About the dual Schanuel conjecture

Schanuel's conjecture predicts a lower bound for the transcendence degree of the values of the complex exponential function. A lesser known "dual" conjecture, stated in a weak form by Schanuel, and fully formulated by Zilber, predicts that the graph of the exponential function must intersect generically all "free rotund" algebraic varieties. This would have strong consequences (i.e., quasi-minimality) for the model theory of complex exponentiation. I will discuss the recent positive results on this problem, which seems to be tractable for curves and surfaces (including joint work in Zannier and work in progress with Masser -- and provided one removes or replaces the word "generically").