Group-based Phylogenetic Models

A phylogenetic model describes the evolution of genes. A group-based phylogenetic model assumes extra structure on the parameters of the model that is related to an abelian group. The joint probabilities of a group-based phylogenetic model are functions in the parameters of the model. We study polynomial inequalities that cut out the set of all joint probabilities for a time-reversible group-based model. We apply this knowledge to the study of boundaries and maximum likelihood estimation on group-based models. In particular, we use the degree of an algebraic variety and numerical algebraic geometry to obtain the maximum likelihood estimate exactly for small group-based models. This talk is based on joint work with Dimitra Kosta.