Abstract
The Breuil--M'ezard conjecture relates the mod p geometry of moduli spaces of n-dimensional potentially crystalline (or semi-stable) Galois representations in terms of the mod p representation theory of GL_n. In this talk I will explain a proof of this result for two dimensional crystalline representations with sufficiently small Hodge--Tate weights (roughly <= p/e for e the ramification degree). The main idea is to relate the geometry of these moduli spaces to degenerations of products of flag varieties in an affine grassmannian, and to prove a version of Breuil--M\'ezard for these degenerations.
