I will describe the combinatorics of Schubert curves: one-dimensional intersections of Grassmannian Schubert varieties defined with respect to tangent flags of the rational normal curve. The real geometry of a Schubert curve is given by a map $\omega$ on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve naturally covers $\mathbb{RP}^1$, with $\omega$ as the monodromy operator.

I will give a local, simpler algorithm to compute $\omega$ without having to rectify the tableau. Certain steps in the algorithm are in bijection with Pechenik and Yong's 'genomic tableaux', which enumerate the K-theoretic Littlewood-Richardson coefficient of the Schubert curve. This bijection leads to purely combinatorial proofs of some numerical and geometric results relating the K-theory and real geometry of the curve. This is joint work with Maria Gillespie.

Time permitting, I'll say how the story seems to shape up for orthogonal Grassmannians (joint with Maria Gillespie and Kevin Purbhoo).