  We provide new proofs of two key results of p-adic Hodge theory: the
Fontaine-Wintenberger isomorphism between Galois groups in characteristic 0 and
characteristic p, and the Cherbonnier-Colmez theorem on decompletion of (phi,
Gamma)-modules. These proofs are derived from joint work with Liu on relative
p-adic Hodge theory, and are closely related to Scholze's study of perfectoid
algebras and spaces.
