Gysin maps, duality and Schubert classes
  We establish a Gysin formula for Schubert bundles and a strong version of the duality theorem in Schubert calculus on Grassmann bundles. We then combine them to compute the fundamental classes of Schubert bundles in Grassmann bundles, which yields a new proof of the Giambelli formula for vector bundles. 
Lionel Darondeau and Piotr Pragacz
math.AG math.AT math.CO