This award will provide support for a group of mathematicians and computer scientists working on problems related to geometry. The broad objectives (i) to develop and understanding of geometry and algorithms in geometry, and (ii) to carry out work on the computation of geometric structures and on the sythesis and analysis of images of these structures. The role of the computer in this effort is both for explication as well as an experimental tool to advance the research. The novelty of this project derives from the use of a supercomputer in the generation and display of geometric constructs such as fractal sets, Dehn surgery space, Julia sets Mandelbrot sets and pleated surfaces. A welding of the mathematical theories of Kleinian groups, Teichmuller theory, analytic dynamics, combinatorial group theory and 3-manifold theory lies at the foundation of this project. In addition it will mark the first time that mathematical researchers located at diverse universities and in several countries will be combining their talents on such a large project. Large scale communication links will be used extensively in this joint effort. Currently available algorithms for the purposes envisioned here are insufficient. In addition to the planned theoretical and graphical experimentation, work will be done in developing advanced algorithms addressing problems of combinatorial complexity and more general language design to establish portable software and advance understanding of fundamental data structures. After an initial set-up period, graduate students and postdoctoral associates will be incorporated into the project. The interdisciplinary research experience will provide a model for future large-scale initiatives in the mathematical sciences.
