群呈示及其相关领域的历史研究
发布时间：2018-11-20 18:03
【摘要】：群的呈示这个概念首次出现于德国数学家代克1882年《群论研究》的论文中,是群最早的抽象定义.以群的呈示这个概念为基础,近百年来很多数学分支交叉发展、互相影响.在这个过程中,戴恩结合两位伟大的数学家庞加莱和希尔伯特的思想,创始了组合群论这个新的数学分支.涉及到该领域的研讨会议、科研团体越来越多,国际数学家大会上也常出现此领域的主题,可见该领域在近现代数学中占有重要位置.本论文在搜集、阅读、整理、分析大量的原始文献、历史研究文献、经典数学著作的基础上,综合运用数学史的研究方法,以群的呈示发展过程中的重要历史转折点、卓越的数学家及其贡献为切入点,对群的呈示在分别由伯恩赛德与戴恩等人引导的方向上的发展进行了历史分析.群的呈示沿前者的发展过程中,时间与伯恩赛德相关问题构成其两条主线.群的呈示沿后者的发展过程中,时间与学术传承构成其两条线索.本论文的主要内容如下:1.从群的呈示与多面体群和离散群之间的联系,分析了群的呈示的思想起源.以代克的工作为基础,从群的呈示这个概念的本质、演变发展、与自由群之间的关系等方面,深入细致地分析了群的呈示的概念.阐述了凯莱和克莱因对代克的影响.通过对文献、术语的统计分析,介绍了群的呈示在中国的发展情况.2.以对伯恩赛德在此领域的贡献进行深入细致的分析为中心,对伯恩赛德工作的背景进行了阐述.对伯恩赛德的生平进行了详细考察.在与凯莱的群的图表示对比分析的基础上,分析了伯恩赛德的群的图表示.3.阐述了伯恩赛德问题、广义伯恩赛德问题、限制性伯恩赛德问题的发展.统计涉及该领域的国际数学家大会,分析了该领域产生的影响.介绍了在该领域做出重要贡献的前苏联数学家之间的联系.4.分析了庞加莱的《位置分析》中的重要思想和基本群的概念.阐述了梯采的《多维流形的拓扑不变量》中对群的呈示的发展的主要贡献.简单介绍了梯采的生平.以戴恩问题和群的图为重点,对戴恩在此领域的贡献进行了深入细致的探究.5.分析了戴恩的学生尼尔森与马格努斯对此领域的贡献,以及他们对此领域的传播发展的影响.马格努斯对组合群论的发展和传播都有突出贡献,本论文对他的生平进行了详细考察.6.在对瑞德迈斯特和施莱尔的工作进行考察的基础之上,分析了瑞德迈斯特-施莱尔方法、自由积、融合自由积的提出.阐述了他们在此领域后续发展中的影响.7.通过小消去理论、几何群论、编码理论的例证,简要阐述了与群的呈示相关的其他领域的情况.
[Abstract]:The presentation of group is the earliest abstract definition of group, which first appeared in the paper "study of Group Theory" by German mathematician Dek in 1882. Based on the concept of group presentation, many branches of mathematics have developed and interacted with each other in the last hundred years. In the process, Dayne combined the ideas of two great mathematicians, Poincare and Hilbert, to create the new branch of combinatorial group theory. There are more and more scientific research organizations involved in this field, and the subject of this field is often appeared in the International Congress of mathematicians, which shows that this field plays an important role in modern mathematics. On the basis of collecting, reading, sorting out and analyzing a large number of original documents, historical research documents and classical mathematical works, this paper synthetically applies the research methods of mathematical history to show an important historical turning point in the course of the development of group presentation. Excellent mathematicians and their contributions as the starting point, the presentation of the group in the direction led by Burnside and Dayne, etc., respectively, the historical analysis of the development. In the course of the development of group presentation, the related problems of time and Burnside form its two main lines. In the course of the development of the group, time and academic heritage constitute its two clues. The main contents of this thesis are as follows: 1. From the relation between the presentation of groups and polyhedron groups and discrete groups, the origin of the ideas of presentation of groups is analyzed. Based on Deke's work, the concept of group presentation is analyzed in detail from the aspects of the essence, evolution and development of the concept, and the relationship between the group and the free group. The influence of Kelley and Klein on Deke is expounded. Through the statistical analysis of literature and terminology, this paper introduces the development of group presentation in China. 2. Based on the detailed analysis of Burnside's contribution in this field, the background of Burnside's work is expounded. The life of Burnside was examined in detail. On the basis of comparative analysis of graph representation of group with Kelley, the graph representation of group of Burnside is analyzed. 3. 3. The development of Burnside problem, generalized Burnside problem and restricted Burnside problem are expounded. The international Congress of mathematicians involved in this field analyzes the impact of the field. This paper introduces the relationship between the mathematicians of the former Soviet Union who have made important contributions in this field. 4. The important ideas and the concept of basic group in Poincare's position Analysis are analyzed. The main contributions to the development of the presentation of groups in the topological invariants of Multidimensional Manifolds are described. The life history of ladder mining is introduced briefly. With the focus on Dayne problem and group map, Dayne's contribution in this field has been explored in detail. 5. The contribution of Dayne's students Nielsen and Magnus to this field and their influence on the communication and development of this field are analyzed. Magnus has made outstanding contributions to the development and dissemination of combinatorial group theory. Based on the investigation of the work of Redmeister and Schlyle, this paper analyzes the presentation of the Redmeister Schlyle method, the free product and the fusion free product. This paper expounds their influence in the follow-up development in this field. 7. 7. Through the examples of small elimination theory, geometric group theory and coding theory, this paper briefly describes other fields related to the presentation of groups.
【学位授予单位】：河北师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O152
[Abstract]:The presentation of group is the earliest abstract definition of group, which first appeared in the paper "study of Group Theory" by German mathematician Dek in 1882. Based on the concept of group presentation, many branches of mathematics have developed and interacted with each other in the last hundred years. In the process, Dayne combined the ideas of two great mathematicians, Poincare and Hilbert, to create the new branch of combinatorial group theory. There are more and more scientific research organizations involved in this field, and the subject of this field is often appeared in the International Congress of mathematicians, which shows that this field plays an important role in modern mathematics. On the basis of collecting, reading, sorting out and analyzing a large number of original documents, historical research documents and classical mathematical works, this paper synthetically applies the research methods of mathematical history to show an important historical turning point in the course of the development of group presentation. Excellent mathematicians and their contributions as the starting point, the presentation of the group in the direction led by Burnside and Dayne, etc., respectively, the historical analysis of the development. In the course of the development of group presentation, the related problems of time and Burnside form its two main lines. In the course of the development of the group, time and academic heritage constitute its two clues. The main contents of this thesis are as follows: 1. From the relation between the presentation of groups and polyhedron groups and discrete groups, the origin of the ideas of presentation of groups is analyzed. Based on Deke's work, the concept of group presentation is analyzed in detail from the aspects of the essence, evolution and development of the concept, and the relationship between the group and the free group. The influence of Kelley and Klein on Deke is expounded. Through the statistical analysis of literature and terminology, this paper introduces the development of group presentation in China. 2. Based on the detailed analysis of Burnside's contribution in this field, the background of Burnside's work is expounded. The life of Burnside was examined in detail. On the basis of comparative analysis of graph representation of group with Kelley, the graph representation of group of Burnside is analyzed. 3. 3. The development of Burnside problem, generalized Burnside problem and restricted Burnside problem are expounded. The international Congress of mathematicians involved in this field analyzes the impact of the field. This paper introduces the relationship between the mathematicians of the former Soviet Union who have made important contributions in this field. 4. The important ideas and the concept of basic group in Poincare's position Analysis are analyzed. The main contributions to the development of the presentation of groups in the topological invariants of Multidimensional Manifolds are described. The life history of ladder mining is introduced briefly. With the focus on Dayne problem and group map, Dayne's contribution in this field has been explored in detail. 5. The contribution of Dayne's students Nielsen and Magnus to this field and their influence on the communication and development of this field are analyzed. Magnus has made outstanding contributions to the development and dissemination of combinatorial group theory. Based on the investigation of the work of Redmeister and Schlyle, this paper analyzes the presentation of the Redmeister Schlyle method, the free product and the fusion free product. This paper expounds their influence in the follow-up development in this field. 7. 7. Through the examples of small elimination theory, geometric group theory and coding theory, this paper briefly describes other fields related to the presentation of groups.
【学位授予单位】：河北师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O152
【共引文献】
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