Singularities of mappings, equivariant cohomology, and pipe dreams

Given a generic smooth map f:X→ Y, one can describe the locus where the differential of f drops rank in terms of the homotopy class of f: there is a universal formula for the homology class of the locus in terms of characteristic classes of the vector bundles TX and f^{*}(TY).

I'll explain why one should expect a universal formula to exist, and reinterpret it as a class in equivariant cohomology associated to a certain irreducible variety. That turns this topology question, and related ones, into ones in algebraic geometry, and proves abstractly that the corresponding degeneracy locus formulae have positive coefficients. Then (in two special cases) I'll degenerate the variety into many pieces, and see the coefficients as counting the number of pieces.

This work is joint with Ezra Miller and Mark Shimozono. While I'll assume a little algebraic topology, all the equivariant cohomology, algebraic geometry, and pipe dreams necessary will be introduced in the talk.

Refreshments will be served at 2:45 p.m. in the Faculty Lounge, Math Annex (Room 1115).