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In this talk, I will discuss the spectral properties of a two-layer neural network after one gradient step and their applications in random ridge regression. We consider the first gradient step in a two-layer randomly initialized neural network with the empirical MSE loss. In the proportional asymptotic limit, where all the dimensions go to infinity at the same rate, and an idealized student-teacher setting, we will show that the first gradient update contains a rank-1 ''spike'', which results in an alignment between the first-layer weights and the linear component of the teacher model. By verifying a Gaussian equivalent property, we can compute the prediction risk of ridge regression on the conjugate kernel after one gradient step. We will present two scalings of the first step learning rate. For a small learning rate, we compute the asymptotic risks for the ridge regression estimator on top of trained features which does not outperform the best linear model. Whereas for a sufficiently large learning rate, we prove that the ridge estimator on the trained features can go beyond this ``linear'' regime. Our analysis demonstrates that even one gradient step can lead to an advantage over the initial features. Our theoretical results are mainly based on random matrix theory and operator-valued free probability theory, which will be summarized in this talk. This is recent joint work with Jimmy Ba, Murat A. Erdogdu, Taiji Suzuki, Denny Wu, and Greg Yang.