Quasisymmetric Maps-Parametrization, Extension and Factorization

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The project features new approaches to long-standing problems in quasiconformal analysis. Quasiconformal maps have played a pivotal role in the development of classical function theory. Quasisymmetric maps have recently found important applications in geometric group theory, structure of manifolds and analysis on fractals. However a large number of fundamental questions remain: quasisymmetric parametrization of metric spaces by Euclidean spaces; extension of quasisymmetric maps to an ambient space; factorization of quasiconformal maps into maps of small dilatation. Extension and smoothing are simpler when the dilatation is small; a manifold carrying a quasiconformal structure of small dilatation is smoothable; if a quasiconformal map can be factored into maps of small dilatation, then the factors can be extended,then smoothed. The PI proposes to study this circle of problems. Classical geometric topology is rich with examples of spaces which were proved homeomorphic to Euclidean spaces only with great ingenuity, e.g., the double suspension of homology 3-spheres and certain decomposition spaces. This project deals with quasisymmetric parametrization of such spaces, which resemble the Euclidean spaces not only topologically, but also geometrically and measure-theoretically. The findings will lead to a better understanding of the general theory.Quasiconformal and quasisymmetric maps have been studied for their mathematical beauty as well as potential scientific applications to objects lacking a smooth structure that occur naturally in physics and biology. Recently, there have been exciting discoveries in applying quasiconformal mappings to study images of brain cortical surfaces, to determine the conductivity of a body, and to study percolation and crystal growth. This proposal deals with intrinsic properties of these mappings at the interface of geometric function theory and classical geometric topology. In addition to theoretical advances, the findings will shed light on some of these more practical problems. The richness of the examples from topology will broaden the participation of graduate students in research activities in geometric analysis.