Nonlinear Bourgain's Projection Theorem

This lecture by the instructor covers a nonlinear version of Bourgain's Projection Theorem and its applications. The theorem addresses the Erdős-Volkmann conjecture and Bourgain's sum-product theorem, providing insights into projections, distance sets, and direction sets. The lecture delves into the motivation behind seeking nonlinear versions of the theorem and discusses various applications, including one-dimensional smooth families and pinned distance sets. The discretized versions of the theorem are explored, highlighting the importance of non-concentration assumptions and transversality. The proof methodology involves entropy bounds, multi-scale decompositions, and Moran measures. The lecture concludes with a discussion on the intricacies of the proof and the significance of the results.