Schubert calculus and Gelfand-Zetlin polytopes

Our goal is to give an interpretation of the Schubert calculus for a
full flag variety in terms of the geometry of polytopes. For this we
use the notion of Pukhlikov-Khovanskii ring. This ring can be
constructed for any convex polytope; it was initially introduced to
describe the cohomology ring of a smooth toric variety. It turns out
that it also can be used for some non-toric varieties. In particular,
the cohomology ring of a full flag variety can be identified with the
Pukhlikov-Khovanskii ring of a Gelfand-Zetlin polytope. This
identification provides a new approach to Schubert calculus. I will
discuss some new results in this direction, recenlty obtained in a
joint work with Valentina Kirichenko and Vladlen Timorin.