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Quantum phases of matter exhibit a richness of structure beyond classical phases: while Landau’s theory dictates that classical phases are governed purely by broken and unbroken symmetries of a model Hamiltonian, quantum phases are distinguished by symmetry and by ``patterns of entanglement’’. Proper formalization of these physical ideas from the condensed matter community and the resultant classification for quantum phases provides a fascinating wide-open mathematical program of research. In this talk, we discuss the classification of 1D Symmetry Protected Topological (SPT) phases of matter, which include topological insulators as an application. The physical conjecture due to Chen-Gu-Wen ’11 has been partially proven by Ogata ’19 for the case of Hamiltonians with translation-invariance and unique gapped ground state. This talk is largely expository and little knowledge of quantum systems is assumed.

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