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Thursday, February 4, 2016

**Abstract:** Given a mod $p$ representation of the absolute Galois group of $\mathbb{Q}_\ell$ , consider the universal framed deformation ring $R$ parametrising its lifts. When $\ell$ and $p$ are distinct I will explain a relation between the mod $p$ geometry of $R$ and the mod $p$ representation theory of $\mathrm{GL}_n(\mathbb{Z}_\ell)$, that is parallel to the Breuil-Mézard conjecture in the $\ell = p$ case. I will give examples and say something about the proof, which uses automorphy lifting techniques.