Quasisymmetric Uniformisation

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A classical result in 2-dimensional geometry is for certain surfaces there exists a conformal map - these maps have good properties for example they preserve angles - to a 2-dimensional model space. Quasisymmetric maps and quasiconformal maps are generalisations of conformal maps. They offer more flexibility than conformal maps as they are not completely determined by their behaviour on an open set. Quasiconformal and quasisymmetric maps can be studied on any metric space and one need not assume differentiability of a map. They occur when one generalises the classical uniformisation result to 2-dimensional metric spaces. Another instance is in geometric group theory. There is an interesting open question relating a group theory to analysis in the sense that a group gives rise to a metric space on which it turns out that it is most interesting to study up to quasisymmetric equivalence. The project consists of expanding known results on quasisymmetric uniformization. A possible method is to work with quasivisual approximations of spaces, a concept recently developed by Bonk-Meyer.