Explore a 57-minute colloquium talk from NIST researcher Ian B. Spielman at the Centre for Quantum Technologies examining how topological invariants classify gapped quantum systems in equilibrium and their unexpected behavior in dynamical systems. Delve into the relationship between dimensionality, symmetries, and topological invariants, discovering why the conventional wisdom about topology's robustness breaks down in far-from-equilibrium evolution. Learn about experimental studies using ultracold atomic BEC in a 1D bipartite lattice, with specific focus on dynamically induced symmetry breaking and Floquet engineering of topological Dirac bands. Examine key findings published in Physical Review Letters, including the surprising ways symmetries and topology can change during quantum system evolution.