Signe Lundqvist: Point-line configurations from the point of view of rigidity theory

Abstract.

In this talk, we will discuss point-line configurations, which are structures that consist of points and straight lines in the real projective plane. Each point-line configuration has an underlying combinatorial incidence geometry.

We are interested in the space of all point-line configurations that are realisations of the same underlying incidence geometry. Specifically, we are interested in determining if there are finitely many realisations of a given incidence geometry as points and straight lines, or if there is a continuous deformation of a point-line configuration that preserves the incidences between the points and lines (a projective motion of the point-line configuration). We will see that theorems in the projective plane, such as Pappus and Desargues classical theorems, make characterising which point-line configurations have a projective motion an interesting but challenging problem.

This talk is based on joint work with Leah Berman, Bernd Schulze, Brigitte Servatius, Herman Servatius, Klara Stokes and Walter Whiteley.