Celebratio Mathematica

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, 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
Scott Morrison	Related

V. Jones: “Some unitary representations of Thompson’s groups \( F \) and \( T \),” J. Comb. Algebra 1 : 1 (2017), pp. 1–44. MR 3589908 Zbl 06684911 ArXiv 1412.7740 article

V. F. R. Jones: “A no-go theorem for the continuum limit of a periodic quantum spin chain,” Comm. Math. Phys. 357 : 1 (2018), pp. 295–317. MR 3764571 Zbl 1397.82025 ArXiv 1607.08769 article

V. F. R. Jones: “Scale invariant transfer matrices and Hamiltonians,” J. Phys. A 51 : 10 (2018). article no. 104001, 27 pages. MR 3766219 Zbl 1387.82010 ArXiv 1706.00515 article

Given a direct system of Hilbert spaces \( s\mapsto \mathcal{H}_s \) (with isometric inclusion maps \[ \iota_s^t:\mathcal{H}_s\rightarrow \mathcal{H}_t \] for \( s\leq t \)) corresponding to quantum systems on scales \( s \), we define notions of scale invariant and weakly scale invariant operators. In some cases of quantum spin chains we find conditions for transfer matrices and nearest neighbour Hamiltonians to be scale invariant or weakly so. Scale invariance forces spatial inhomogeneity of the spectral parameter. But weakly scale invariant transfer matrices may be spatially homogeneous in which case the change of spectral parameter from one scale to another is governed by a classical dynamical system exhibiting fractal behaviour.

V. Aiello, R. Conti, and V. F. R. Jones: “The Homflypt polynomial and the oriented Thompson group,” Quantum Topol. 9 : 3 (2018), pp. 461–472. MR 3827807 Zbl 1397.57022 ArXiv 1609.02484 article

Roberto Conti	Related
Valeriano Aiello	Related

V. F. R. Jones: Irreducibility of the Wysiwyg representations of Thompson’s groups. Preprint, June 2019. ArXiv 1906.09619 techreport

V. F. R. Jones: “On the construction of knots and links from Thompson’s groups,” pp. 43–66 in Knots, low-dimensional topology and applications: Knots in Hellas (Olympia, Greece, 17–23 July 2016). Edited by C. C. Adams, C. M. Gordon, V. F. R. Jones, L. H. Kauffman, S. Lambropoulou, K. C. Millett, J. H. Przytycki, R. Ricca, and R. Sazdanovic. Springer Proceedings in Mathematics & Statistics 284. Springer (Cham, Switzerland), 2019. MR 3986040 Zbl 1423.57013 ArXiv 1810.06034 incollection

Renzo Luigi Ricca	Related
Radmila Sazdanović	Related
Józef Henryk Przytycki	Related
Cameron McAllan Gordon	Related	Volume
Kenneth C. Millett	Related
Louis H. Kauffman	Related
Colin C. Adams	Related
Sofia S. F. Lambropoulou	Related

@incollection {key3986040m,
	AUTHOR = {Jones, Vaughan F. R.},
	TITLE = {On the construction of knots and links
		 from {T}hompson's groups},
	BOOKTITLE = {Knots, low-dimensional topology and
		 applications: {K}nots in {H}ellas},
	EDITOR = {Adams, Colin C. and Gordon, Cameron
		 McA. and Jones, Vaughan F. R. and Kauffman,
		 Louis H. and Lambropoulou, Sofia and
		 Millett, Kenneth C. and Przytycki, Jozef
		 H. and Ricca, Renzo and Sazdanovic,
		 Radmila},
	SERIES = {Springer Proceedings in Mathematics
		 \& Statistics},
	NUMBER = {284},
	PUBLISHER = {Springer},
	ADDRESS = {Cham, Switzerland},
	YEAR = {2019},
	PAGES = {43--66},
	DOI = {10.1007/978-3-030-16031-9_3},
	NOTE = {(Olympia, Greece, 17--23 July 2016).
		 ArXiv:1810.06034. MR:3986040. Zbl:1423.57013.},
	ISSN = {2194-1009},
	ISBN = {9783030160302},
}

A. Brothier and V. F. R. Jones: “Pythagorean representations of Thompson’s groups,” J. Funct. Anal. 277 : 7 (October 2019), pp. 2442–2469. MR 3989149 Zbl 07089431 ArXiv 1807.06215 article

A. Brothier and V. F. R. Jones: “On the Haagerup and Kazhdan properties of R. Thompson’s groups,” J. Group Theory 22 : 5 (2019), pp. 795–807. MR 4000616 Zbl 07104291 ArXiv 1805.02177 article

A machinery developed by the second author produces a rich family of unitary representations of the Thompson groups \( F \), \( T \) and \( V \). We use it to give direct proofs of two previously known results. First, we exhibit a unitary representation of \( V \) that has an almost invariant vector but no nonzero \( [F,F] \)-invariant vectors reproving and extending Reznikoff’s result that any intermediate subgroup between the commutator subgroup of \( F \) and \( V \) does not have Kazhdan’s property (T) (though Reznikoff proved it for subgroups of \( T \)). Second, we construct a one parameter family interpolating between the trivial and the left regular representations of \( V \). We exhibit a net of coefficients for those representations which vanish at infinity on \( T \) and converge to 1 thus reproving that \( T \) has the Haagerup property after Farley who further proved that \( V \) has this property.

year	title	people

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Vaughan F. R. Jones

On Jones’ connections between
subfactors, conformal field theory,
Thompson’s groups and knots

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On Jones’ connections between subfactors, conformal field theory, Thompson’s groups and knots

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On Jones’ connections between
subfactors, conformal field theory,
Thompson’s groups and knots