“The set of k-planes in n-space satisfying the following list of closed conditions” defines a cycle in the Grassmannian (of all k-planes), about which one can ask (to begin with) two kinds of questions: (1) what does this cycle look like as a variety, e.g. how singular is it? and (2) what is its homology class inside the Grassmannian.

It turns out that every cycle is (uniquely) a positive combination of

“Schubert classes”, the classes of “Schubert varieties”. In the first

lecture I’ll define these, and explain enough equivariant cohomology (from the ground up) to compute the Schubert classes and explain their connection to semistandard Young tableaux. (It will turn out, for inductive purposes, to be useful to go beyond Grassmannians to flag manifolds.) I’ll also give a big picture of the problems of Schubert “calculus”, which concerns the structure constants for the cohomology product, and admits many generalizations that are still open.

In the second lecture, we’ll look at the singularities of Schubert varieties, using Bott-Samelson manifolds to help define a Gröbner basis for local patches on Schubert varieties; the resulting combinatorics is that of ”subword complexes”. Then I’ll talk about recent work of [Maulik–Okounkov], [C. Su], [Aluffi–Mihalcea], [Huh], and [me–Zinn–Justin] about D-modules/Chern–Schwarz–MacPherson classes of Schubert varieties.