We report on recent progress in the field of Schubert calculus and its recently uncovered relation to quantum integrable systems. We shall see how the latter provide many explicit combinatorial formulae (``puzzle rules'') for intersection numbers for partial flag varieties, and their extensions to e.g. equivariant K-theory, all of which generalize the classical Littlewood--Richardson numbers. We shall also discuss the connection with the work of Okounkov et al on quantum integrable systems and the equivariant cohomology of Nakajima quiver varieties. This is joint work with A. Knutson.
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