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The paper by Agnihotri and Woodward, Eigenvalues of products of unitary matrices and quantum Schubert calculus, uses a Narasimhan-Seshadri correspondence between parabolic bundles and unitary connections to determine the possible spectrum of a product of two (special) unitary matrices of known spectrum. They start with a triple of unitary matrices with product $1$, N-S relate that to bundles on $\mathbb P^1$ with parabolic structure at three points, classify those bundles as maps of the $\mathbb P^1$ into a Grassmannian, and end up at quantum Schubert calculus of Grassmannians. Maybe not the most obviously natural source of parabolic bundles, but a wonderful application.