Mathematical Sciences: Geometry Supercomputer Project

该奖项将为一群致力于几何相关问题的数学家和计算机科学家提供支持。主要目标是(1)发展和理解几何学和几何学中的算法，以及(2)开展几何结构的计算和这些结构的图像的合成和分析的工作。计算机在这一努力中的作用既是为了解释，也是推动研究的实验工具。这个项目的新颖性来自于使用超级计算机来生成和显示几何结构，如分形集、Dehn手术空间、Julia集、Mandelbrot集和褶皱曲面。Kleian群的数学理论、TeichMuller理论、解析动力学、组合群论和3-流形理论的融合是本项目的基础。此外，这将标志着分布在不同大学和几个国家的数学研究人员将首次在如此大的项目中结合他们的才华。在这一联合努力中，将广泛使用大规模通信链路。目前可用于这里设想的目的的算法是不够的。除了计划的理论和图形实验外，还将开展工作，开发解决组合复杂性问题的先进算法和更通用的语言设计，以建立可移植的软件和增进对基本数据结构的理解。在最初的建立阶段之后，研究生和博士后助理将被纳入该项目。跨学科的研究经验将为未来数学科学的大规模倡议提供模式。

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.