Speaker: David Bryant 
Institution: Dept. Mathematics and Statistics, University of Otago
URL: www.maths.otago.ac.nz/~dbryant

Title: An Introduction to Diversity Theory

Abstract:
The Hahn-Banach theorem states when a bounded linear functional can be 
extended to the entire space. In the 50s and 60s, mathematicians looked into 
how the theorem might be generalised to arbitrary metric spaces: for which 
spaces can we always extend bounded functions on the subspace to functions on 
the whole space? The real line is one such metric space, the Euclidean plane 
is not. The resulting theory of hyperconvexity and injective hulls has (not 
surprisingly) been rediscovered multiple times and has also (perhaps 
surprisingly) become a cornerstone of phylogenetic mathematics with widespread 
applications.

I will give a rushed introduction to this theory, and then describe recent 
work (with Paul Tupper, Simon Fraser) extending it to a generalisation of 
metric spaces called diversities. Diversities assign values to all finite 
subsets of points, instead of just pairs, and satisfy the 
`triangle inequality' d(A,C) \leq d(A,B) + d(B,C) for all non-empty B. I will 
sketch some of the directions this theory is heading, as well as some 
applications.

Details are available at http://arxiv.org/abs/1006.1095