Schanuel's conjecture predicts a lower bound for the transcendence
degree of the values of the complex exponential function. A lesser
known "dual" conjecture, formulated independently by Schanuel and by
Zilber, is the following: 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 U.
Zannier and work in progress with D. Masser -- and provided one removes
or replaces the word "generically").