This 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 Merit Using 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 Impacts The 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.
One of the most abstract areas of mathematics, representation theory, is ? study of fundamental symmetries underlying mathematical objects. In the past few decades, representation theory played a critical role in shaping new insights in modern physical theories – quantum mechanics and quantum field theory. While traditional technical studies in history of mathematics are aimed at a narrow group of specialists, this project took a new approach, linking the history of representation theory to broader trends in science and society. This project focused on the international dimension of activities of the mathematical community. The Principal Investigator conducted more than 70 interviews with mathematicians from the United States, Russia, France, Britain, Israel, Japan, and other countries. This study helped reconstruct sprawling networks of intellectual influence and exchange, identify social mechanisms of the spread of fashionable ideas, and examine the interplay of local and global factors in shaping research agendas. This project highlighted a diversity of research styles and communication patterns in different national and disciplinary contexts. Stressing the role of social factors, this study linked various writing conventions, presentation styles, and ethical norms prevalent in different research communities to specific social mechanisms for exchange and transmission of knowledge that emerged under different types of social constraints. This study also outlined the role of representation theory as a vehicle of translation between different conceptual mathematical languages, facilitating the use of methods and tools across different fields. This project’s central case study was the development of representation theory in the Soviet Union in the 1960-70s, in the context of oppressive policies of the Soviet regime. Soviet officials subjected the mathematics community to serious constraints, including discriminatory policies in university admissions, hiring, and publishing; severe limitations on foreign travel; and even restricted physical access to research institutions and universities. While some mathematicians actively enforced these policies, others opposed them by creating a semi-private social infrastructure for mathematical instruction and research. This infrastructure included open seminars on advanced mathematical topics; a network of specialized mathematical high schools; informal educational organizations; "creative" editorial policies, broadening the scope of some mathematical publications; the organization of pure mathematics research groups under the auspices of physical research institutes and computation centers; and the practice of conducting mathematical communications outside of formal institutions – in private apartments, at summer dachas, or during nature walks. This parallel social infrastructure fostered intellectual exchanges across disciplinary lines, in particular, between mathematics and physics. It was in this context that crucial breakthroughs were made, turning highly abstract mathematical notions into working tools for modern physicists. This project reconstructed key components of this informal social infrastructure, drawing on unpublished materials, such as Gelfand seminar notes, handwritten in Russian. To complement the study of primary sources with oral history, the Principal Investigator organized a Gelfand Recollections session at the Gelfand Centennial conference (2013). The videos from the session will soon be made publicly available on the conference website. The approach developed under this project facilitated an ongoing shift in the development of history of mathematics as a discipline – from narrow technical studies to socially and culturally informed analyses, which stress the broad cultural connections of mathematics and the role of social factors in shaping the activities of the mathematics community. This approach was manifested in the session, "What It Means to Be a Mathematician: Post-WWII Mathematics in Search of Identity," which the Principal Investigator organized at the 2013 annual meeting of the History of Science Society, with papers contributed by three graduates students. Looking at a variety of contexts (the United States, France, the Soviet Union, and international organizations), the panelists examined the norms and structures of postwar mathematical communities as the mathematicians adapted to the evolving political and institutional environment and strove to define their role in the postwar academic world. This project’s results have been disseminated in two publications and seven presentations at academic conferences and will be used in a course on cultural history of mathematics at MIT. The Principal Investigator is currently preparing a monograph on this topic, accessible to a wide audience.
