Explore the interplay between topology, symmetry, and interactions in quantum materials through this 45-minute conference talk by Qiong Ma from Boston College. Delve into recent advancements in probing symmetry and quantum geometry using nonlinear techniques. Examine the effects of correlations on topological insulators, semimetals, and superconductors, and investigate how symmetry and strong correlations produce correlated topological phases in Kondo systems. Discover the connections between higher-order topology and fracton systems, and learn about the potential of topology as a nonperturbative framework for understanding strongly correlated gapless phases. Gain insights into the unifying themes of electronic topology across various materials and correlation strengths, as part of the Kavli Institute for Theoretical Physics' conference on quantum crystals and correlated topological phases.