Mathematical Sciences: Groups, Algorithms, and Geometries

该奖项支持团体，算法和 几何学 有限的概率和渐近性质 将研究组：随机元素的性质，或 一个给定的置换群的随机子群。 一些结果 将被应用于算法问题， 置换群 多项式时间与并行的研究 （复杂性类NC）群论算法将是 持续期间内的 新的实用群论算法 将获得，在某些情况下，基于在 多项式时间情形 所有这些工作将使详细的 利用有限群的分类和性质。 一些 这些算法中的大多数都依赖于几何方法。 其他几何 项目将继续进行，包括调查 多个平面、设计、广义四边形或其他 几何，基于其自同构的特殊性质 组 这个项目是在有限群理论的一般领域 以及这个领域的计算机辅助计算方面。 人们对使用计算机回答问题的兴趣越来越大 代数中的理论问题，反过来说，代数是相当 这对算法的开发很有用。 该项目使 认真运用象征主义的理论和实践 计算来解决群论中的真实的问题。

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.