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The Hasse-Minkowski theorem says that a quadratic form over a global field admits a nontrivial zero if it admits a nontrivial zero everywhere locally. Over more general fields of arithmetic and geometric interest, the failure of the local-global principle is often controlled by auxiliary structures of interest: 2-torsion points of the Jacobian and elements of Tate-Shafarevich groups for quadratic forms over function fields of curves, and the Brauer group over function fields of surfaces. I will explain recent work with V. Suresh on constructing failures of the local-global principle for quadratic forms over function fields of higher dimension varieties. These counterexamples are controlled by higher unramified cohomology groups and involves the study of certain Calabi-Yau varieties of generalized Kummer type that originally arose from number theory.