Schubert calculus and quiver varieties

Since the 1970s we have known that the structure constants for intersection theory on compact complex homogeneous spaces (such as Grassmannians) are nonnegative, but our only formulae for these constants (outside special cases) are essentially as alternating sums. The most effective tool to date for giving manifestly positive formulae are the "puzzles" that Terry Tao and I introduced, but the connection to quantum integrable systems observed by Paul Zinn-Justin made it clear that the puzzles should be solving a richer problem. This turns out to involve Nakajima quiver varieties, and has shed light even on the original problem of intersecting three cells in a Grassmannian. This work is joint with Paul Zinn-Justin.