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Description

Thom studied cohomology classes given by the locus of points where a given map between manifolds has rank at most r. Rephrasing and generalizing, Buch and Fulton considered collections of vector bundles and maps between them, organized by a quiver (directed graph). Instead of a single integer r, the kinds of degeneracies are described by (equivalence classes of) representations of the quiver. There exist universal formulae of a combinatorial flavor for the cohomology classes of degeneracy loci, given by evaluating a universal polynomial (that depends only on the quiver) at the Chern roots of the bundles, the classical example being the Giambelli-Thom-Porteous formula. We indicate a method for computing these universal polynomials that involves the Grobner deformation of the orbit closure of a quiver representation. Our method leads naturally to beautiful combinatorial formulae, which give geometric explanations for the formulae and which apply to the computation of classical intersection numbers and genus zero 3-point Gromov-Witten invariants for the flag variety. This is joint work with Allen Knutson and Ezra Miller.