A brief introduction to Combinatorial Geometry

ABSTRACT

Point, lines, and circles are some of the fundamental entities from geometry. In this talk we discuss the underlying combinatorics of point configurations and their dual structures: arrangements of lines and arrangements of great-circles. By slightly relaxing the geometric restrictions ("lines" dont have to be straight and "circles" dont have to be round), we obtain so-called pseudopoint configurations, arrangements of pseudolines and (great-)pseudocircles, respectively. While the original settings cannot be axiomized via finitely many forbidden subconfigurations, we can indeed find such a purely combinatorial describtion for the relaxed "pseudo" setting which allows us to make investigations using computer assistance, and in particular, using SAT attacks. Last but not least, we discuss so-called simple topological drawings, which have the same combinatorial properties as straight-line drawings. Since on top of each point set, we can place a straight-line drawing of the complete graph, simple topological drawings can be seen as a further generalization of point configurations (they indeed also generalize pseudopoint configurations).

CONTACT

Stefan Szeider