Geometric Function Theory and Dynamics

提案：DMS-9970398主要研究者：Steffen R.罗德简介：罗德将调查失真性能的共形和拟共形映射，使用的方法从几何函数理论以及从动力系统。拟共形映射是弱可微的同胚，它们是“几乎共形的”，在这个意义上，它们最多使角度扭曲一个有界因子。这些映射推广了共形映射，并在几何、几何函数论、复动力学和偏微分方程中自然出现。两个具体的问题罗德将调查估计衍生的共形映射的磁盘承认一个拟共形扩展到复杂的平面和找到界限的Hausdorff维数的拟共形图像的圆圈（所谓的quasicircles）。这两个问题是密切相关的布伦南猜想的程度可积性的衍生物的共形映射的单位磁盘。罗德还将研究几何性质，如孔隙率或wiggliness集在欧几里得空间，连同估计其Hausdorff维数。保形映射在数学内外的许多领域都有应用，包括控制理论、热传导、流体动力学和复动力学等。共形映射的一个标准用法是将坐标从一个区域转换到一个更简单的区域，比如一个圆盘，在这个圆盘上，一个问题可以从一个新的角度来看待，希望在这个新的角度中，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在空气动力学中获得了巨大的好处，其中保角映射有助于获得翼型的原始设计轮廓。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，分形在许多科学分支中的出现导致了一个自然问题，即研究由高度非光滑的分形型曲线包围的区域的保角映射性质。罗德研究的核心目标，一方面是借助共形映射获得对分形曲线更深层次的认识，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