Uniformization of Metric Spaces and Quasiconformal Removability

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

The goal of this project is to develop methods and geometric tools for understanding the geometry of fractal spaces. Fractal spaces appear in description of many natural phenomena such as in lightning bolts, growth models of plants and crystals, snowflakes, coastlines, and river networks. The questions that the project plans to study have applications whenever storage of three-dimensional information (landscapes, faces, human brain surface) in a two-dimensional image is desired without loss of information. While in the case of "smooth" objects (objects that are not modeled using fractals) the corresponding mathematical theory is well understood, this is not the case for fractal objects, which require the development of new techniques. This project aims to develop mathematical theory for such fractal spaces. Another focus of this project is the study of removable fractal sets. Fractal sets appear sometimes as boundaries of otherwise "smooth" objects, and for many problems it is useful to know that these fractals are removable, in the sense that their presence can be ignored for some purposes. Removability of fractal sets has applications in mathematical problems that require "gluing" together two functions, or two dynamical systems, or two surfaces, and could result in the better understanding of dynamical systems in physics. This project consists of three parts, concerning the uniformization of Sierpinski carpets, the uniformization of two-dimensional metric surfaces, and the problem of removability of fractal sets for conformal maps. Continuing earlier work, the PI will study problems related to the uniformization of Sierpinski carpets by square Sierpinski carpets and the PI will study the regularity of the uniformizing map, which is already known to be quasisymmetric or discrete quasiconformal. The PI will also work in questions related to Hausdorff dimension distortion under the uniformizing map and in generalizations of this planar uniformization theory to abstract Sierpinski carpets. Another focus is the problem of uniformization of two-dimensional metric surfaces. In this direction, the PI will investigate possible generalizations of uniformization theorems for two-dimensional surfaces by Euclidean space and concentrate efforts on weakening the existing geometric assumptions. Finally, the PI will work on extending earlier results on the removability of fractal sets, by finding topological criteria for fractal sets (resembling the Sierpinski gasket or carpet) to be non-removable, studying the equivalence of Sobolev removability and conformal removability, and exploring the connections of removability to the problem of rigidity of circle domains. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.