Recent work of Bhatt-Lurie and Drinfeld has constructed a category of p-adic motives (aka prismatic F-gauges) for schemes in positive and mixed characteristic. Roughly, these correspond to a notion of 'variations of Hodge structures' in integral p-adic Hodge theory. In this talk, I will review this notion and explain how to completely describe it in the case of Frobenius liftable schemes in positive characteristic . This description is in terms of (big) Fontaine-Laffaile modules, a somewhat classical coefficients system in p-adic Hodge theory and is closely related to recent results of Ogus and Terentiuk--Vologodsky--Xu.
While the result is of a classical flavour, our techniques use some recent conceptual advances in derived geometry, due to various authors, which we will explain if time permits.