K3 Modular Parametrization and Calabi-Yau Threefold Moduli

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A series of generalizations of the Weierstrass normal
form for elliptic curves to the case of K3 surfaces will be presented.  These have already been applied to better
understand F-theory/Heterotic string duality. 
We will see how they also resolve a long-standing question of which
"mirror-compatible" variations of Hodge structure over the
thrice-punctured sphere can arise from families of Calabi-Yau threefolds.