Applications of Group Theory to Large Scale Computation

这个项目解决了有限群的计算问题，目前超出了当前的范围。 算法和现有技术。这些群可以表示为置换 群或定义在有限域上的矩阵群。的问题 兴趣涉及群论结构和群的应用 理论 研究的一个中心动机是建立 有限环上矩阵群的表示 领域的 这是大规模国际努力的一个重要部分， 开发一套计算工具，用于执行矩阵群 计算。 要开发的工具可以应用于物理问题， 科学、工程和数学中处理 群的高维矩阵表示。 第二个研究领域 求解最小字的计算技术 有限群的问题 这个问题出现在许多环境中，例如 互连网络中的路由和编码理论，并且已知 在计算上很困难。 这项研究将导致新的算法， 使用组自同构来修剪可能的搜索树 解决方案 第三个研究领域是软件体系结构的开发 用并行计算机来处理大规模计算问题， 群论，如极大矩阵和置换的构造 表示。 如集成、交互式环境将使 研究人员迅速克服最初的学习障碍，有效地利用 平行性 他们将能够将联合收割机 工作站在他们的主页（如学生实验室），与已经熟悉的 语言，如GAP或LISP。

This project addresses computational issues for finite groups that are currently beyond the scope of current algorithms and existing technology. The groups are represented either as permutation groups or as matrix groups defined over finite fields. The questions of interest concern both group theoretic structure and applications of the group theory. A central motivation for the research is the construction of representations for matrix groups defined over finite fields. This is an important part of a large-scale international effort to develop a suite of computational tools for performing matrix group computations. The tools to be developed have application to problems in the physical sciences, engineering, and those parts of mathematics dealing with high dimensional matrix representations of groups. A second research area computational techniques for solving the minimum word problem for finite groups. This problem arises in many contexts, such as routing in interconnection networks and coding theory, and is known to be computationally difficult. The research will lead to new algorithms which use group automorphisms to prune the search tree of possible solutions. A third research area is the development of a software architecture to bring parallel computers to bear on large scale problems in computational group theory, such as the construction of very large matrix and permutation representations. Such as integrated, interactive environment will enable researchers to quickly overcome the initial learning barrier for effective use of parallelism. They will be able to combine readily available networks of workstations at their home site (such as student labs), with an already familiar language, such as GAP or LISP.