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 82-minute presentation serves as part of a comprehensive series designed for advanced mathematics students and researchers interested in deepening their understanding of geometric structures and topological properties.