Dominique Maldague, a mathematician affiliated with Cambridge University and UCLA, gives the third installment of a lecture series delivered at the Institute for Advanced Study's Women and Mathematics program. She starts from the classical use of Fourier series to build solutions to equations like the wave and Schrodinger equations, then shows how imposing restrictions on the frequencies involved turns these series into geometric objects that can be studied directly. From there she connects Fourier restriction theory to three seemingly unrelated areas: the distribution of prime numbers, the size of arithmetic progressions inside sets, and the Kakeya problem, which asks how small a set can be while still containing a line segment pointing in every direction. The talk runs 79 minutes and assumes a working knowledge of harmonic analysis, tracing the shared machinery that lets one perspective illuminate number theory, combinatorics, and fractal geometry at once.