Applications of Group Theory to Large Scale Computation

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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.