A Coxeter group is a group generated by a set S subject to defining
relations of the form: sē=1 for all s in S, and
(st)mst=1 for some or all pairs s,t in S.
Remarkably, every Coxeter group can be faithfully represented as a group
of linear transformations on a real vector space. In studying automorphisms
of Coxeter groups - or, more generally, the isomorphisms between Coxeter groups -
the challenge is to use group-theoretic data to characterize the group
This talk will describe progress on this, as yet unfinished, project.