Gysin homomorphism for resolutions of Grassmann and flag degeneracies

Cohomology classes represented by cycles of singularities of generic holomorphic maps are universal characteristic classes represented as polynomials (aka Thom polynomials) in the Chern classes of manifolds participating in the mapping. One of the possible ways to compute these polynomials is to use resolutions of singularity cycles and to apply the Gysin homomorphism associated with the resolution map.
We present in the talk the formalism of multilinear expansions providing a new formula for the Gysin homomorphism. This formula proves to be very efficient allowing one to extend considerably the list of singularities with explicitly known Thom polynomials.