ABSTRACT
T he “ten-fold way” underlying the classification of non-interacting topological insulators and superconductors maps the infras-
tructure of the “civilized” part of the topological world. With interactions, we step into the wild. The main challenges that stem from the presence of interactions can be organized into three categories. The first set of problems is generated by the lack of a general classification of interacting topological phases (i.e., the interacting analog of the “ten-fold way”), which could be used as the basis for a systematic understanding of these quantum states. Moreover, it is not even clear what type of mathematical structure should underlie such a classification. Nonetheless, significant progress has been made in the past few years involving partial classifications of certain types of interacting topological states based on tensor categories [429], group cohomology theory [82, 200], cobordism [246], and other “scary mathematical beasts.”
