Workshop

A number of key disciplines define the "modern" mathematics of today, but the first to have the adjective "modern" associated with it was the "modern algebra" that Emmy Noether and her school created in the 1920s and 1930s and that led directly, among other things, to what we now know as commutative algebra. In fact, that "modern algebra" has deep nineteenth-century roots in the work, for example, of Richard Dedekind and Leopold Kronecker and profound interconnections with other branches of mathematics such as algebraic number theory and algebraic geometry. Moreover, two alternative formulations have long dominated "modern algebra" and continue to animate research today: the conceptual approach associated with the names of Dedekind and David Hilbert, and a more algorithmic approach espoused by Kronecker and Hermann Weyl. Recent advances in computer-assisted mathematics have naturally revived the latter, but both philosophies provide broad visions of mathematics productive of many important open problems as well as of effective tools for solving them. Historians of mathematics have come to focus seriously on the history of modern algebra only within the last twenty-five years. That history originally tended to be done from the very technical point of view of the history of ideas, an approach typified in, for example, Morris Kline's massive Mathematical Thought from Ancient to Modern Times (1972), although Kline tended to give algebra in general rather short shrift in that work. In 1985, B. L. van der Waerden provided a more synthetic and focused account in A History of Algebra from al-Khwarismi to Emmy Noether, but while this work incorporated some biographical and broader historical analysis, it presented the development of algebra as a series of rather disjointed mathematical vignettes instead of in terms of a coherent historical and mathematical analysis. To date, no work has been written on the history of algebra in the nineteenth and twentieth centuries that ranges widely over the complex and interacting factors that led to modern algebra. Still, the past twenty-five years have witnessed the publication of numerous targeted, specialized studies on a number of these aspects. There have been extensive studies of some of the key figures -- Hermann Grassmann, James Joseph Sylvester, Leopold Kronecker, Sophus Lie, David Hilbert, Georg Frobenius, Emmy Noether -- and there is work progress on Dedekind, Francis Macaulay, and Oscar Zariski, among many others. There have also been studies focused not on individuals but on broader trends such as structuralism and modernity and their reflections in the development of mathematics in general and of algebra in particular. With this sophisticated body of literature to draw from -- a literature that has fruitfully combined a more traditional internalistic approach to the history of mathematics with historiographical techniques informed by biography, sociology, institutional history, philosophy, and other fields -- the time seems ripe to begin to lay the groundwork and to set the parameters for a new synthesis of the history of modern algebra. Some of the topics the workshop will address are: the contribution of David Hilbert to modern algebra, his re-working of the ideas of Dedekind and Kronecker, and the role of his new formulation of the subject in guiding subsequent research; the origins of Emmy Noether's work and its later developments; the spread and development of algebraic thought from Germany to Britain, France, and the United States; the importance and the nature of the connections between commutative algebra and algebraic geometry and algebraic number theory; the philosophy and the teaching of modern algebra; the institutionalization of modern algebra in the nineteenth and early twentieth centuries