Molecular phylogenetics from an algebraic viewpoint

The probabilistic models used in the inference of phylogenetic trees from molecular data are particularly rich in algebraic structure. This was first noticed 20 years ago when certain polynomials called phylogenetic invariants were introduced by Cavender and Felsenstein, and by Lake. Recently, however, there have been considerable advances in our algebraic understanding of these models. We survey some of this work, indicating both how algebra has been exploited for theoretical understanding and the preliminary steps that have been taken toward its use in practical inference.