Projective Structures in Teichmuller Theory and Kleinian Groups

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AbstractAward: DMS-0805525Principal Investigator: David DumasThe principal investigator will explore complex projectivestructures on surfaces and their applications to Teichmullertheory, Kleinian groups, and hyperbolic 3-manifolds. The spaceof all complex projective structures on a fixed surface is acontractible manifold which has two natural but very differentcoordinate systems: A classical analytic approach uses theSchwarzian derivative to identify the moduli space with aholomorphic vector bundle over Teichmuller space, while a morerecent geometric approach builds each projective structure fromhyperbolic and Euclidean pieces in a process known as grafting.The major goals of this project are to understand the relationbetween these two coordinate systems (both theoretically andusing computer experiments), and to use that understanding tostudy Teichmuller spaces and deformations of Kleinian groups.The PI will also explore ways in which the techniques used tostudy complex projective structures could be adapted to otherlow-dimensional geometric structures, such as real projectivestructures.The study of the shapes and configurations of geometric objectshas applications to diverse areas of science and engineering,from understanding the folding of proteins or the formation ofgalaxies to programming autonomous robots that must navigatecomplex terrain. In a mathematical abstraction of this type ofproblem, one studies the space of all possible shapes, or "modulispace", of a geometric object. This project focuses on themoduli space of complex projective Riemann surfaces, a class ofgeometric objects that encode information about 3-dimensionalspaces (hyperbolic manifolds) in 2-dimensional form. Throughboth theoretical study and computational experiments, the PI willdevelop new tools for analyzing these structures, enhance theconnections between 2- and 3-dimensional geometry, and expandapplications of these structures in related fields. The projectwill also produce computer images of the moduli space, displayingits rich structure and complexity in a way that can beappreciated by scientists and non-scientists alike.