The Breuil--Mézard conjecture describes the geometry of
moduli spaces of n-dimensional mod p Galois representations of G_K for K
a finite extension of Qp. This description is in terms of the
respresentation theory of GL_n(k) (k being the residue field of K) and
the conjecture can be viewed as a combinatorial shadow of the p-adic/mod
p Langlands correspondence.
In this talk I will give an overview of the conjecture and then talk
about recent work in which new instances are deduced for crystalline
representations with bounded Hodge types (roughly <= p/e for e the
ramification degree of K). This is done by establishing analogous
Breuil--Mézard phenemona for certain degenerations inside the affine
grassmannian and relating these degenerations to moduli of crystalline
representations.
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