Mappings with Little Smoothness

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Proposal Number: DMS-0200566PI: Mario BonkABSTRACTThe investigation of non-smooth phenomena is one of the main subjects of contemporary Geometric Function Theory.The purpose of this project is to study metric structures on spaces and appropriate nonsmooth classes of mappingsbetween these spaces such as bilipschitz, quasisymmetric,quasiconformal, and quasiregular mappings. Many basic questions in this area are open. For example, a satisfactory characterization of two-dimensional metric spheres that are quasisymmetric to the standardtwo-dimensional sphere is unknown. This problem is relevantin connection to Thurston's hyperbolization conjecture.Progress in the field relies on a combination of geometric and analytic methods that were recently established in the analysis of metric spaces.The particular question of how to map surfaces has a long history. Cartography aims to preserve particular features of a surface under suitable mappings. This has lead to major mathematical developments. In the nineteenth century basic problems of surveying and geodesy motivated Gauss to build up a systematic theoryof curved surfaces which laid the foundation of modern differential geometry. For the investigation of wrinkledand fractal objects new mathematical tools arerequired. A clear understanding of the mathematical concepts for describing nonsmooth phenomena will benefitresearch in other areas and lead to practical applications.For example, methods for attacking the theoretical questions of quasiconformal parametrizations of sphereswere also used for finding algorithms for mapping the surface of the human brain. One of the aims of the project is to involve graduate students in this promising and important area of mathematical research.