Stable homotopy theory of invertible quantum spin systems

In the past decade, A. Kitaev proposed that the set of invertible gapped quantum spin systems would form an \Omega-spectrum. This conjecture is considered to have potentially significant application to the study of SPT phases. Recently, we give a mathematically rigorous realization of this proposal with the language of functional analysis and operator algebra. This gives a unified proof of a series of existing researches. The proof also suggests to understand Kitaev's proposal from the viewpoint of coarse geometry of metric spaces. This association leads us to the concept of localization flow.