Pablo Shmerkin continues MIT's graduate course 18.156, Projection Theory, with the second part of his treatment of the Bourgain projection theorem. The lecture introduces uniform sets and branching functions, the key technical devices used to control how sets project onto lines and the combinatorial structure that makes Bourgain's discretized sum-product and projection estimates work. Building on the prior session's setup, Shmerkin develops the machinery step by step at the chalkboard, aimed at students already fluent in fractal geometry, measure theory, and additive combinatorics. Running 77 minutes, it is one lecture in a full MIT OpenCourseWare sequence on projection theory, a subject connecting geometric measure theory to Fourier analysis and combinatorics. The pace and notation assume real familiarity with the course's earlier material, making this a lecture for students tracking the full sequence rather than a standalone introduction to the topic.