Explore the fourth lecture in a series on Stein's restriction conjecture and the polynomial method in restriction theory. Delve into the intricacies of estimating functions with Fourier transforms supported on hypersurfaces, such as spheres in Rn. Examine the decomposition of these functions into sums over wave packets supported on long, thin tubes. Discover how Guth's introduction of the polynomial method revolutionized the study of tube intersections in restriction theory. Gain a comprehensive understanding of Stein's restriction conjecture and its connection to the Kakeya conjecture. Investigate the application of the polynomial method in addressing these fundamental problems in harmonic analysis and geometric measure theory.