The map that sends a rational curve in a Grassmann variety to its kernel-span pair can be used to express any (3 point, genus zero) Gromov-Witten invariant of a Grassmannian as a classical triple intersection number of Schubert classes on a two-step flag variety. A conjecture of Allen Knutson asserts that any such triple intersection number is equal to the number of triangular puzzles that can be constructed using a list of puzzle pieces, such that the border of the puzzles is determined by the intersection problem. I will discuss a recent proof of this conjecture and how it leads to a positive combinatorial formula for Grassmannian Gromov-Witten invariants. I will also discuss generalizations to equivariant Gromov-Witten invariants. This talk is based on papers with A. Kresch, L. Mihalcea, K. Purbhoo, and H. Tamvakis.
Yale 2013 >