The lecture will be about complete embedded minimal surfaces in R^3 (some immersed ones help the explanations). Before 1978
very little was known and after 1984 progress became rapid. I will
illustrate with pictures, how observed features of known surfaces
led to new, increasingly abstract, constructions. I will not assume
that the audience consists of minimal surface experts, the lecture
is intended to be understandable by graduate students.

I will describe a "part" of the standard GW-invariant, which under
ideal conditions counts genus-one curves without any genus-zero contribution.

In contrast to the standard GW-invariant, the resulting reduced GW-invariant has the expected behavior with respect to certain embeddings. These invariants have applications to computing the standard genus-one GW-invariants of complete intersections as well as some enumerative genus-one invariants of sufficiently positive complete intersections. The former application opens a way to try to verify the mirror symmetry prediction for genus-one curves in Calabi-Yau therefolds.