The Nielsen–Thurston classification of mapping classes and Thurston's theorem for the characterization of rational maps are central theorems in surface topology and complex dynamics, respectively. We give a single proof that unifies the two theorems. Moreover, we adapt mapping class group techniques to develop an algorithm that identifies when a branched self-cover of the plane is equivalent to a polynomial. This is joint work with Jim Belk, Justin Lanier, and Dan Margalit.