(joint with B.Bakker and Y.Brunebarbe) One very fruitful way of studying complex algebraic varieties is by
forgetting the underlying algebraic structure, and just thinking of them as complex analytic spaces. To this
end, it is a natural question to ask how much the complex analytic structure remembers. One
very prominent result in this direction is Chows theorem, stating that any closed analytic subspace of projective space is in fact algebraic. A notable consequence of this result is that a compact complex analytic space admits at most one algebraic structure - a result which is false in the non-compact case.
We explain how we can extend Chows theorem and its generalizations to the non-compact case by
working with complex analytic structures that are ’tame’ in the precise sense defined by o-minimality. This
leads to some very general ’algebraization’ theorems, which we apply to obtain new results in Hodge
Theory. In particular, we use this technology to prove a conjecture of Griffiths on the algebraicity and
quasi-projectivity of images of period maps.