  We survey some recent progress on generalizations of conjectures of Serre
concerning the cohomology of arithmetic groups, focusing primarily on the
"weight" aspect. This is intimately related to (generalizations of) a
conjecture of Breuil and M\'ezard relating the geometry of potentially
semistable deformation rings to modular representation theory. Recently, B.
Levin, S. Morra, and the authors established these conjectures in tame generic
contexts by constructing projective varieties (local models) in mixed
characteristic whose singularities model, in generic cases, those of tamely
potentially crystalline Galois deformation rings for unramified extensions of
$\mathbb{Q}_p$ with small regular Hodge-Tate weights.
