Algebraic Structures into Phylogenetics: Insights from Tropical Convexity and Likelihood Geometry

Abstract

This thesis explores two areas of algebraic statistics: tropical phylogenetics and likelihood geometry. Part I concerns the space of equidistant phylogenetic trees, which is a convex tropical variety. Specifically, I classify the tree topology changes that can occur along a tropical line segment between two tree data points and compute the tropical geometric median (Fermat-Weber points) of weighted tree data. Part II is concerned with the maximum likelihood degree of Brownian motion tree models. The main result is that the maximum likelihood degree of a Brownian motion tree model on the star tree with n leaves is 2^{n+1} - 2n - 3. Another important result in this part is a generalization of the Cayley-Prufer Theorem to complete graphs with a weighting determined by the paths in a tree.