Cyclic Configurations
by
T. Pisanski
Ljubljana

In nineteenth century configurations played an important role in mathematical research. Prominent mathematicians, such as Cayley, Klein, Schönflies, etc. studied their properties. Configurations have always occupied a firm place at the intersection of geometry and combinatorics. Unfortunately, the 20th century growth of both of these mathematical disciplines put configurations out of fashion. In his dissertation in 1894 E. Steinitz proved that any connected v₃ configuration can be drawn in the Euclidean plane with at most one curved line. Based on a construction by H. L. Dorwart and B. Grünbaum from 1992 we are able to produce a sequence of v₃ configurations K(n) with the following property: For any integer m there exists an integer N such that for each n > N one has to remove at least m lines from K(n) in order to obtain an incidence structure that has a realization in the real projective plane (and hence in the Euclidean plane). In the talk we show how to reconcile this result with the theorem of Steinitz. The background theme of the talk is the class of cyclic configurations. Although they are very simple, cyclic configurations have many surprising properties and deep connections to other parts of mathematics. Using Levi graphs (also called incidence graphs) we connect them to computation of the Schur norm of certain matrices via Haar integrals.

Date received: May 26, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafd-20.