| Résume | This is the title of a paper that has recently been accepted for the volume of the journal “Model Theory” dedicated to Boris Zilber on the occasion of his 75th birthday. (The paper can be found on the GTM preprint server or on arXiv.) The conjecture asserts that every definable subset of the complex field expanded by the complex exponential function is either countable or cocountable. In the paper I propose a conjecture concerning the analytic continuation of o-minimally defined complex analytic functions which implies Zilber's conjecture (and much more) and in this talk I will give an outline of the main argument in the paper as well as some further remarks. (I was going to write “as well as some recent progress”, but that would be too strong!)
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