MATH 7390 - Topics in Algebra: Schubert varieties and degeneration

Allen Knutson, fall 2014.

This is an introductory course, which should be accessible to anyone who has taken the graduate algebra sequence; all the algebraic geometry within will be developed from scratch. Exposure to Lie groups will be helpful but not essential (mostly we will concentrate on the general linear group).

Topics planned:

• Projective varieties and their Hilbert polynomials
• Degenerations of algebraic varieties
• Flag manifolds, Schubert varieties, and Schubert polynomials
• Frobenius splitting, and application to reducedness of schemes
• Bott-Samelson manifolds
• Stanley-Reisner theory
• Equivariant cohomology and equivariant K-theory
• Positroid varieties in the Grassmannian
• Schubert calculus on Grassmannians
• Representations of quivers