Zilber?s quasiminimality conjecture predicts that all subsets of the complex numbers that are definable using the language of rings and the exponential function are either countable or cocountable. Building on Zilber?s work, Bays and Kirby have proved that the quasiminimality conjecture would follow from the exponential-algebraic closedness conjecture, also due to Zilber, which predicts sufficient conditions for systems of equations in polynomials and exponentials to have complex solutions. In this talk, I will give an introduction to this topic before presenting some recent work which solves the conjecture for a class of algebraic varieties which corresponds to systems of exponential sums. This turns out to be closely related to tropical geometry, a "combinatorial shadow" of algebraic geometry which reduces some questions about algebraic varieties to questions about polyhedral objects.