Geometric Function Theory and Dynamics

建议：DMS-9970398首席研究员：Steffen R.Rohde摘要：Rohde将使用几何函数论和动力系统的方法研究共形和准共形映射的扭曲性质。拟共形映射是“几乎共形”的弱可微同胚，因为它们至多使角度扭曲一个有界因子。这些映射推广了共形映射，在几何学、几何函数论、复动力学和偏微分方程组中自然产生。Rohde将研究的两个具体问题是估计允许拟共形扩张到复平面的圆盘的共形映射的导数，以及寻找圆(所谓的拟圆)的拟共形映象的Hausdorff维数的界。这两个问题都与Brennan关于单位圆盘上保角映射导数的可积度的猜想密切相关。Rohde还将研究欧几里德空间中集合的几何性质，如孔隙度或摇摆度，以及它们的Hausdorff维度的估计。这类集合在复杂动力学和几何函数理论中都很自然地出现。共形映射在许多领域都有应用，无论是在数学领域内还是在数学之外。这些学科包括控制理论、热传导、流体动力学和复杂动力学，仅举几例。协形映射的一个标准用法是将坐标从一个区域改变到一个更简单的区域，比方说到一个圆盘，在那里可以从一个新的角度来看待一个问题，希望在这个新的角度下，原始问题的解决方案变得更明显。一个著名的例子是这种方法在空气动力学中产生了巨大的红利，在空气动力学中，保形映射在提出翼型的原始设计轮廓方面发挥了重要作用。从保角映射的观点来看，具有光滑边界的区域已经被很好地理解了一段时间。然而，分形学在许多科学分支中的出现导致了研究由高度非光滑的分形型曲线所包围的区域的保角映射性质的自然问题。Rohde研究的核心目标一方面是通过共形映射来加深对分形曲线的理解，另一方面是通过分析具有分形边界的区域的几何来研究关于共形映射的一些问题

Proposal: DMS-9970398Principal Investigator: Steffen R. RohdeAbstract: Rohde will investigate distortion properties of conformal and quasiconformal mappings, using methods from geometric function theory as well as from dynamical systems. Quasiconformal mappings are weakly differentiable homeomorphisms that are "almost conformal," in the sense that they distort angles by at most a bounded factor. These maps generalize conformal mappings and arise naturally in geometry, geometric function theory, complex dynamics, and PDE. Two specific problems Rohde will investigate are to estimate the derivative of a conformal map of a disk that admits a quasiconformal extension to the complex plane and to find bounds for the Hausdorff dimension of quasiconformal images of circles (so-called quasicircles). Both questions are closely related to Brennan's conjecture concerning the degree of integrability of the derivative of a conformal mapping of the unit disk. Rohde will also study geometric properties such as porosity or wiggliness of sets in euclidean space, together with estimates for their Hausdorff dimension. Such sets appear naturally both in complex dynamics and in geometric function theory.Conformal mappings have applications in many areas, both within mathematics and outside it. These include control theory, heat conduction, fluid dynamics, and complex dynamics, to name just a few. One standard use of coformal mapping is to change coordinates from one region to a simpler region, say to a disk, where a problem can be viewed from a new perspective, hopefully one in which the solution to the original problem becomes more readily apparent. A famous instance were this approach paid huge dividends occurred in aerodynamics, where conformal mappings were instrumental in coming up with the original design profiles for airfoils. From the standpoint of conformal mapping, regions with smooth boundaries have been well understood for some time. However, the appearance of fractals in many branches of science led to the natural problem of investigating the conformal mapping properties of regions bounded by highly nonsmooth, fractal-type curves. The core objectives of Rohde's research are, on the one hand, to obtain a deeper understanding of fractal curves by means of conformal mappings, and conversely, to study some problems about conformal mappings by analyzing the geometry of regions with fractal boundaries