Kisin, Mark2011-04-042011-04-04Kisin, Mark. Forthcoming. The structure of potentially semi-stable deformation rings. Proceedings of the International Congress of Mathematicians, Hyderabad, India, August 19-27, 2010.http://nrs.harvard.edu/urn-3:HUL.InstRepos:4794673Inside the universal deformation space of a local Galois representation one has the set of deformations which are potentially semi-stable of given p-adic Hodge and Galois type. It turns out these points cut out a closed subspace of the deformation space. A deep conjecture due to Breuil-M´ezard predicts that part of the structure of this space can be described in terms of the local Langlands correspondence. For 2-dimensional representations the conjecture can be made precise. We explain some of the progress in this case, which reveals that the conjecture is intimately connected to the p-adic local Langlands correspondence, as well as to the Fontaine-Mazur conjecture.en-USConference Paper2011-04-0410.1142/9789814324359_0052