MSI Weekly Bulletin - Week starting Monday 4 August, 2008

It is known that each finitely generated Coxeter group embeds nicely as a subgroup of (various) simply laced Coxeter groups. In this talk, we describe extensions of the notion of root system which allow one to obtain corresponding embeddings of root systems. For example, the "universal symmetric root system of a Coxeter group over commutative rings" embeds nicely in the standard root system (over the integers) of a canonically associated simply laced Coxeter system.

Advances in 3D imaging technology are enabling us to peer inside an enormous range of materials and create the need for quantitative measures of the complex structures being unveiled. The most fundamental description of an object is in terms of its topology: how many pieces, and what types of holes does it have? To answer this question we turn to work from the past decade or so on algorithms for extracting topological information from scattered point data. One of the most important concepts to emerge is that of topological persistence - a method for quantifying the significance of features using good old-fashioned homology. This talk will introduce the theory of persistent homology and describe how we are adapting it to the study of 3D digital images. Perhaps unsurprisingly, this is not as simple as it first appears. It turns out the problems stem from topological issues of a different kind - that of adjacency in digital grids.