Explore the challenging question of P vs. NP in this undergraduate computational complexity theory lecture from Carnegie Mellon University. Delve into negative results, Ladner's theorem, simulations, and algorithm design as Professor Ryan O'Donnell guides you through the intricacies of this fundamental problem in computer science. Gain insights into the proof techniques and ideas behind this difficult question, and understand why it remains one of the most significant unsolved problems in mathematics and computer science.