Home /  The History of Algebra in the Nineteenth and Twentieth Centuries

Workshop

The History of Algebra in the Nineteenth and Twentieth Centuries April 21, 2003 - April 25, 2003
Registration Deadline: April 25, 2003 over 21 years ago
To apply for Funding you must register by: January 21, 2003 almost 22 years ago
Parent Program: --
Organizers Jeremy J. Gray and Karen Hunger Parshall
Speaker(s)

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Description
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
Keywords and Mathematics Subject Classification (MSC)
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
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To apply for funding, you must register by the funding application deadline displayed above.

Students, recent PhDs, women, and members of underrepresented minorities are particularly encouraged to apply. Funding awards are typically made 6 weeks before the workshop begins. Requests received after the funding deadline are considered only if additional funds become available.

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For information about recommended hotels for visits of under 30 days, visit Short-Term Housing. Questions? Contact coord@slmath.org.

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Schedule, Notes/Handouts & Videos
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Apr 21, 2003
Monday
08:00 AM - 05:00 PM
  On the history of the Frobenius- and Tchebotarev-Density
Gunther Frei
08:45 AM - 09:00 AM
  Welome and introduction
Loa Nowina-Sapinski
09:00 AM - 10:00 AM
  Very full of symbols: Duncan F. Gregory, the calculus of operations, and the Cambridge Mathematical Journal
Sloan Despeaux
10:00 AM - 10:30 AM
  Morning Tea
10:30 AM - 11:30 AM
  Between algebra and analysis: The impact of the French school of Idéologie on Charles Babbage’s work on functional equations
Eduardo Ortiz
11:30 AM - 01:00 PM
  Lunch
01:00 PM - 02:00 PM
  F. S. Macaulay: An English schoolmaster and abstract algebra
Jeremy Gray (University of Warwick)
02:00 PM - 03:00 PM
  Quotient rings of noncommutative rings
Severino Coutinho
04:10 PM - 05:10 PM
  The mathematical legacy of James Joseph Sylvester (1814-1897)
Karen Parshall (University of Virginia)
Apr 22, 2003
Tuesday
09:00 AM - 10:00 AM
  On the early history of commutative algebra with special emphasis on divisibility theories and the foundations of algebraic geometry
Olaf Neumann
10:00 AM - 10:30 AM
  Morning Tea
10:30 AM - 11:30 AM
  Kronecker’s ‘Fundamentalsatz der Allgemeinen Arithmetik’ and its relations to the fundamental theorem of algebra
Harold Edwards
11:30 AM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  From Algebra (1895) to Moderne Algebra (1930), changing conceptions of a discipline: A guided tour using the Jahrbuch uber die Fortschritte der Mathematik
Leo Corry
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:30 PM
  Discussion: National contexts in the formation of mathematics
Karen Parshall (University of Virginia)
04:30 PM - 05:30 PM
  Reception
Apr 23, 2003
Wednesday
09:00 AM - 10:00 AM
  Defining a mathematical research school: The case of algebra at the University of Chicago, 1892-1950
Karen Parshall (University of Virginia)
10:00 AM - 10:30 AM
  Morning Tea
10:30 AM - 11:30 AM
  Leonard Dickson’s algebraic research and its institutional context: A case study of A. Adrian Albert
Della Fenster
11:30 AM - 01:00 PM
  Lunch
01:00 PM - 02:00 PM
  Historical notes on Emmy Noether’s 1932 ICM lecture on simple algebras and algebraic number theory
Charles Curtis (University of Oregon)
02:00 PM - 03:00 PM
  On the local-global principle in arithmetic and algebra
Joachim Schwermer
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:30 PM
  Discussion: Institutions and their role in the formation of mathematics
Karen Parshall (University of Virginia)
Apr 24, 2003
Thursday
09:00 AM - 10:00 AM
  On the theory of density of primes and the irreducibility of polynomials in the nineteenth century
Gunther Frei
10:00 AM - 10:30 AM
  Morning Tea
10:30 AM - 11:30 AM
  The rising tide: Grothendieck on simplicity and generality
Colin McLarty
11:30 AM - 01:00 PM
  Lunch
01:00 PM - 02:00 PM
  Shaping Bourbaki’s algebra
Liliane Beaulieu
04:10 PM - 05:00 PM
  Kurt Heegner: An outsider in number theory (UC Berkeley Math Dept. Colloquium, at Evans Hall, Room 60)
Norbert Schappacher
Apr 25, 2003
Friday
09:00 AM - 10:00 AM
  Bartel van der Waerden’s work on algebraic geometry
Norbert Schappacher
10:00 AM - 10:30 AM
  Morning Tea
10:30 AM - 11:30 AM
  On the arithmetization of algebraic geometry: The case of Oscar Zariski
Silke Slembek
11:30 AM - 01:00 PM
  Lunch
01:00 PM - 01:50 PM
  Discussion: What is, or was, modern algebra?
Karen Parshall (University of Virginia)
02:00 PM - 02:50 PM
  Discussion: Conclusions
Karen Parshall (University of Virginia)
04:00 PM - 05:00 PM
  Partition: Hardy and Ramanujan in Berkeley (at the Sibley Auditorium in the Bechtel Engineering Center, UC Berkeley)
Jeremy Gray (University of Warwick)