Period morphisms in p-adic Hodge theory

In classical Hodge theory, variations of Hodge structure give rise to varying periods, and then to period morphisms.  In p-adic Hodge theory, there are plenty of periods but (until recently) not so many period morphisms.  It turns out these exist in some ridiculous generality, and they have some interesting applications – you can read more here! (The linked thing was originally meant to be a blog post.)

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