Geometric positivity in the cohomology of homogeneous spaces and   generalized Schubert calculus
  We describe recent work on positive descriptions of the structure constants of the cohomology of homogeneous spaces such as the Grassmannian, by degenerations and related methods. We give various extensions of these rules, some new and conjectural, to K-theory, equivariant cohomology, equivariant K-theory, and quantum cohomology. 
Izzet Coskun and Ravi Vakil