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

点击翻译按钮获取中文摘要

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.