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