Operator algebraic approach to topological orders

Topological phases of matter are hot topics in mathematical physics. In quantum statistical mechanics, dynamics are given by linear operators called Hamiltonians with certain localities. Hamiltonians with a spectral gap between the lowest spectrum and the rest are said to be gapped. Such gapped Hamiltonians can be considered to be in a physically "normal" phase. We say two Hamiltonians are equivalent if they can be connected smoothly without closing the gap. Finding invariants of this classification is an exciting mathematical problem. In this talk, I would like to explain the operator algebraic approach to this problem.