A conjecture of Fontaine--Mazur predicts that many Galois representations arise from geometry. In some cases this conjecture has been proven; an important ingredient in the arguments is an understanding of the geometry of some deformation spaces of Galois representations. I will give an overview of this link, and then will explain an extension of a method, originally due to Kisin, which produces a resolution of these spaces via semi-linear algebra. As a result we can say something about the irreducible components of crystalline deformation rings with Hodge--Tate weights between 0 and p, for unramified extensions of Qp.
Links:
[1] http://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/de/node/3444
[3] http://www.mpim-bonn.mpg.de/de/node/246