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

*Dr. Paul Tupper*

### 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).

### About the Speaker

Paul Tupper graduated from Simon Fraser University with a BSc in Mathematics in 1997, and went on to do a PhD in the Scientific Computing-Computational Mathematics Program at Stanford. After two years as a post-doc in Physics at McGill, Dr Tupper took a position as Assistant Professor in Mathematics and Statistics at McGill. In 2008 he joined SFU as Associate Professor and Canada Research Chair of Applied Mathematics. Dr Tupper's interests span numerical analysis, molecular dynamics, phylogenetics, mathematical modelling in linguistics. In 2005 he won 1st place in the Leslie Fox Prize competition for his work on ergodicity and simulation of Hamiltonian systems. Asked how it feels to be a professor in the same department where he was a student, Dr Tupper replies "It is a thrill to be able to work beside my boyhood heroes, in such an excellent department."