Mathematical Sciences: Geometric Structures, Discontinuous Groups, Moduli Spaces and Surface Symmetries

首席研究员将研究空间上的几何结构，但将主要研究莫比乌斯结构。空间上的莫比乌斯结构是作用于空间上的群作用，它来源于球体的保角变换的标准结构。在二维情况下，莫比乌斯结构等价于一维复射影结构。Kulkarni教授将研究三类具有莫比乌斯结构的空间，并试图理解这些空间的基本几何结构。本研究的主题是通过在空间上放置额外的结构来理解一个抽象的空间。理解这个数学工具的一个简单方法是想象一条简单的闭合曲线——可能以某种方式扭曲。现在，如果在曲线上划出相等的距离从而创造了一个度量空间，我们就可以讨论曲线上点之间的关系。因此，曲线被赋予了“附加结构”。首席研究员将尝试做类似的事情，但结构要复杂得多。

The principal investigator will investigate geometric structures on spaces but will focus much of the research on Mobius structures. A Mobius structure on a space is a group action which acts on the space and comes from the standard structure of conformal transformations of the sphere. In the two dimensional case a Mobius structure is equivalent to a one dimensional complex projective structure. Professor Kulkarni will study three classes of spaces with Mobius structure and attempt to understand the underlying geometry of these spaces. The theme of this research is to understand an abstract space by placing additional structure on the space. One simple way of understanding this mathematical tool is to think of a simple closed curve - perhaps twisted in some way. Now if one marks off equal distances on the curve thus creating a metric space, one can talk about the relationship between points on the curve. Thus the curve has been endowed with "additional structure". The principal investigator will attempt to do something like this but with a much more complicated structure.