The Culture of Abstract Algebra: Conceptual Languages and Working Styles in Representation Theory in the Second Half of the 20th Century

这个合作项目将研究影响表征理论研究方向、范围和优先级的社会和文化因素，表征理论是抽象代数中发展最活跃、技术难度最大的领域之一。通过口述历史访谈、话语分析和原始资料研究，本研究将从两个主要方面探讨数学界的文化实践：代数理论作为概念语言的作用，以及不同数学风格在表征理论中的相互作用。本项目将考察实际研究实践中的模式是否符合以下文体对立：问题解决和概念主义；抽象与有形；概念化与计算；极简主义和自由选择；以及猜测和严谨。以莫斯科国立大学Gelfand研讨会为例，该项目将考察国家背景在塑造数学实践中的作用，并探索不同研究学派后代之间跨文化接触的后果。潜在的更广泛的影响该项目的结果可能对对交流模式和研究风格之间的联系感兴趣的历史学家和科学哲学家有用，或者在纯数学和理论物理之间的跨学科联系中，或者在塑造数学实践的地方和国家背景中所起的作用。该项目还将有助于向更广泛的受众开放深奥的抽象数学实践，促进在更广泛的文化中更好地理解和欣赏高等数学，并帮助吸引有才华的高中生进入数学领域。提出的分析数学实践，使高度抽象的概念“有形”，将有助于设计新的教育技术，以帮助学生？掌握高等数学结构。通过从科学文化史的角度提供对近代数学实践的可理解的描述，该项目还将促进数学家和人文主义者之间更广泛的文化对话。

IntroductionThis collaborative project will examine the social and cultural factors affecting the direction, scope, and priorities of recent research in representation theory, one of the most actively developing and technically difficult areas of abstract algebra.Intellectual MeritUsing oral history interviews, discourse analysis, and the study of primary sources, this study will address cultural practices of the mathematical community in two major aspects: The role of algebraic theories as conceptual languages, and the interaction of different mathematical styles in representation theory. This project will examine whether patterns in actual research practice correspond to the following stylistic oppositions: Problem-solving and conceptualism; abstraction and tangibility; conceptualization and computation; minimalism and freedom of choice; and speculation and rigor. Using the Gelfand seminar at Moscow State University as a case study, this project will examine the role of a national context in shaping mathematical practice and will explore the ramifications of cross-cultural contacts between the descendants of different research schools.Potential Broader ImpactsThe project results are likely to be useful to historians and philosophers of science interested in connections between communication modes and research styles, or in the cross-disciplinary contacts between pure mathematics and theoretical physics, or in the role of local and national contexts in shaping mathematical practice. The project will also serve to facilitate opening the esoteric practice of abstract mathematics to a wider audience, fostering better understanding and appreciation of advanced mathematics in broader culture, and helping attract talented high school students to the field of mathematics. The proposed analysis of mathematical practices that render highly abstract concepts "tangible" would help devise new educational techniques to help student?s master advanced mathematical constructions. By providing an accessible account of recent mathematical practice from the perspective of cultural history of science, this project will also facilitate a broader cultural dialogue between mathematicians and humanists.