Combinatorial positivity in the Schubert calculus via dual equivalence
graphs

   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.

   I will attempt to explain the problem in the Schubert
calculus, and its reduction to showing Schur-positivity of
a quasi-symmetric generating function, then discuss the
context of Assaf's new method of dual equivalence from Macdonald
polynomials, and finally explain how her method applies to
resolve this problem in the combinatorics of the Schubert
calculus.  This is joint with with Nantel Bergeron and
Sami Assaf.