In this talk, we present an analog of dual equivalence for affine permutations. Exploiting the connection between affine permutations and n-cores, this establishes the Schur positivity of the strong Schur functions introduced by Lam, Lapointe, Morse and Shimozono, which are generalizations of the k-Schur functions introduced by Lapointe, Lascoux and Morse. Time permitting, we will show how this approach may ultimately lead to an explicit connection between Macdonald polynomials (and, more generally, LLT polynomials) and k-Schur functions. \\ \noindent This is joint work with Sara Billey at the University of Washington.