Geometric Function Theory and Dynamics

摘要：Rohde将利用几何函数理论和动力系统的方法研究共形和拟共形映射的畸变特性。拟共形映射是弱可微的同胚，它们是“几乎共形”的，从某种意义上说，它们最多将角度扭曲一个有界因子。这些映射推广了保角映射，自然出现在几何、几何函数理论、复杂动力学和PDE中。Rohde将研究的两个具体问题是：估计一个圆盘的共形映射的导数，该映射允许在复平面上进行拟共形扩展；以及找到圆的拟共形像（所谓的拟圆）的Hausdorff维的边界。这两个问题都与布伦南关于单位盘的保形映射的导数的可积度的猜想密切相关。罗德还将研究几何性质，如欧几里得空间中集合的孔隙度或波动性，以及它们的豪斯多夫维数的估计。这种集合在复杂动力学和几何函数理论中都很自然地出现。保角映射在数学内外的许多领域都有应用。这些包括控制理论，热传导，流体动力学和复杂动力学，仅举几例。共形映射的一种标准用法是将坐标从一个区域更改为一个更简单的区域，比如磁盘，在这个区域中，可以从一个新的角度来看待问题，希望在这个角度中，原始问题的解决方案变得更加明显。一个著名的例子是这种方法支付了巨大的红利发生在空气动力学，其中保形映射是在拿出原始设计轮廓的翼型工具。从保角映射的角度来看，具有光滑边界的区域已经被很好地理解了一段时间。然而，分形在许多科学分支中的出现导致了研究高度非光滑分形型曲线边界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对分形曲线有更深的理解，另一方面是通过分析具有分形边界的区域的几何来研究有关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