SFU Canada Research Chairs Seminar Series: "From Distance to Diversity: Extending the Concept of a Metric Space"

Abstract

In mathematics and related disicplines a metric space is a mathematical abstraction of our everyday concept of space with a corresponding notion of distance. For each pairs of points in a metric space there is a number which represents the distance between the points. Metrics spaces play an important role in phylogenetics, in which the genetic history of a collection of species is reconstructed from the species' genomes. One important construction in the theory of metric spaces is the tight span. The tight span of a metric space can be thought of as a generalization of the idea of a convex hull in linear spaces and is the basis for much work in the study and visualization of finite metric spaces. Motivated by problems in phylogenetics, we have developed a generalization of the concept of metric spaces, which we call diversities. In a diversity, every subset of points in the space corresponds to a number, not just pairs, and there is a more general version of the triangle inequality. Besides encompassing a number of interesting examples as special cases, diversities have a natural tight span construction with corresponding theory. I will give an introduction to metric spaces, phylogenetics, and tight spans, and then present some of our own work on diversities. Much of the talk will be accessible to a general audience. This is joint work with David Bryant (University of Otago, New Zealand).