Algebraic Geometry Seminar
David Anderson
Ohio State
Schubert polynomials, strong transversality, and positivity
Abstract: Given a matrix of homogeneous polynomials, one often wants to specify rank bounds on various submatrices – the result is called a degeneracy locus. An old problem, considered by many 19th century mathematicians, asks for a formula for the degree of such a variety. It turns out that the universal such formulas – the Schubert polynomials – have an incredibly rich algebra and combinatorics in their own right.
I’ll describe recent work on Schubert polynomials, including developments by Lam, Lee, and Shimozono, as well as joint work with William Fulton. These polynomials possess a striking and subtle positivity property. As I’ll explain, this positivity is an artifact of a new Kleiman-Bertini-type transversality theorem, applied to subvarieties of flag varieties.
Wednesday February 26, 2025 at 3:00 PM in 1227 SEO