Groups with One Defining Relation

主要研究者建议在这里研究由一个单一的关系定义的群体。这些群体出现的基本群体在代数拓扑，作为discrete群的变换在分析和几何和抽象群论。这种研究的重要性在于它与数学的其他分支，以及计算机科学和逻辑学的许多联系。在其他一些问题上的进展也会给我们带来启示。这些包括双曲线和自动组，可计算性，遗传和其他算法的理论，并可能导致更深入地了解作为一个整体的群。这一建议代表了第一步，在一个整体的研究关系群，使用的步骤数在一个所谓的马格努斯分解，作为基础归纳。首席研究员，连同他的许多同事，将开始这项研究的重点放在一个相关群体的planarityone.These是非常特殊的HNN扩展的一个自由群，出现在第一步的theMagnus故障上面提到的。 群是一种数学结构，其设计部分是为了捕捉对称性的内在本质。因此，群在几何学、化学、粒子物理学、密码学以及许多数学中起着极其重要的作用。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。物理世界中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，如进化，可以用来解开特定群体的性质。因此，群论为更好地理解这些过程提供了一个试验场。在即将进行的工作中，主要研究人员将特别利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。这种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.