Phylogenetics is the area of research concerned with finding the genetic relationship between species. The relationship can be represented by a phylogenetic tree, which is a simple, connected, acyclic graph equipped with some statistical information. This furnishes a certain polynomial map and we are interested in polynomials, called phylogenetic invariants, which vanish for every choice of model parameters. The set of phylogenetic invariants forms a certain algebraic object and we want to compute this object explicitly. One of the reasons that we want an explicit description of these polynomials is because it is claimed by Casanellas and Fernandez-Sanchez that using the entire set of phylogenetic invariants is an efficient phylogenetic reconstruction method. More importantly, phylogenetic invariants were used by Allman and Rhodes to study the problem of identifiability of tree topology for a number of phylogenetic models. In other words, given a distribution of observations that a certain model predicts, is it possible to uniquely determine all the parameters of the model? It is an important question since, if a tree is not uniquely determined by an expected joint distribution, then we cannot use that model for inference.
This presentation will explore in some detail group-based models and their invariants. (The content is drawn from the joint work with Sonja Petrovic, UIC , email@example.com)