Conformal Mappings and Loewner Evoluation

提案编号：DMS-0201435PI：Donald Matt和Steffen RohdeABSTRACT保角映射和洛夫纳演化马歇尔和罗德将研究由洛夫纳微分方程生成的保形映射，以及相关主题。Loewner方程描述了在连续递减的单连通平面区域上与共形映射相关的流。Schramm最近发现了随机Loewner演化SLE，即由一维布朗运动驱动的Loewner方程，开辟了涉及共形映射、概率论和数学物理的研究的新领域。Loewner方程也与数值共形映射的算法有关。共形映射在数学内外的许多领域都有应用，如控制理论、热传导、流体力学和复动力学。它们经常将坐标从一个区域转换到一个更简单的区域，如圆盘。具有光滑边界的区域被很好地理解。然而，在许多科学分支中出现的分形图导致了从共形映射的观点来研究由高度非光滑的、分形曲线所包围的区域的自然问题。近年来，由随机过程产生的分形曲线，例如在统计物理学中，受到了极大的关注。马歇尔和罗德研究的核心是通过共形映射更好地理解随机分形曲线，反过来通过分析分形曲线所包围的区域来研究关于共形映射的一些问题。

Proposal Number: DMS-0201435PI: Donald Marshall and Steffen RohdeABSTRACTConformal mapping and Loewner evolutionsMarshall and Rohde will investigate conformal mappings generated by the Loewner differential equation, and related topics. TheLoewner differential equation describes the flow associated withthe conformal mappings onto a continuously decreasing sequence ofsimply connected planar domains. It relates a sequence of domainsto a real-valued function, the driving term of the equation.Schramm's recent discovery of the stochastic Loewner evolutionSLE, the Loewner equation driven by one-dimensional Brownianmotion, has opened up a new area of investigations involvingconformal mappings, probability theory and mathematical physics.The Loewner equation is also related to an algorithm fornumerical conformal mapping.Conformal mappings have applications in many areas, both within and outside of mathematics, such as control theory, heatconduction, fluid dynamics, and complex dynamics. They are oftenused to change coordinates from one region to a simpler regionlike a disc. Regions with smooth boundaries are well understood.However, the appearance of fractals in many branches of science led to the natural problem of investigating regions bounded by highly nonsmooth, fractal curves, from the conformal mappingpoint of view. In recent years, fractal curves generated byrandom processes arising, for instance, in statistical physics,have received enormous attention. The core of Marshall's andRohde's research is to better understand random fractal curvesby means of conformal mappings, and conversely to study someproblems about conformal mappings by analyzing domains boundedby fractal curves.