I will give a definition of a certain category of "log quasicoherent" sheaves on a logarithmic variety which uses Falting's "almost mathematics" and which has the property that log differential forms and log polyvector fields are the Hochshild homology (appropriately understood) and Hochschild cohomology, respectively, of this category. This implies a certain "noncommutative Hodge theory" associated to a log variety in mixed characteristic. I will also explain (if there is time left over) a relationship of the proof of the main results to mirror symmetry.
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Ahmed Abbes