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Richard Rimanyi : Thom polynomials (Nov 7, 2007 4:25 PM)

In certain situations global topology may force singularities. For example, the topology of the Klein bottle forces self-intersections when mapped into 3-space. Any map of the projective plane must have at least cusp singularities when mapped into the plane. The topology of a manifold may force any differential form on it to degenerate at certian points. In a family of vector bundles over a complex curve some must degenerate to a non-stable bundle (in the GIT sense), depending on the topology of the family. In a family of vector bundle maps---arranged according to a directed graph (quiver)---some may be forced to degenerate. In families of linear spaces some have special incidence with some other fixed ones (Schubert calculus). These degenerations are governed by a unified notion in equivariant cohomology, the Thom polynomial of "singularities". In the lecture I will review Thom polynomials, computational strategies (interpolation, localization, Grobner basis), show examples and applications.

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