Schubert calculus and puzzles

Abstract

These are notes for four lectures given at the Osaka summer school on Schubert calculus in 2012, presenting the geometry from the unpublished arXiv:1008.4302 giving an extension of the puzzle rule for Schubert calculus to equivariant $K$-theory, while eliding some of the combinatorial detail. In particular, §3 includes background material on equivariant cohomology and $K$-theory.

Since that school, I have extended the results to arbitrary interval positroid varieties (not just those arising in Vakil's geometric Littlewood-Richardson rule), in the preprint [Kn2].