Mathematician Dominique Maldague, affiliated with Cambridge University and UCLA, gives this Institute for Advanced Study lecture on Fourier restriction theory and its surprising reach into other fields. She starts from the classical use of Fourier series to solve PDEs like the wave and Schrodinger equations, then shows how imposing frequency conditions turns these series into geometric objects worth studying in their own right. From there the talk connects restriction theory to the distribution of prime numbers, the size of arithmetic progressions inside sets, and the Kakeya problem in fractal geometry, tracing a single analytic technique across number theory, combinatorics, and geometric measure theory. Delivered as part of the IAS Women and Mathematics program, the lecture is aimed at an audience with solid background in analysis and is a clear example of how one mathematical framework can unify problems that look unrelated on the surface.