For log smooth rigid analytic varieties, Kummer pro-étale B_dR cohomology is isomorphic to log de Rham cohomology, and logarithmic B_dR^+ cohomology gives degeneration of Hodge-Tate and Hodge-log de Rham spectral sequences when proper.
Perfectoid Spaces: A survey
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This paper, written in relation to the Current Developments in Mathematics 2012 Conference, discusses the recent papers on perfectoid spaces. Apart from giving an introduction to their content, it includes some open questions, as well as complements to the results of the previous papers.
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On the Kummer pro-\'etale cohomology of $\mathbb B_{\operatorname{dR}}$
For log smooth rigid analytic varieties, Kummer pro-étale B_dR cohomology is isomorphic to log de Rham cohomology, and logarithmic B_dR^+ cohomology gives degeneration of Hodge-Tate and Hodge-log de Rham spectral sequences when proper.