Character varieties allow one to view all geometric structures on a fixed topology at once. Their use has produced many significant results regarding the hyperbolic geometry and topology of 3-manifolds using PSL(2,C)-character varieties. A less well studied area is PGL(4,R) character varieties. I believe that an understanding of these objects will further advance the study of 3-manifolds. Crucial aspects of this understanding involve computation of concrete examples and description of local and global properties of the relevant varieties. Both of these projects would be well suited for joint investigation by a new postdoctoral scholar and myself.
In addition to helping a postdoctoral candidate perform cutting edge research I would provide mentorship on submission of successful research grant proposals (I currently hold an NSF individual grant). I would also assist the scholar in preparation of their work for publication in prestigious academic journals.