Hong Wang of New York University gives the third lecture in this Emmy Noether Lecture Series session at the Institute for Advanced Study, continuing a study of projection theory: how the Hausdorff dimension of a set in R^n behaves under orthogonal and nonlinear projections onto lower-dimensional spaces. She builds on discretized and multi-scale techniques originated by Bourgain and extended by Shmerkin and Orponen, showing how to locate structure inside an arbitrary set and use it to control the exceptional set of directions where dimension fails to be preserved. The talk assumes familiarity with the earlier parts of the series and moves through the technical machinery at research level, aimed at an audience already working in geometric measure theory and harmonic analysis.