Most properties of topological phases of matter TPMs in 2D are captured by the algebraic models: unitary modular tensor categories (UMTCs). A classification of UMTCs would provide something like a periodic table for TPMs, and this problem has been pursued from many different perspectives. In this lecture we will review what is known, some open problems and a few of the successful techniques from category theory, algebraic geometry, number theory and representation theory.
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