# Phylogenetic Networks

## TL;DR

This paper discusses the challenges and mathematical questions involved in inferring phylogenetic networks, which are more accurate than trees for representing complex evolutionary histories involving hybridization and gene flow.

## Contribution

It introduces basic concepts of phylogenetics and explores open research questions in the combinatorics and algebraic modeling of phylogenetic networks.

## Key findings

- Identifies open questions in network combinatorics
- Connects phylogenetic models to algebraic varieties
- Highlights the importance of networks over trees in complex evolution

## Abstract

Phylogenetics is the study of the evolutionary relationships between organisms. One of the main challenges in the field is to take biological data for a group of organisms and to infer an evolutionary tree, a graph that represents these relationships. Developing practical and efficient methods for inferring phylogenetic trees has lead to a number of interesting mathematical questions across a variety of fields. However, due to hybridization and gene flow, a phylogenetic network may be a better representation of the evolutionary history of some groups of organisms. In this chapter, we introduce some of the basic concepts in phylogenetics, and present related undergraduate research projects on phylogenetic networks that touch on areas of graph theory and abstract algebra. In the first section, we describe several open research questions related to the combinatorics of phylogenetic networks. In the second, we describe problems related to understanding phylogenetic statistical models as algebraic varieties.