The principal investigator employs computer visualization tools to aid in the study of a number of questions in low-dimensional differential and algebraic geometry and topology. In particular, these questions have to do with fundamental domains for discrete groups of isometries of various three-dimensional geometries, moduli spaces of polygons in each of the three-dimensional space forms, and direct constructions of families of compactifications of semisimple Lie groups of low dimension. Interactive software has been developed to explore fundamental domains on the sphere at infinity for cyclic groups of isometries of three-dimensional hyperbolic space and this is extended to the interior of hyperbolic space, other groups and other geometries. Moduli spaces of polygons are investigated to determine aspects of their own geometry and because of their relationship to moduli spaces of vector bundles over punctured Riemann surfaces and some symplectic and geometric-invariant theory quotients. Images have also been made of moduli spaces of certain algebraic compactifications of the group of isometries of the hyperbolic disk and further images are generated of moduli spaces of other algebraic compactifications as well as of the compactifications themselves. An important component of this project is the direct involvement of undergraduate students: while the mathematics is of various levels of abstraction, the computer tasks are accessible to serious students and provide them with an unusual opportunity for hands-on involvement in current mathematical research.
The fantastic interactive three-dimensional graphics that can be seen in many of today's commercial computer games are the result of the phenomenal advances that have occurred in recent years in the sophistication of computer software and the speed and power of computer hardware. But in a larger sense they can be attributed to the increase of engineering knowledge based on hard science and, ultimately, on deep mathematical understanding. It is thus somewhat surprising that these computational and graphics tools have been used very little in pure mathematical research. The principal investigator uses interactive computer visualization to perform mathematical experiments in a variety of different fields that all have some component relating to two-, three- or four-dimensional geometry; as this is pure mathematics, experimental evidence must then be justified in a purely abstract setting. An important part of this project is the direct involvement of undergraduate students: here again it is possible to take advantage of the very high level computer abilities that many undergraduates now possess to create the software that drives the mathematical explorations, while at the same time giving these undergraduates the extremely rare opportunity to have immediate experience of current mathematical research.
