Explore a comprehensive lecture from the Simons Semester on Dynamics where Professor Antti Käenmäki from the University of Oulu delves into recent advancements in the dimension drop conjecture. Learn why the conjecture can be easily proven for self-similar sets with positive Hausdorff measure, and understand Hochman's groundbreaking result demonstrating how exponential separation prevents dimension drop. Examine the verification of the conjecture for self-similar sets defined by algebraic parameters, while discovering the current limitations in addressing the complete conjecture. The nearly two-hour presentation provides a thorough mathematical exploration of this complex topic in dynamical systems theory.