A phylogeny is a mathematical model of the common evolutionary history of a group of species or genes. While phylogenies are inferred from various data types, in the era of advanced genome sequencing technology the data used to infer the phylogeny is primarily nucleotide or protein sequence data. Phylogeny-inference problems are most often expressed as optimization problems that are in complexity classes such as NP, so polynomial-time heuristics are developed to approximate solutions to these problems. These heuristics can be evaluated using traditional methods such as upper bounds on running time, average case analysis, and statistical consistency. In this talk, we discuss (1) the alternative approach of evaluating polynomial-time heuristics through the lens of geometry and (2) connections between algebraic geometry, statistics, and phylogenetics. No previous knowledge of phylogenetics will be assumed for this talk.