I will give an overview of how oscillatory integrals arise in analytic number theory, especially the theory of L-functions and automorphic forms. Usually the integrals we are faced with are multiple-yet-low dimensional, so that they are approachable by repeated one-dimensional stationary phase estimates. I will present some lemmas that allow one to pass information on the uniformity with respect to the other variables from one dimension of the stationary phase to the next, which has been useful to us in estimating oscillatory integrals one dimension at a time. (Two further lectures by this speaker will take place on July 11 and July 12 in the Lipschitz-Saal, Endenicher Allee 60, at the Hausdorff School "Uniformity and Stability of Oscillatory Integrals."

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