Geometric Function Theory and Loewner Evolutions

摘要几何函数理论与Loewner演化rohde将继续研究由Loewner微分方程生成的保角映射及相关课题。loewner微分方程描述了与保形映射相关的流到连续增加的单连通平面区域序列上。它将这样一个域序列编码成一个实值函数，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证明临界渗流簇收敛于SLE(6)等结果，开辟了一个涉及共形映射、概率论和数学物理的新研究领域。统计物理学中被推测收敛于SLE的临界晶格过程的列表正在不断增长，并且不同于布朗运动的驱动项正在研究中。自相似集（所谓的“分形”），如范·科赫雪花，在数学中扮演着重要的角色，因为它们既可以作为物理现象的模型，也可以用于数学分析（动力学、遍历理论、共形映射）。最近在数学、概率和统计物理的各个分支中反复出现的“随机分形”（与分形相似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。Schramm的SLE是这个方向上最令人兴奋的发展之一，它验证了物理学家早期的许多预测。Rohde研究的核心是更好地理解随机分形，并通过共形映射来回答关于随机分形的几个问题。

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.