In this article, which is dedicated to my friend and colleague Boris Zilber on the occasion
of his 75th birthday, I put forward a strategy for proving his quasiminimality
conjecture for the complex exponential field. That is, for showing that every subset of
definable in the expansion of the complex field by the complex exponential function is
either countable or cocountable. In fact the strategy applies to any expansion of the
complex field by a countable set of entire functions (in any number of variables) and
is based on a certain property — an analytic continuation property — of the
o-minimal structure obtained by expanding the ordered field of real numbers by the
restrictions to compact boxes of the real and imaginary parts of the functions in the
given set.
In a final section I discuss briefly the (rather limited) extent of our unconditional
knowledge in the area.
