logo The conjectures of Fontaine-Mazur and Breuil-Mezard

I will explain an intimate connection between the Fontaine-Mazur conjecture, the Breuil-Mezard conjecture and the p-adic local Langlands correspondence.
Using recent work of Colmez and Berger-Breuil, one can prove both conjectures for representations which becomes semi-stable over an abelian extension.