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.