In this project we wish in particular to explore the interface between linear algebra, geometry, and topology, that arises from the embedding of finite metric spaces into ambient spaces with different geometric and topological features. A fundamental question is how many of these features are picked up and thus represented by such embeddings. This area of research is particularly well suited for computer experiments. A significant number of metric spaces can be effectively examined, and the influence on the algebraic properties of the metric by parameters such as the curvature of the ambient space can be studied, to strengthen intuition and to generate hypotheses. To understand a finite metric space is a matter of understanding its distance matrix. Thus, in order to find good metric invariants for the recognition program, one natural place to search and experiment is in the set of invariants for distance matrices of finite metric spaces.
For example we have introduced the invariant notion of strictly negative type for such spaces and show that this property implies unique realization of the so called infinity-extent (i.e., of the transfinite diameter). Finite metric spaces that have this property include all trees and all finite subsets of the Euclidean spaces. We show that the strictly negative type finite subspaces of spheres are precisely those which do not contain two or more pairs of antipodal points. We also recently proved that all finite subspaces of the hyperbolic spaces are of strictly negative type.
Although the first paper from this project does not contain any sign of it, the main results of it were initially observed via relatively simple computer experiments.
A similar strategy is applied in an ongoing project concerning the embeddability of discrete metric subspaces - in particular so-called leaf spaces - into constant curvature standard spaces. Some interesting phenomena relating the possible curvatures of the receiving space to metric properties of the domain space have been observed using computer experiments, but are not yet properly understood.