Algebraic geometry poses many positivity challenges to enumerative combinatorics. Two notable such challenges are Macdonald's positivity conjecture, and structure constants in the Schubert Calculus. This talk will explain how Assaf's solution to the first, through her new method for showing symmetry and Schur-positivity of quasi-symmetric generating functions, may be applied to resolve a (by now old) problem of the positivity of some of the structure constants in the Schubert calculus of the flag manifold.

This is joint with with Nantel Bergeron and Sami Assaf.