Explore advanced concepts in Riemannian geometry through this graduate-level lecture delivered by Professor Spiro Karigiannis from the University of Waterloo at the Fields Institute. Delve into sophisticated topics in geometry and topology, building upon foundational knowledge of Riemannian geometry to examine more complex theoretical frameworks and applications. The 83-minute presentation serves as part of a comprehensive series, offering deep insights into advanced mathematical concepts at the intersection of differential geometry and topology.