A very old problem asks for the degree of a variety defined by rank conditions on matrices. A version of this problem was studied by Giambelli, and his solution is one of the cornerstones of Schubert calculus. The story of the modern approach begins in the 1970’s, when Kempf and Laksov proved that the cohomology class of a degeneracy locus for a map of vector bundles is given by a Giambelli-type determinant in their Chern classes.

Since then, many variations have been studied — for example, when the vector bundles are equipped with a symplectic or quadratic form, the formulas become Pfaffians. In this series of lectures, I will describe some recent extensions of these results — beyond determinants and Pfaffians, and beyond ordinary cohomology — including my joint work with W. Fulton, as well as work of several others.

These formulas are closely related to Schubert polynomials, which were introduced combinatorially by Lascoux and Schutzenberger in 1982, and I will discuss some of the combinatorics involved. There are many delicate combinatorial and algebraic questions connected with this subject, some of which remain unanswered.