A fundamental theorem in classical Hodge theory is
the isomorphism between de Rham and Betti cohomology for
complex manifolds; this follows directly from the Poincare
lemma. The *p*-adic analogue of this comparison was the subject
of a series of conjectures made by Fontaine in the early
'80s. In the last three decades, these conjectures have been
proven by various mathematicians, and have had an enormous
influence on arithmetic algebraic geometry. In my talk, I will
first discuss Fontaine's conjectures, and why one might care
about them. Then I will talk about some work in progress that
leads to a simple conceptual proof of these conjectures based
on general principles in *derived* algebraic geometry, and some
classical geometry with curve fibrations.

The work presented builds on ideas of Beilinson who proved one
of Fontaine's conjectures this way. The key new result that
allows us to take this story further is that derived de Rham
and crystalline cohomology are isomorphic for lci varieties in
characteristic *p* (as well as a relative version for *p*-adic
schemes).

The talk will be introductory in nature: no background in
*p*-adic Hodge theory or derived algebraic geometry will be
needed.