Proposal: DMS-9970398 Principal Investigator: Steffen R. Rohde
Abstract: 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