Schubert calculus (intersection theory on Grassmannians) was used, and Knutson-Tao puzzles were introduced, to characterize around 1997 the spectrum of a sum of two Hermitian matrices. The two approaches turned out to be directly related. I'll introduce a combinatorial recasting of the puzzles using pipes which makes clear that a natural puzzle piece was missed: it relates to the cohomology of \emph{cotangent bundles} to Grassmannians, which are examples of quiver varieties. The quiver variety geometry then informs the combinatorics.

This work is joint with Paul Zinn-Justin and Iva Halacheva.