  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.
