  We present a semicontinuity result, proven in recent joint work with Morrow
and Scholze, relating the mod $p$ singular cohomology of a smooth projective
complex algebraic variety X to the de Rham cohomology of a smooth
characteristic $p$ specialization of X: the rank of the former is bounded above
by that of the latter. The path to this result passes through $p$-adic Hodge
theory and perfectoid geometry, so we survey the relevant aspects of those
subjects as well.
