Schubert calculus is a part of enumerative "configuration space" algebraic geometry. For over a decade, it has been actively studied by combinatorialists and geometers. In particular, for combinatorialists, it is attractive due to its elementary combinatorial challenges. In this talk, I'll focus on one such problem, and its solution.
I'll formulate a nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of [Fomin-Kirillov '94] in the basis of stable Grothendieck polynomials for partitions. This gives a common generalization, as well as new proofs of the rule of [Fomin-Greene '98] for the expansion of the stable Schubert polynomials into Schur polynomials, and the K-theoretic Grassmannian Littlewood-Richardson rule of [Buch '02]. This solves some problems that arose from work of [Buch-Fulton '99].
At the core of the proof is a combinatorial algorithm that we introduce. This is a generalization of the [Robinson '38] and [Schensted '61] and [Edelman-Greene '87] tableau insertion algorithms. This is joint work with Anders Buch, Andrew Kresch, Mark Shimozono and Harry Tamvakis. See math.CO/0601514.
I hope to end (or have coffee) by briefly surveying the two main open combinatorial problems in Schubert calculus/varieties; some progress is represented in, e.g., joint work with Allen Knutson ( math.CO/0407051) and Alexander Woo ( math.AG/0603273).