In this project, the PI will explore complex projective structures on surfaces and their applications to other areas of analysis and geometry. The space of all complex projective structures on a surface is a contractible manifold which has two natural but very different coordinate systems; one involves a classical complex-analytic construction (the Schwarzian derivative), while the other comes from the operation of grafting, which assembles a projective surface from Euclidean and hyperbolic pieces. The major research goals of this project are to understand the relationship between the two coordinate systems for the moduli space of projective structures and to use this understanding to solve problems in related areas. The project will also include a significant educational component focusing on undergraduate mathematics at the University of Illinois at Chicago (UIC) through two targeted programs: First, the PI will organize a yearly undergraduate research symposium, which will include lectures by senior mathematicians and by undergraduates reporting on their own research projects. Second, the PI will create a new course at UIC for undergraduate mathematics and mathematical computer science majors on "experimental mathematics", discussing the way computer software (both custom and off-the-shelf) is used as an exploratory tool in mathematical research. The PI will teach the course for the first time in the 2010-2011 academic year, emphasizing computer exploration of the geometry of curves and surfaces.
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 vehicles 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 of mathematics. 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.
