Oberwolfach Seminar: The Erlangen Program, Myths and Realities: Geometry and Group Theory, 1870-1920

Date

October 18th - October 24th, 2009

Organizers

Igor Dolgachev, Ann Arbor

David Rowe, Mainz

Klaus Volkert, Köln

Duco van Straten, Mainz

Programme

This is an interdisciplinary seminar that focuses on an important topic in the history of mathematics. As every geometer knows, Felix Klein's "Erlangen Program" ("Vergleichende Betrachtungen über neuere geometrische Forschungen") argued that group theory provided an ideal framework for unifying geometrical research. In Klein's time geometers did not sharply distinguish between algebraic and differential geometry, nor did they tend to emphasize whether they were working over the real or complex numbers. Classical geometers often used imaginary coordinates to describe various phenomena, but distinguished these from real solutions. Within this rather chaotic setting, the 23-year-old Klein came up with the idea that the various approaches to geometry could be understood as the invariant theories associated with the various kinds of transformation groups that act on a given manifold. In this seminar we will reassess the historical significance of Klein's "Erlangen Program" by exploring three different phases in which geometry was enriched by the theory of groups.

During the first phase this interaction mainly took place within projective algebraic geometry, a field in which Cayley, Sylvester, Hesse, and Clebsch linked invariant theory with the properties of systems of conics and higher order curves. Klein and Sophus Lie also approached group theory through the higher-dimensional spaces of line- and sphere-geometry. By exploring this background and Klein's collaboration with Lie we will reassess the significance of both for the Erlangen Program.

The second phase, marked by the emergence of Italian algebraic geometry, did much to promote the ideas in the Erlangen Program, though its newfound popularity also created friction between Klein and Lie. We will consider the relevance of such personal rivalries and tensions between leading mathematical schools in Germany as well as the larger national interests that pitted German mathematicians against their counterparts in France. In this context the work of Henri Poincaré deserves special attention, particularly since he arrived at a group-theoretic vision for geometrical research, though without knowledge of the Erlangen Program.

During the third phase of reception, beginning ca. 1910, Klein promoted the Erlangen Program in connection with Einstein's special theory of relativity after his colleague Minkowski recast Einstein's theory as the 4-dimensional geometry associated with the Lorentz group. Later, in the context of relativistic cosmology, Klein used projective methods to clarify a crucial issue in the debate between Einstein and de Sitter over the status of singularities in de Sitter's matter-free model of the universe. Klein included all his work on relativity together with his early geometrical investigations in his Collected Works (vol. 1, 1921) under the title "Zum Erlanger Programm."

Prerequisites

Basic knowledge of algebraic geometry and group theory; interest in classical mathematics.

Literature

To be announced

Deadline for applications

September 1st, 2009

The seminars take place at the Mathematisches Forschungsinstitut Oberwolfach. The number of participants is restricted to 24. The Institute covers accommodation and food. Travel expenses cannot be reimbursed. Applications including

• full name and address, including e-mail address
• present position, university
• name of supervisor of Ph.D. thesis
• a short summary of previous work and interest

should be sent as hard copy or by e-mail (.ps or .pdf file) to:

    Prof. Dr. Gert-Martin Greuel
    Universität Kaiserslautern
    Fachbereich Mathematik
    Erwin Schrödingerstr.
    67663 Kaiserslautern, Germany