Celebratio Mathematica

V. F. R. Jones: Actions of finite groups on the hyperfinite type \( \mathrm{II}_1 \) factor. Memoirs of the American Mathematical Society 237. American Matheammatical Society (Providence, RI), 1980. Republication of Jones’ 1979 PhD thesis. MR 587749 Zbl 0454.46045 book

V. F. R. Jones: “Index for subfactors,” Invent. Math. 72 : 1 (1983), pp. 1–25. A lecture based on this was published in Fields Medallists’ lectures (1997). MR 696688 Zbl 0508.46040 article

V. F. R. Jones: “A polynomial invariant for knots via von Neumann algebras,” Bull. Am. Math. Soc. 12 : 1 (January 1985), pp. 103–111. A lecture based on this was published in Fields Medallists’ lectures (1997). MR 766964 Zbl 0564.57006 article

V. F. R. Jones: “Hecke algebra representations of braid groups and link polynomials,” Ann. Math. (2) 126 : 2 (September 1987), pp. 335–388. This was republished in New Developments in the Theory of Knots (1990). MR 908150 Zbl 0631.57005 article

V. F. R. Jones, S. Morrison, and N. Snyder: “The classification of subfactors of index at most 5,” Bull. Am. Math. Soc. (N.S.) 51 : 2 (2014), pp. 277–327. MR 3166042 Zbl 1301.46039 ArXiv 1304.6141 article

A subfactor is an inclusion \( N\subset M \) of von Neumann algebras with trivial centers. The simplest example comes from the fixed points of a group action \( M^G \subset M \), and subfactors can be thought of as fixed points of more general group-like algebraic structures. These algebraic structures are closely related to tensor categories and have played important roles in knot theory, quantum groups, statistical mechanics, and topological quantum field theory. There is a measure of size of a subfactor, called the index. Remarkably, the values of the index below 4 are quantized, which suggests that it may be possible to classify subfactors of small index. Subfactors of index at most 4 were classified in the 1980s and early 1990s. The possible index values above 4 are not quantized, but once you exclude a certain family, it turns out that again the possibilities are quantized. Recently, the classification of subfactors has been extended up to index 5, and (outside of the infinite families) there are only 10 subfactors of index between 4 and 5. We give a summary of the key ideas in this classification and discuss what is known about these special small subfactors.

Noah Snyder	Related
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Vaughan F. R. Jones

Memories of Vaughan Jones

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