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Spectral methods — those based on eigenvectors and singular vectors — have become increasingly popular for data analysis in various fields such as recommendation systems, biology, psychology, and social science. They are simple, computationally efficient, and often exhibit remarkably strong empirical performance. However, their theoretical properties remain relatively underexplored. In this talk, we present sharp statistical guarantees for the performance of spectral methods, enabled by new spectral perturbation tools. In the first part of the talk, we analyze spectral clustering under sub-Gaussian mixture models and show that it achieves exponentially small error rates. In the second part, we turn to the phase synchronization problem and prove that a spectral method achieves exact minimax optimality. The spectral perturbation tools developed in these works are of independent interest and may have broader applications to other low-rank matrix problems.