Groups with One Defining Relation

具有一个定义关系的组

• 批准号: 0202382
• 负责人: Gilbert Baumslag
• 金额: $ 15.64万
• 依托单位: CUNY City College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0202382&HistoricalAwards=false
• 关键词: Groups; Defining; Relation

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.

主要研究者在这里提议研究由单个关系定义的小组。类似小组是代数拓扑的基本组，作为分析和几何学和抽象组理论中转换的离散组。这种研究的重要性在于它与其他数学的其他关系以及计算机科学和逻辑的许多联系。进度将在许多其他主题上脱颖而出。这些包括双曲线和自动组的理论，可计算性，遗传性和其他算法，并可能会更深入地理解有限呈现的群体。该提案代表了对单余集的整体研究的第一步，使用所谓的Magnus分解中的步骤数，作为诱导的基础。首席研究员以及他的许多同事将通过将注意力集中在一定的一体式群体上来发起这项研究。这些是在Themagnus崩溃的第一步中出现的自由群体的非常特殊的HNN扩展。 组是数学结构，部分设计为捕获对称性的内在性质。因此，组在粒子物理学，化学和加密术中以及在大部分数学中扮演着极为重要的角色。它们在密码学中的用途尚未被利用； RSA加密可以被视为该方向上的第一步。物理世界的许多问题都可以转化为群体理论中的问题。另一方面，许多过程，例如进化，都可以用来揭示特定群体的性质。舒格组理论为更好地理解这些过程提供了一个测试基础。在要进行的工作中，主要研究者将特别使用群体理论来尝试改善计算机科学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越来越智能计算机的一种手段。