SES 0312697 Thomas W. Hawkins, Boston University "Georg Frobenius and the History of Linear Algebra"
This proposal is to write an intellectual biography of the mathematician Georg Frobenius (1849-1917), who was trained in the mathematical school of Kronecker and Weierstrass at the University of Berlin in the 1870s and eventually returned to Berlin to play a prominent role in the direction the school took during 1892-1917. Although his impact upon present-day mathematics has been considerable, his work and its influence have been little studied by historians. Because Frobenius was not one of the mathematical giants of his era but a more typical mathematician, the proposed study will contribute to a more balanced understanding of how mathematics actually progresses. The proposed study will focus upon Frobenius's many contributions involving the theory or application of linear algebra. Since such work grew out of his investigation of a diverse spectrum of mathematical fields throughout his career, a good overview of his mathematical activity is a by-product. Considerable attention will be given to the context of his work, with three types of context being distinguished: prior, collateral, and subsequent developments. "Prior developments" includes those mathematical developments essential for an understanding of the disciplinary ideals of the Berlin school of Kronecker and Weierstrass that played an important role in guiding Frobenius in his selection of problems and in his approach to resolving them. "Collateral developments" refers to developments that are independent of Frobenius's work but led to more or less the same result. Frobenius was involved in an unusually large number of instances of multiple discovery that fall within the scope of the proposed study. They will be studied as a means of gaining a greater appreciation of both the special characteristics of Frobenius's work (vis `a vis that of his co-discoverers) and of the fact that he was very much a man of his mathematical times, abreast of a wide spectrum of 19th-century developments and typical in his interests albeit untypical in the Berlin-style manner in which he pursued those interests. Comparison of his work with that of his co-discoverers will also contribute to an understanding of how and why his work has impacted present-day mathematics. This involves consideration of the "subsequent developments" that, for reasons to be analyzed, incorporated the work and ideas of Frobenius thereby shaping present-day mathematics. The goal of the project is a publishable book that will shed light on the emergence of linear algebra from diverse areas of 19th-century mathematics and will clarify the central role played therein by Frobenius. On another level, it will represent an historically informed characterization of Berlin-style algebra, from its roots in the work of Weierstrass and Kronecker to its continuation at the hands of Frobenius's prize student, Issai Schur. Such a characterization should prove illuminating by comparison with the abstract algebraic style that emerged from Gottingen somewhat later. The book can have a broader impact in two ways, both of which are facilitated by its focus on linear algebra, a basic part of mathematics that is studied and utilized by mathematicians and a broad spectrum of scientists. On a pedagogical level, it can provide students of mathematics with a vicarious experience of what is actually involved in the human endeavor of mathematical research. It can also help expand the small infrastructural base of historians of mathematics by engaging the interest of mathematicians and scientists in history of mathematics, and encouraging them to collaborate with historians.