An entire map is said to be post-singularly finite if the forward orbit of its set of singular values is finite. Such maps play a crucial role in understanding natural families of entire maps. Motivated by previous work of Devaney, Goldberg and Hubbard, we ask the following question:

Given a post-singularly finite entire function f, can f be realized as the limit of a sequence of post-singularly finite polynomials?

In joint work with Nikolai Prochorov and Bernhard Reinke, using techniques from Teichmüller theory, we show how we may answer this question in the affirmative.