the fontaine-mazur conjecture predicts which p-adic galois representations arise geometrically. a few years back, emerton and kisin made astounding progress in the two-dimensional case, by employing the p-adic langlands correspondence for gl(2) over $q_p$, which is very well-understood by now. a key point was the existence of ``locally algebraic" vectors. for groups of higher rank, even a conjectural generalization remains elusive. however, there is a conjecture of breuil & schneider, which gives a weak (but precise) analogue for gl(n). roughly it says that a certain filtration exists if and only if a certain lattice exists. in his thesis, hu completely proved one direction, and produced the expected filtration (by translating its existence into the so-called emerton condition). we will report on progress in the other direction, and in many cases prove the existence of gl(n)-stable lattices in locally algebraic representations constructed from p-adic hodge theoretical data. this argument is global in nature; the ultimate integral structure comes from p-adic modular forms. we hope to also hint at a formalism, in which an eigenvariety for u(n) parametrizes a correspondence between semisimple galois representations and banach-hecke modules with a unitary gl(n)-action, and discuss local-global compatibility "at p" in this context. in particular, we'll settle the breuil-schneider conjecture for dr representations which ``come from an eigenvariety".