Quasisymmetric Uniformisation

二维几何中的一个经典结果是，对于某些曲面，存在到二维模型空间的共形映射-这些映射具有良好的性质，例如它们保持角度。拟对称映射和拟共形映射是共形映射的推广。它们提供了比共形映射更大的灵活性，因为它们不完全由它们在开集上的行为决定。拟共形映射和拟对称映射可以在任何度量空间上进行研究，并且不需要假设映射是可微的。它们发生时，一个一般化的经典一致化结果2维度量空间。另一个例子是几何群论。有一个有趣的悬而未决的问题有关的一组理论分析的意义上说，一组产生一个度量空间上，它原来是最有趣的研究拟对称等价。该项目包括扩大已知的结果准对称一致化。一种可能的方法是使用空间的准视觉近似，这是Bonk-Meyer最近提出的一个概念。

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.