群论统一数学的历史研究

For nearly two centuries, mathematics has shown rapidly exponential development. With its contents increasing, modern mathematics is inevitably differentiated into many branches. But instead of mathematics’ splitting, the connection among mathematical branches is very strong, which reveals that there exists a great trend of unification in mathematics. In 1872, a variety of geometries were connected from the viewpoint of group theory by German mathematician Klein for the first time, which created a successful sample of the mathematical unification by group theory. By the end of the 19th century, with the application of group theory in many fields, including number theory, algebra, geometry and analysis, group theory had a tendency to unify mathematics, During this process, the group representation theory played a central role..Based on the unity of mathematics, this project focuses on the following three aspects: concept and theory: linear group, continuous group, combinatorial group. Interdisciplinarity: group theory and topology, group theory and analysis, group theory and number theory. Scientists biography: Dehn, Chevalley, etc. Being the major outcome of this projection, a clarification of the thread of different branches in mathematics, as well as the effect and influence of group theory, will help us understand the unity of mathematics.

近两个世纪以来，数学呈现出指数式的飞速发展。随着数学内容的不断增加，现代数学不可避免地分化为许多分支。但是，数学并没有被分割，各分支之间的联系十分密切。这说明，数学存在着某种统一的趋势。纵观世界上的数学强国，没有一个是只注重单一学科而忽视数学统一性的。特别是1872年以来，德国数学家克莱因采用群的观点首次将多种几何学联系在一起，开创了群论统一数学的成功范例。到19世纪末，群论在数论、代数、几何与分析等学科分支都找到了应用，大有统一数学的趋势。在此过程中，群表示论起到了核心的作用。.本项目以数学的统一性为着眼点，拟从概念理论、学科交叉、人物评传三个方面对线性群、连续群、组合群等概念与理论，群论与拓扑、分析、数论等学科交叉以及戴恩、谢瓦莱等关键的数学家进行研究。通过本项目的实施，将有助于厘清这一时期不同学科的发展脉络，明晰群在不同学科发展中的作用和影响，加深人们对数学统一性的认识。

课题组严格按照计划书开展研究，顺利完成了研究任务。课题组成员及研究生在国内重要学术刊物上共计发表研究论文17篇、会议论文1篇（均标注本基金资助成果）。其中包含CSSCI论文10篇，北大中文核心11篇，EI论文1篇。通过本项目的实施，将有助于厘清这一时期不同学科的发展脉络，明晰群在不同学科发展中的作用和影响，加深人们对数学统一性的认识。取得的相关成果对于丰富近现代数学史料、开展数学史及数学文化教育具有重要意义和价值。