Quasiconformal Mappings in Geometry and Analysis

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ABSTRACTQuasi-conformal and related mappings form the largest class of maps that can be studied by analytic methods. Accordingly, their theory and their applications lie at the intersection of geometry and analysis and have connections to many other areas of mathematics and physics. While fundamental questions at the foundation of the theory of quasi-conformal maps remain open, recent advances in analysis on metric spaces have enlarged the range of applicability of these maps. Their theory may now lead to solutions of previously inaccessible problems in other fields such as geometric group theory. The purpose of this project is to explore current trends in the area. Specific topics of research include quasi-symmetric uniformization, dynamics on fractal spheres, the quasi-conformal Jacobian problem, quasi-regular maps and elliptic manifolds.The roots of this subject can be traced back to Gauss's work on cartography and surface geometry in the first half of the 19th century. He coined the phrase ``conformal map" and derived equations that govern the theory of planar quasi-conformal mappings. The full importance of this theory was only realized about a century later. By now quasi- conformal mappings have developed into one of the major tools in contemporary Geometric Function Theory.