The deformation of algebraic cycles in characteristic p 
-- Matthew Morrow, April 17, 2015
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The problem of deforming vector bundles or algebraic cycles has
classically been relatively well understood (at least
conjecturally) in characteristic zero, and to a lesser extent
in mixed characteristic, whereas the situation in equal
characteristic p has been less satisfactory. I will explain a
new theory of infinitesimal crystalline cycle classes in
characteristic p, which allows general results of the following
form to the proved: given an algebraic cycle on a subvariety Y
of a variety X in characteristic p, the cycle extends to the
formal completion of X along Y if and only if its crystalline
cycle class extends. This leads to generalisations of results
of de Jong, infinitesimal Lefschetz theorems, and a
characteristic p analogue of the Deformational Hodge
Conjectures introduced by Bloch-Esnault-Kerz.