
Lecture 24: Sharp Projection Theorems, Part 3 - Combining Different Scales
Lawrence Guth continues MIT's 18.156 Projection Theory course with the third installment on sharp projection theorems. The lecture centers on how Ren and Wang proved the full Furstenberg conjecture by applying the Ahlfors-David regular case at many different scales, a technique that lets local estimates be assembled into a global sharp result. Guth works through the mechanics of this multi-scale combination, showing how controlling behavior at one scale constrains what can happen at others. Part of a graduate-level sequence on projection theorems in geometric measure theory, the lecture assumes familiarity with earlier sessions covering the AD regular case and the setup of the Furstenberg set problem, and builds directly on that foundation toward the full conjecture's resolution. Runtime runs 77 minutes, consistent with a full seminar-style session rather than a survey talk.