We discuss recent results on inverse spectral problems with sparse data on closed Riemannian manifolds. We then present two applications of these results. First, we show generic uniqueness results for inverse problems for evolutionary PDEs with a single passive measurement. Second, we present uniqueness results for an analogue of the anisotropic Calderón problem for fractional Schrödinger operators on closed manifolds. This talk is based on joint works with Ali Feizmohammadi and Gunther Uhlmann.