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The purpose of these lectures is to explain Beilinson's proof of the p-adic - de Rham comparison theorems using derived de Rham complexes.
(1) Review of the Betti - de Rham comparison theorem, and outline of the method
(2-3-4) Preliminaries on cotangent and derived de Rham complexes, a new look at BdR
(5-6) Statement of the p-adic Poincaré lemma and construction of the comparison map
(7-8) Proof of the p-adic Poincaré lemma. Sketch of proof of the comparison theorem.
If there is time left, sketch of crystalline variants.
References:
A. Beilinson, p-adic periods and derived de Rham cohomology, J. of the AMS, 25 (3), 715-738 (2012)
A. Beilinson, On the crystalline period map,http://arxiv.org/pdf/1111.3316.pdf | 
