Algorithms for finite groups and their associated Cayley graphs have a wide range of applications in mathematics and computer science. These applications range from problems related to the classification of finite simple groups to the mixing rate of Markov chains, the design of interconnection networks for large interacting arrays of CPUs, the graph isomorphism problem (of relevance to chemical documentation), and quantum mechanics.

The primary focus of the working group is the design, analysis, and implementation of algorithms for handling finite permutation and matrix groups. Recent algorithms [1-2] set up a framework to reduce computational problems in arbitrary permutation and matrix groups to problems about finite simple groups. In turn, the algorithmic questions about finite simple groups lead to theoretical problems in these groups, mostly about statistical properties and about the structure of subgroups generated by random elements satisfying certain restrictions. The classification theorem for finite simple groups [3] asserts that the nonabelian finite simple groups fall into three infinite classes (alternating groups, classical groups of Lie type, and exceptional groups of Lie type) plus a set of 26 additional ``sporadic'' simple groups. A major project now is to develop a theory of simple groups of exceptional Lie type and of sporadic simple groups which can be exploited in the design of efficient algorithms, similar to the algorithmic handling of classical groups of Lie type [1] and alternating groups [2].

The currently most active area of computational group theory is the design and implementation of algorithms dealing with matrix groups over finite fields. There are a lot of subproblems, suitable for students at all levels (advanced undergraduate, beginning graduate, advanced graduate). Examples for each level:

• Given a long list of group elements such that one of them is in a proper normal subgroup (but we do not know which one), construct one group element in the proper normal subgroup. An efficient practical algorithm for this problem would speed up the reduction of the handling of arbitrary matrix groups to the handling of simple groups.
• Investigate how the quality of the random element generator influences randomized recognition algorithms for simple groups. In particular, investigate the reliability of the random generator ``product replacement algorithm'' in this setting.
• Design and implement algorithms for the decomposition of rank one exceptional groups of Lie type as a short product of cyclic groups. This problem is relevant for the constructive recognition of these simple groups. Implementations and experiments will be carried out in the GAP computer algebra system.

A secondary focus of the working group is the use of symmetry in the sciences. An interesting example is provided by spiral waves which appear in abundance in physics and chemistry. One famous experiment where rigidly-rotating spiral waves have been observed is the Belousov-Zhabotinsky reaction (a chemical reaction that involves the bromination and oxidation of malonic acid by catalysts). In certain parameter regimes, the spiral waves begin to meander or to drift, tracing out flower-like patterns. The package EZSPIRAL (written by Dwight Barkley) will allow us to compute and study meandering spirals on the computer. The observed phenomena, as well as other related ones, can be understood by combining elementary group theory with dynamical systems techniques. First, we will learn about the Euclidean symmetry group of the plane. Next, we consider planar differential equations and learn about Hopf bifurcations by exploring and experimenting with numerical tools. Combining this knowledge will help us to understand meandering and drifting spirals. These projects are accessible to both advanced undergraduate and beginning graduate students. Other projects will involve exploring the impact of boundary conditions on the behavior of spiral waves: this is an interesting topic related to the role that arteries play for sustaining cardiac arrhythmias.

This working group is especially well suited for the involvement of undergraduate and graduate students. Participating in the implementation of the algorithms, and the study of spiral waves, can start without much theoretical background, and it may lead to accessible problems in group theory and in the theory of computing.