Recent work of Dylan Thurston gives a "positive criterion" for when a post-critically finite branched self-cover of the sphere is equivalent to a rational map. In this talk we will discuss this theorem and then apply it to give a new proof of a theorem of Rees, Shishikura, and Tan about the mateability of quadratic polynomials when one polynomial is in the main molecule. These methods may be a step in understanding the mateability of higher degree post-critically finite polynomials and demonstrate how to apply the positive criterion to classical problems. Joint with J Powell, R Winarski, and J Yang. Speaker(s): Caroline Davis (Indiana University)