mirror symmetry and the breuil-mezard conjecture abstract: the breuil-mezard conjecture predicts the existence of hypothetical "breuil-mezard cycles" that should govern congruences between mod p automorphic forms on a reductive group g. most of the progress thus far has been concentrated on the case g = gl_2, which has several special features. i will talk about joint work with bao le hung on a new approach to the breuil-mezard conjecture, which applies for arbitrary groups (and in particular, in arbitrary rank). it is based on the intuition that the breuil-mezard conjecture is analogous to homological mirror symmetry.