Title: Discrete Geodesic Geometry
Abstract: A path system in a graph G is a collection of paths containing exactly one path between every pair of vertices in G. We call a path system consistent if it is closed under taking subpaths. A consistent path system is said to be metric if it can be realized as a system of geodesics with respect to some positive edge weights. It turns out that the class of consistent path systems is extremely rich, going well beyond the metric case. In this talk, we will explore what is currently known about these objects and highlight some of the many questions that remain open.
This talk is based on joint work with Nati Linial and Maria Chudnovsky.