let $v$ be a vector bundle over a smooth projective curve of rank $n$. we are interested in understanding the set of sub-bundles of rank $r$ with maximal degree. when we impose some numerical conditions, this set happens to be finite. in this talk, we will go over chronological developments in finding the number of maximal sub-bundles. surprisingly, these numbers are related to the gromov-witten invariants of grassmannian which are computed by the vafa-intriligator formula.