Classification of subfactors in theory of operator algebras and its applications

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I have obtained a classification result in operator algebraic approach to conformal field theory. A one-dimensional conformal field theory in the operator algebraic formulation is called a conformal net, and I have shown, with Longo, that if the symmetry group is a diffeomorphism group, then the central charge of a conformal net is defined and if furthermore it is less than 1, then it is completely classified with an invariant labeled with pairs of A-D-E Dynkin diagrams. This is a complete solution to a well known problem. I fully use theory of alpha-induction and modular invariants I have studied before. Next, I have studied classification theory of two-dimensional conformal nets and obtained a complete classification result with Longo for the case with central charge less than 1. In the one-dimensional classification, the diagrams D 2n+1 and E_7 did not appear, but now they. do appear. Strictly speaking, in the two-dimensional classification results, I have assumptions that a net has parity symmetry and is maximal, and these two are equivalent to the condition of the mu-index being 1, but these two conditions can be easily dropped, and we would have merely more combinatorial complecity. We use the above-mentioned one-dimensional classification for this, and the new point is theory of 2苗ohomology group of a tensor category. A key step is a proof of 2-chomology vanishing for tensor categories related to the Virasoro algebra