Explore the Hopcroft--Paul--Valiant Theorem in this graduate-level computational complexity theory lecture from Carnegie Mellon University's Course 15-855. Delve into the intricacies of this fundamental theorem, its implications, and related concepts such as simulation, k-tapes, graph representations, low space simulation, and the pebble game. Learn about key tools like the Depth Reduction Lemma and public strategy. Gain insights from Professor Ryan O'Donnell's expert instruction, complemented by suggested reading of the original paper. This comprehensive 1-hour 20-minute video lecture, part of the Fall 2017 series, offers an in-depth exploration of advanced complexity theory concepts for graduate students and researchers in computer science.