Extending work of Bombieri and Pila on counting lattice points on convex curves [4], Pila and Wilkie proved a strong counting theorem on the number of rational points in a more general class of sets definable in an o-minimal structure on the real numbers [37]. Following a strategy proposed by Zannier, the Pila-Wilkie upper bound has been leveraged against Galois-theoretic lower bounds in works by Daw, Habegger, Masser, Peterzil, Pila, Starchenko, Yafaev and Zannier [6, 18, 25, 31, 36, 38] to prove theorems in diophantine geometry to the effect that for certain algebraic varieties the algebraic relations which may hold on its "special points" are exactly those coming from "special varieties". Of these results, Pila's unconditional proof of the André-Oort conjecture for the j-line is arguably the most spectacular and will be the principal object of this resumé. Readers interested in a survey with more details about some of the other results along these lines, specifically the Pila-Zannier reproof of the Manin-Mumford conjecture and the Masser-Zannier theorem about simultaneous torsion in families of elliptic curves, may wish to consult my notes for the Current Events Bulletin lecture [42]. Acknowledgements. I wish to thank M. Aschenbrenner, J. Pila and U. Zannier for their advice and especially for suggesting improvements to this text.
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