Algebraic geometry: a window into phylogenetics

Abstract

Evolution by natural selection explains how species change and diversify over time. These relationships are often represented by phylogenetic trees, which illustrate how different species descend from common ancestors. In modern biology, such trees are reconstructed using molecular data, particularly DNA sequences. To model how DNA sequences evolve along a phylogenetic tree, mathematical models predict the probabilities of observing various nucleotide patterns across species. Under certain assumptions, these probabilities can be expressed as polynomials in the model parameters. This naturally leads to algebraic geometry, a branch of mathematics that studies the solution sets of polynomial equations, known as algebraic varieties. This thesis provides an introduction to algebraic geometry, covering fundamental concepts such as affine and projective varieties, dimension theory, and constructions like joins of varieties. These concepts are then applied in phylogenetics, focusing on a particular class of widely used evolutionary models known as equivariant models. In this context, the thesis explores phylogenetic mixtures, which model situations in which genetic data arise from multiple evolutionary trees. It explains how these mixtures can be described by linear equations and outlines a general method for constructing them for any number of species and any equivariant model. Finally, it addresses the identifiability problem, which concerns whether the evolutionary history can be uniquely inferred from the data, and presents a known lower bound on the number of trees beyond which neither the tree topologies nor the continuous parameters can be generically identified. Overall, this thesis aims to provide an introduction to algebraic geometry and explores how these mathematical tools can offer new perspectives on evolutionary models, opening a small window into a vast landscape.