Generalized Branched Coverings and Parameterizations

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This project involves investigations into the global conformal geometry of Riemannian manifolds and the local quasiconformal geometry of nonsmooth manifolds. Questions and proposed methods arise from analysis, geometry, and topology. The main topics are the global rigidity phenomena of mappings distorting the infinitesimal conformal geometry and natural local parameterizations of nonsmooth quasiconformal geometries. New methods that go beyond the range of nonlinear potential theory in the context of global rigidity questions will be introduced, and new geometric structures on quasiconformal manifolds will be explored.Quasiconformal methods give information about the geometrical properties of spaces simultaneously at all scales. These methods have their roots in complex analysis and in the geometry of the complex plane. The methods, however, can be used both in higher dimensions and in spaces where analysis based on traditional calculus is not available. Mathematical areas of application for these methods include such subjects as Teichmuller theory, topology and geometry of manifolds, geometric group theory, nonlinear geometric analysis, and nonsmooth calculus on manifolds. Quasiconformal mappings have recently begun to play a serious role in applied areas as well. These include fluid dynamics, elasticity, and even the analysis of nanostructures. This project focuses on the following fundamental question: When can a given geometry of a space be understood as a possibly highly distorted Euclidean geometry?