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。它们提供了比保角映射更多的灵活性，因为它们不是完全由它们在开集上的行为决定的。拟共形和拟对称映射可以在任何度量空间上研究，并且不需要假设映射是可微的。当人们将经典的均衡化结果推广到2维度量空间时，它们就会发生。另一个例子是几何群论。有一个有趣的悬而未决的问题，将群论与分析联系起来，从这个意义上说，一个群产生了一个度量空间，在这个度量空间上，研究准对称等价是最有趣的。该项目包括扩展关于准对称均匀化的已知结果。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.