R. Padmanabhan

Geometry Using Group Theory

Date Thursday, October 16, 2014

While visualization is the source of all geometry, group theory is a part of abstract algebra. However, these two topics interact beautifully as expounded by Felix Klein in his famous Erlanger Program recalled by Eric in his talk last week. We plan to use the familiar calculus of reflections as a backdrop to demonstrate the expressive power of group theory to systematically build geometric concepts by just using the algebra of groups. This will show that group theory and plane geometry are but two sides of the same concept. The axioms we use were introduced by Frederich Backhmann to describe the Hjelmslev planes.

This talk was originally given way back in 2006 in the Rings & Modules Seminar.

References [1] Bachmann, F. Aufbau der Geometrie aus dem Spiegelungswbegriff, Springer Verlag, Berlin, 1983. [2] Bachmann, F et al. Absolute Geometry, Foundations of Mathematics, Volume II, MIT Press, Cambridge, 1983. [2] Szczerba, L. W., "Interpretations of elementary theories" pages 129-145 in Logic and Foundations of Mathematics, Reidel, Boston 1977 [3] Tarski, A., "What is elementary geometry?", Studies in Logic and the Foundations of Mathematics, p.16-29, North- Holland, 1959