Subdivisions of combinatorial objects arise in various contexts, part of which are investigated in the following project:
Polyhedral subdivisions of point configurations are dissections of the convex hull of a finite point configuration in Euclidean space into finitely many polytopes; all vertices of the polytopes have to be in the point configuration. If in a subdivision all polytopes are simplices it is a triangulation. Certain Topological spaces constructed from subdivision posets allow for a unified description of a variety of phenomenons from order theory, model theory, and the theory of discriminants.
Elementary statements about the topology (e.g., connectivity) or metrics (e.g., diameter) yield basic theoretical building blocks for the design and analysis of flip algorithms in computational geometry.
In this project we investigate subdivision spaces of elementary point configurations.
