Abstract: There exist a number of perhaps surprising connections between certain random surface models of
direct probabilistic interest and seemingly distant geometric questions concerning algebraic curves. On the
one hand, for specific boundary conditions, random surfaces turn out to be counting algebraic curves in some
particular algebraic 3-folds. On the other hand, the probabilistic analysis of such random surface models
requires a different kind of geometric input that may be seen, in some sense, as the "mirror" of the original
enumerative problem. In my lectures, I will try to explain how this works in both directions, introducing the
necessary background material along the way.