This award supports work on groups, algorithms and geometries. Probabilistic and asymptotic properties of finite groups will be studied: the nature of random elements, or of random subgroups, of a given permutation group. Some results obtained will be applied to algorithmic questions concerning permutation groups. The study of polynomial-time and parallel (complexity class NC) group-theoretic algorithms will be continued. Additional new practical group-theoretic algorithms will be obtained, in some cases based on methods developed in polynomial-time situations. All of this work will make detailed use of the classification and properties of finite groups. Some of these algorithms depend on geometric methods. Other geometric projects will be continued including investigations into the number of planes, designs, generalized quadrangles, or other geometries, based upon special properties of their automorphism groups. This project is in the general area of finite group theory and the computer-aided computational aspects of this field. There is a growing interest in using computers to answer theoretical questions in algebra and conversely, algebra is quite useful for the development of algorithms. This project makes serious use of both the theory and practice of symbolic calculations to solve real problems in group theory.
