Principal Investigator: David Dumas
The principal investigator will explore complex projective structures on surfaces and their applications to Teichmuller theory, Kleinian groups, and hyperbolic 3-manifolds. The space of all complex projective structures on a fixed surface is a contractible manifold which has two natural but very different coordinate systems: A classical analytic approach uses the Schwarzian derivative to identify the moduli space with a holomorphic vector bundle over Teichmuller space, while a more recent geometric approach builds each projective structure from hyperbolic and Euclidean pieces in a process known as grafting. The major goals of this project are to understand the relation between these two coordinate systems (both theoretically and using computer experiments), and to use that understanding to study Teichmuller spaces and deformations of Kleinian groups. The PI will also explore ways in which the techniques used to study complex projective structures could be adapted to other low-dimensional geometric structures, such as real projective structures.
The study of the shapes and configurations of geometric objects has applications to diverse areas of science and engineering, from understanding the folding of proteins or the formation of galaxies to programming autonomous robots that must navigate complex terrain. In a mathematical abstraction of this type of problem, one studies the space of all possible shapes, or "moduli space", of a geometric object. This project focuses on the moduli space of complex projective Riemann surfaces, a class of geometric objects that encode information about 3-dimensional spaces (hyperbolic manifolds) in 2-dimensional form. Through both theoretical study and computational experiments, the PI will develop new tools for analyzing these structures, enhance the connections between 2- and 3-dimensional geometry, and expand applications of these structures in related fields. The project will also produce computer images of the moduli space, displaying its rich structure and complexity in a way that can be appreciated by scientists and non-scientists alike.
