Anomalous Subvarieties—Structure Theorems and Applications

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The theory of the exponential differential equations of semiabelian varieties (2007, unknown): “) For a discussion of results and conjectures of this form, see [Zil02] and [BMZ07].”
Applications of the hyperbolic Ax–Schanuel conjecture (2017, unknown): “Habegger and Pila also extended the Pila-Zannier strategy to the Zilber-Pink conjecture for products of modular curves.”
Ax–Schanuel for the $j$-function (2014, unknown): “On the Zilber-Pink conjecture generally see [6, 24, 25, 29, 30].”
O-minimality and the André-Oort conjecture for $\mathbb{C}^{n}$ (2011, result): “…by Zannier and implemented in three recent papers: a new proof of the Manin-Mumford conjecture by Pila-Zannier; a proof of a special (but new) case of Pink’s relative Manin-Mumford conjecture by Masser-Zannier; and new proofs of certain known results of André-OortManin-Mumford type by Pila.”
On the Bounded Height Conjecture (2008, background): “4(b) [4]) that Xoa is Zariski open in X, so by virtue of Corollary 1.”