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

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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.