Explore the persistent homology of function spaces in this 53-minute lecture by Shmuel Weinberger. Delve into the central topic of algebraic topology, focusing on specific examples where geometric considerations provide illuminating insights. Learn about nonquantitative theorems, Reine Thom, free loop spaces, and geometric group theory. Gain a deeper understanding of how these concepts relate to the homology of function spaces and their persistent homology. Engage with the discussion and examples presented to enhance your knowledge of this fundamental area in algebraic topology.