The discovery of quantized Hall conductance in the early 1980s showed how quantum phases of matter can be distinguished by topological invariants. This kicked off a decades-long quest to understand subtle distinguishing topological aspects of quantum matter. In this talk, I will describe a new set of topological invariants that arise in quantum systems with crystalline symmetry. Analogous to the quantized Hall conductance, these new invariants dictate quantized responses of the system to crystal defects. Our understanding of how to extract these invariants from quantum systems has provided the first new ways to color Hofstadter's famous fractal butterfly in over 40 years.
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