是在这个方向上最令人兴奋的发展之一，并导致验证了许多预测早期的物理学家。罗德研究的核心是更好地理解随机分形，并通过共形映射回答有关它们的几个问题。

AbstractGeometric function theory and Loewner evolutionsRohde will continue to investigate conformal mappings generated by theLoewner differential equation and related topics. The Loewnerdifferential equation describes the flow associated with the conformalmappings onto a continuously increasing sequence of simply connectedplanar domains. It encodes such a sequence of domains into a real-valued function, the driving term of the equation. The recent discovery of the stochastic Loewner evolution SLE by Oded Schramm (the driving term is one-dimensional Brownian motion), together with results such as Smirnov's proof of convergence of critical percolation clusters to a SLE(6), has opened up a new area of investigations involving conformal mappings, probability theory and mathematical physics. The list of critical lattice processes from statistical physics that are conjectured to converge to SLE is constantly growing, and driving terms different from Brownian motion are under investigation.Self-similar sets (so called "fractals") such as the van Koch snowflakehave played an important role in mathematics because they serve both astoy models for physical phenomena, and are ameanable to mathematicalanalysis (dynamics, ergodic theory, conformal mapppings). The more recent recurrent appearance of "random fractals" (sets that resemble fractals but are only statistically self-similar) in various branches of mathematics, probability and statistical physics necessitates a mathematical foundation allowing rigorous analysis. Schramm's SLE is one of the most exciting developments in this direction and has resulted in verifications of numerous predictions made earlier by physicists. The core of Rohde's research is to better understand random fractals and to answer several questions about them by means of conformal mappings.