Groups with One Defining Relation

主要研究者建议研究由单一关系定义的群，这种群在代数拓扑学中作为基本群出现，在分析和几何中作为离散变换群出现，在抽象群论中作为离散变换群出现。这种研究的重要性在于它与数学的其他分支以及与计算机科学和逻辑的许多联系。进展将在其他一些问题上显现出来。这些理论包括双曲和自动群理论、可计算性、遗传和其他算法，并可能导致对有限表示群作为一个整体的更深层次的理解。这一建议代表了对一亲组的整体研究的第一步，使用所谓的马格努斯分解中的步骤数作为归纳的基础。首席研究人员和他的许多同事将开始这项研究，将注意力集中在平面性一的一个相关群上。这些是在上面提到的Magnus崩溃的第一步中出现的自由群的非常特殊的HNN扩展。群是数学结构，其设计部分是为了捕捉对称性的内在本质。因此，基团在几何学、晶体学、粒子物理学、化学和密码学以及许多数学中都扮演着极其重要的角色。它们在密码学中的用途还有待开发；RSA加密可以被视为朝着这个方向迈出的第一步。物理世界的许多问题都可以转化为群论中的问题。另一方面，许多过程，例如进化，可以用来揭开特定群体的本质。群论为更好地理解这些过程提供了试验场。在即将开展的工作中，主要研究人员将特别使用群论来尝试改进计算机科学中最令人兴奋的新方面之一，即所谓的遗传算法。这样的算法似乎是设计更智能计算机的一种手段。

The principal investigator proposes here to study groups defined by a single relation.Such groups arise as fundamental groups in algebraic topology, as discretegroups of transformations in analysis and geometry and in abstract grouptheory. The importance of such a study is in its many connections to otherbranches of mathematics, as well as to computer science and logic. Progress will shedlight on a number of other subjects. These include the theories of hyperbolic and automatic groups, computability, genetic and other algorithms and may lead to a deeper understanding of finitely presented groups as awhole. This proposal represents a first step in an overall study of one-relatorgroups, using the number of steps in a so-called Magnus breakdown, as thebasis for induction. The principal investigator, together with his many colleagues,will initiate this study by focussing attention on one-relator groups of planarity one.These are the very special HNN extensions of a free group that arise in the first step of theMagnus breakdown referred to above. Groups are mathematical structures which are designed, in part, to capture theintrinsic nature of symmetry. Consequently groups play an extremely important rolein geometry, in chrystallography, in particle physics, in chemistry and in cryptographyas well as in much of mathematics. Their uses in cryptography are yet to be exploited; RSA encryption may be viewed as a first step in this direction. Many of the problems of the physical world can be translated into problems in group theory. On the other hand, many processes, such as evolution, can be used to unravel the nature of particular groups. Thusgroup theory provides a testing ground for a better understanding of these processes.The principal investigator, in the work to be undertaken, will, in particular, use group theory to attempt to improve on one of the most exciting new aspects of computerscience, so-called genetic algorithms. Such algorithms appear to be a means for designing ever more intelligent computers.