The great references given on *Ilya's question* make me wonder about the current status of the many conjectures and open questions in *Illusie's survey* from 1994 on *crystalline cohomology*. Obviously (just compare Illusie's survey from 1975 with that above or with *Chambert-Loir's survey* from 1998), there is very intense work on that and the connections between the various cohomology theories attacking the case "l=p". Some more recent surveys only on Fontaine's p-adic Hodge theory are already linked to in the answers to Ilya's question, *Le Stum's book* (*Errata*) covers rigid chohomology. Among the open issues mentioned in Illusie's survey are finiteness theorems, crystalline coefficients, geometric semistability, the identity of characteristic polynomials of the Frobenius of different theories,... What is the current status of these? Which new theories have been created the past decade, how fit they together and which new questions emerged?

Edit: U. Jannsen talked *recently* on "a refinement of crystalline cohomology by using the theory of so-called gauges as introduced earlier by Mazur and Kato and certain syntomic sheaves." Unfortunately I found no preprint on that. **Edit:** Jannsen on (*slides*) "a cohomology theory in characteristic p which reﬁnes the crystalline cohomology – and works well for torsion" and "a sheaf theory which generalizes the Dieudonné theory – and works well for torsion."

Edit: Go Yamashita talked about "La Theorie de Hodge p-adique pour varietes ouverts" avoiding Falting's almost etale extensions. Unfortunately I found no text where one can read that.

Edit: A short note by Bhargav Bhatt and Aise Johan de Jong on a shortened proof of the comparison theorem between crystalline and de Rham cohomology.

Edit: A new proof of the semistability conjecture by Beilinson and a definition of derived crystals by Gaitsgory and Rozenblyum.

Edit: A p-adic derived de Rham cohomology by Bhargav Bhatt, giving "derived de Rham descriptions of the usual period rings and related maps in p-adic Hodge theory" and "a new proof of Fontaine's crystalline conjecture and Fontaine-Jannsen's semistable conjecture".

Edit: A "a new cohomology theory in characteristic p>0, the so called F-gauge cohomology, a cohomology with values in the category of so-called F-gauges, which refines the cristalline cohomology" by Fontaine, Jannsen.

Edit: An other very interesting and very nice to read history of mathematics talk by Illusie "Grothendieck at Pisa: crystals and Barsotti-Tate groups":

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