Celebratio Mathematica — Jones

[1]V. F. R. Jones: “Quantum mechanics over fields of non-zero characteristic,” Lett. Math. Phys. 1 : 2 (1975–1976), pp. 99–103. MR 418670 article

[2] V. Jones: “Sur la conjugaison de sous-facteurs de facteurs de type \( \mathrm{II}_1 \)” [On the conjugation of subfactors of type \( \mathrm{II}_1 \) factors], C. R. Acad. Sci. Paris Sér. A-B 284 : 11 (1977), pp. A597–A598. MR 430803 Zbl 0344.46128 article

[3] F. R. V. Jones: Actions of finite groups on the hyperfinite \( \mathrm{II}_1 \) factor. Ph.D. thesis, University of Geneva, 1979. Advised by A. Haefliger. Republished as Memoirs of the American Mathematical Society 237 (1980). phdthesis

[4] V. F. R. Jones: “An invariant for group actions,” pp. 237–253 in Algèbres d’opérateurs [Operator algebras] (Les Plans-sur-Bex, France, 13–18 March 1978). Edited by P. de la Harpe. Lecture Notes in Mathematics 725. Springer (Berlin), 1979. MR 548118 Zbl 0497.46044 incollection

[5] T. Giordano and V. Jones: “Antiautomorphismes involutifs du facteur hyperfini de type \( \mathrm{II}_1 \)” [Involutive antiautomorphisms of hyperfinite type \( \mathrm{II}_1 \) factors], C. R. Acad. Sci. Paris Sér. A-B 290 : 1 (1980), pp. A29–A31. MR 564145 Zbl 0428.46047 article

[6] V. F. R. Jones: “A \( \mathrm{II}_1 \) factor anti-isomorphic to itself but without involutory antiautomorphisms,” Math. Scand. 46 : 1 (1980), pp. 103–117. MR 585235 Zbl 0432.46062 article

[7] 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

[8] V. F. R. Jones: “Isomorphisms of automorphism groups of type \( \mathrm{II} \) factors,” pp. 211–219 in Topics in modern operator theory (Timişoara and Herculane, Romania, 2–12 June 1980). Edited by C. Apostol, R. G. Douglas, B. Sz.-Nagy, and D. Voiculescu. Operator Theory: Advances and Applications 2. Birkhäuser (Basel and Boston), 1981. MR 672823 Zbl 0463.46049 incollection

Constantin Gelu Apostol	Related
Béla Szőkefalvi-Nagy	Related
Ronald George Douglas	Related

[9] V. F. R. Jones: “Prime actions of compact abelian groups on the hyperfinite type \( \mathrm{II} \) factor,” J. Oper. Theory 9 : 1 (1982), pp. 181–186. MR 695946 Zbl 0508.46043 article

This note is intended as an appendix to Ocneanu’s paper on amenable group actions. It seems likely that, using Ocneanu’s results one may obtain a satisfactory classification of all actions of a separable compact abelian group on the hyperfinite type \( \mathrm{II}_1 \) factor \( R \). (Finite group actions are classified in [Jones 1980]). Here we restrict ourselves to prime actions (i.e. ones whose fixed point algebra is a factor) because the classification is easy to understand, and, using Ocneanu’s work and Takesaki duality, easy to prove. An understanding of abelian actions will certainly be important for the classification of classical group actions as the maximal torus is a compact abelian group.

[10] A. Connes and V. Jones: “A \( \mathrm{II}_1 \) factor with two nonconjugate Cartan subalgebras,” Bull. Am. Math. Soc. (N.S.) 6 : 2 (1982), pp. 211–212. MR 640947 Zbl 0501.46056 article

A maximal abelian subalgebra \( A \) of the von Neumann algebra \( M \) is called a Cartan subalgebra if the normalizer \[ N(A) = \{\text{unitaries } u\in M \text{ with } uAu^* = A\} \] generates \( M \) as a von Neumann algebra (see [Feldman and Moore 1977]). It is a corollary of the paper by Connes, Feldman and Weiss [1982] that any two Cartan subalgebras of the hyperfinite type \( \mathrm{II}_1 \) factor \( R \) are conjugate by an automorphism of \( R \). In this paper we construct the first example of a (separable) \( \mathrm{II}_1 \) factor with two nonconjugate Cartan subalgebras, showing that the correspondence between ergodic theory (up to orbit equivalence) and von Neumann algebras is many to one.

[11] R. H. Herman and V. F. R. Jones: “Period two automorphisms of \( \mathrm{UHF} \) \( C^* \)-algebras,” J. Funct. Anal. 45 : 2 (1982), pp. 169–176. MR 647069 Zbl 0511.46057 article

[12] V. F. R. Jones: “Central sequences in crossed products of full factors,” Duke Math. J. 49 : 1 (March 1982), pp. 29–33. MR 650367 Zbl 0492.46049 article

Kazidan’s property \( T \) for groups is used by Connes in [1980] to provide the first examples of type \( \mathrm{II}_1 \) factors with countable fundamental group. The proof involves showing that the group \( \operatorname{Out} M \), the quotient of the group \( \operatorname{Aut} M \) of all automorphisms by the normal subgroup \( \operatorname{Int} M \) of inner automorphisms, is discrete. The factors thus constructed are automatically full, that is to say \( \operatorname{Int} M \) is closed in the topology of pointwise strong convergence on \( \operatorname{Aut} M \). This is in distinct contrast to, say, the hyperfinite type \( \mathrm{II}_1 \) factor \( R \) for which \( \mathrm{R} \) is dense in \( \operatorname{Aut} R \), but \( \operatorname{Out} R \) contains any separable locally compact group.

In [1981], Marie Choda used the property \( T \) technique to show, among other things, that the crossed product of any of Connes’ factors by a countable group of outer automorphisms is again full. We shall show that if \( M \) is any full \( \mathrm{II}_1 \) factor and \( G \) is a countable group of outer automorphisms whose image in \( \operatorname{Out} M \) is discrete, then the crossed product \( M\times G \) is a full factor. Choda’s result then follows from [Connes 1980]. We shall use two characterizations of full \( \mathrm{II}_1 \) factors: the first ([Connes 1974]) is that all central sequences (i.e., bounded sequences \( \{x_n\} \) for which \[ \lim_{n\to\infty}[x_n,y] = 0 \] in the strong topology for all \( y\in M \)) are trivial (i.e., there is a sequence \( \lambda_n \) of scalars such that \[ \lim_{n\to\infty}(\lambda_n-x_n) = 0 \] in the strong topology). The second is given by Theorem 2.1 of [Connes 1976].

In [1976], J. Phillips conjectured that if the group in question is cyclic, then the crossed product \( M\times G \) is full iff \( G \) is discrete in \( \operatorname{Out} M \) (\( M \) is full). We prove this conjecture.

[13] V. F. R. Jones: “L’indice d’un sous-facteur d’un facteur de type \( \mathrm{II} \)” [The index of a subfactor of a type \( \mathrm{II} \) factor], C. R. Acad. Sci. Paris Sér. I Math. 294 : 12 (1982), pp. 391–394. MR 659729 Zbl 0492.46048 article

[14] V. F. R. Jones: “Actions of discrete groups on factors,” pp. 167–177 in Operator algebras and applications (Kingston, ON, 14 July–2 August 1980), part 2. Edited by R. V. Kadison. Proceedings of Symposia in Pure Mathematics 38. American Mathematical Society (Providence, RI), 1982. MR 679504 Zbl 0503.46045 incollection

[15] V. Jones and S. Popa: “Some properties of MASA’s in factors,” pp. 89–102 in Invariant subspaces and other topics (Timişoara and Herculane, Romania, 1–11 June 1981). Edited by C. Apostol, R. G. Douglas, B. Szőkefalvi-Nagy, and D. Voisulescu. Operator Theory: Advances and Applications 6. Birkhäuser (Basel and Boston), 1982. MR 685457 Zbl 0508.46041 incollection

Sorin Teodor Popa	Related
Constantin Gelu Apostol	Related
Ronald George Douglas	Related

[16] R. H. Herman and V. F. R. Jones: “Models of finite group actions,” Math. Scand. 52 : 2 (1983), pp. 312–320. MR 702960 Zbl 0551.46043 article

[17] V. F. R. Jones: A converse to Ocneanu’s theorem, 1983. MR 715556 Zbl 0547.46045 arcticle

[18] 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

[19] N. Habegger, V. Jones, O. Pino Ortiz, and J. Ratcliffe: “Relative cohomology of groups,” Commentarii Math. Helvet. 59 : 1 (1984), pp. 149–164. MR 743948 Zbl 0544.20047 article

Nathan Habegger	Related
Oscar R. Pino-Ortiz	Related
John G. Ratcliffe	Related

[20] V. Jones: “Groups de tresses, algèbres de Hecke et facteurs de type \( \mathrm{II}_1 \)” [Braid groups, Hecke algebras and type \( \mathrm{II}_1 \) factors], C. R. Acad. Sci., Paris, Sér. I 298 : 20 (1984), pp. 505–508. An expanded English version of this was published in Geometric methods in operator algebras (1986). MR 753900 Zbl 0597.20034 article

[21] V. F. R. Jones and M. Takesaki: “Actions of compact abelian groups on semifinite injective factors,” Acta Math. 153 (1984), pp. 213–258. MR 766264 Zbl 0588.46042 article

The Clifford algebra construction allows one to associate with any unitary representation of a group an action of the group on an algebra in a functorial way. If the representation is infinite dimensional one must allow infinite dimensional algebras and one is led immediately to consider actions of groups on factors not of type \( \mathrm{I} \). Using this approach, Blattner showed in [1958] that any separable locally compact group has a faithful action, by outer automorphisms and the identity, on the hyperfinite type \( \mathrm{II}_1 \) factor \( \mathscr{R} \). It was certainly not obvious at the time that one might hope to say much more about the actions even of finite cyclic groups on \( \mathscr{R} \), but largely thanks to work of Connes, much progress has been made. This paper adds another step in a continuing project by giving a detailed description of all actions of a compact abelian group on semifinite injective factors.

[22] V. Jones and A. Connes: “Property \( T \) for von Neumann algebras,” Bull. Lond. Math. Soc. 17 : 1 (January 1985), pp. 57–62. MR 766450 Zbl 1190.46047 article

[23] 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

[24] F. Goodman, P. de la Harpe, and V. Jones: “Classification des matrices entières non négatives de petites normes” [Classification of non-negative integral matrices with small norms], C. R. Acad. Sci., Paris, Sér. I 300 : 14 (1985), pp. 463–465. MR 788972 Zbl 0624.15013 article

Frederick Michael Goodman	Related
Pierre de la Harpe	Related

[25] V. Jones: “Index for subrings of rings,” pp. 181–190 in Group actions on rings (Brunswick, ME, 8–14 July 1984). Edited by S. Montgomery. Contemporary Mathematics 43. American Mathematical Society (Providence, RI), 1985. MR 810651 Zbl 0607.46033 incollection

[26] V. Jones: “A new knot polynomial and von Neumann algebras,” Notices Am. Math. Soc. 33 : 2 (March 1986), pp. 219–225. MR 830613 article

[27] A. Connes: “Seminar Bourbaki, vol. 1984/85” [Index of subfactors, Hecke algebras and knot theory (after Vaughan Jones)], pp. 289–308. Astérisque 133–134. Société Mathématique de France (Paris), 1986. Exposé no. 647. MR 837226 Zbl 0597.57005 incollection

[28] V. F. R. Jones: “Braid groups, Hecke algebras and type \( \mathrm{II} \) factors,” pp. 242–273 in Geometric methods in operator algebras (Kyoto, July 1983). Edited by H. Araki and E. G. Effros. Pitman Research Notes in Mathematics 123. Longman Scientific & Technical, 1986. A brief, French language version was published in C. R. Acad. Sci., Paris, Sér. I 298:20 (1984). MR 866500 Zbl 0659.46054 incollection

Huzihiro Araki	Related
Edward George Effros	Related

[29] C. C. King: “Vaughan Jones and knot theory: A New Zealand mathematician unravels a new invariant which links diverse sciences in an unforeseen thread,” NZMS Newslett. 37 (August 1986), pp. 28–32. MR 988593 article

[30] V. F. R. Jones and K. Schmidt: “Asymptotically invariant sequences and approximate finiteness,” Am. J. Math. 109 : 1 (February 1987), pp. 91–114. MR 878200 Zbl 0638.28014 article

[31] R. H. Herman and V. F. R. Jones: “Central sequences in crossed products,” pp. 539–544 in Operator algebras and mathematical physics (Iowa City, IA, 17–21 June 1985). Edited by P. E. T. Jørgensen and P. S. Muhly. Contemporary Mathematics 62. American Mathematical Society (Providence, RI), 1987. MR 878399 Zbl 0624.46044 incollection

Palle E. T. Jørgensen	Related
Paul Scott Muhly	Related
Richard Howard Herman	Related

[32] 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

[33] V. Jones: “Subfactors of type \( \mathrm{II}_1 \) factors and related topics,” pp. 939–947 in Proceedings of the International Congress of Mathematicians (Berkeley, CA, 3–11 August 1986), vol. 2. Edited by A. M. Gleason. American Mathematical Society (Providence, RI), 1987. MR 934296 Zbl 0674.46034 incollection

[34] V. F. R. Jones: “On knot invariants related to some statistical mechanical models,” Pac. J. Math. 137 : 2 (1989), pp. 311–334. MR 990215 Zbl 0695.46029 article

[35] F. M. Goodman, P. de la Harpe, and V. F. R. Jones: Coxeter graphs and towers of algebras. MSRI Publications 14. Springer (New York), 1989. MR 999799 Zbl 0698.46050 book

Frederick Michael Goodman	Related
Pierre de la Harpe	Related

[36] D. M. Goldschmidt and V. F. R. Jones: “Metaplectic link invariants,” Geom. Dedicata 31 : 2 (1989), pp. 165–191. MR 1012438 Zbl 0678.57007 article

[37] V. F. R. Jones: “On a certain value of the Kauffman polynomial,” Commun. Math. Phys. 125 : 3 (1989), pp. 459–467. MR 1022523 Zbl 0695.57003 article

[38] V. F. R. Jones: “Notes on subfactors and statistical mechanics,” pp. 1–25 in Braid group, knot theory and statistical mechanics, II. Edited by C. N. Yang and M. L. Ge. Advanced Series in Mathematical Physics 9. World Scientific (River Edge, NJ), 1989. Also published in Int. J. Mod. Phys. A 5:3 (1990) and Braid group, knot theory and statistical mechanics, II (1994). MR 1062421 Zbl 0725.46038 incollection

A lot has been made in the last few years of connections between knot theory, statistical mechanics, field theory and von Neumann algebras. Because of their more technical nature, the von Neumann algebras have tended to be neglected in surveys. This is not an accurate reflection of their fundamental role in the subject, both as a continuing inspiration and as the vehicle of the discovery of the original ties between statistical mechanics and knot theory. In this largely expository article, we attempt to redress this balance by talking almost entirely about Neumann algebras and prove they occur as algebras of transfer matrices in statistical mechanical models. We shall focus mostly on the Potts model and the model of Fateev and Zamolodchikov, with a brief exposition of how vertex models are related to type \( \mathrm{III} \) factors.

Mo Lin Ge	Related
Chen Ning Yang	Related

[39] R. H. Herman: “Vaughan F. R. Jones,” Notices Am. Math. Soc. 37 : 9 (1990), pp. 1211–1213. article

[40] V. F. R. Jones: “Hecke algebra representations of braid groups and link polynomials,” pp. 20–73 in New Developments in the theory of knots. Edited by T. Kohno. Advanced Series in Mathematical Physics 11. World Scientific (Singapore), 1990. Republished from Ann. Math. 126:2 (1987). incollection

[41] V. F. R. Jones: “Notes on subfactors and statistical mechanics,” Int. J. Mod. Phys. A 5 : 3 (1990), pp. 441–460. Also published in Braid group, knot theory and statistical mechanics, II (1989) and Braid group, knot theory and statistical mechanics, II (1994). MR 1034604 Zbl 0737.46042 article

A lot has been made in the last few years of connections between knot theory, statistical mechanics, field theory and von Neumann algebras. Because of their more technical nature, the von Neumann algebras have tended to be neglected in surveys. This is not an accurate reflection of their fundamental role in the subject, both as a continuing inspiration and as the vehicle of the discovery of the original ties between statistical mechanics and knot theory. In this largely expository article, we attempt to redress this balance by talking almost entirely about Neumann algebras and prove they occur as algebras of transfer matrices in statistical mechanical models. We shall focus mostly on the Potts model and the model of Fateev and Zamolodchikov, with a brief exposition of how vertex models are related to type \( \mathrm{III} \) factors.

[42] V. F. R. Jones: “Baxterization,” pp. 701–713 in Proceedings of the conference on Yang–Baxter equations, conformal invariance and integrability in statistical mechanics and field theory (Canberra, Australia, 10–14 July 1989), published as Int. J. Mod. Phys. B 4 : 5. Issue edited by M. N. Barber and P. A. Pearce. World Scientific (Singapore), 1990. Other versions of this were published in Int. J. Mod. Phys. A 6:12 (1991) and Differential geometric methods in theoretical physics (1990). MR 1064744 Zbl 0723.57003 incollection

Paul A. Pearce	Related
Michael N. Barber	Related

[43] P. de la Harpe and V. Jones: “Paires de sous-algèbres semi-simples et graphes fortement réguliers” [Pairs of semi-simple subalgebras and strongly regular graphs], C. R. Acad. Sci., Paris, Sér. I 311 : 3 (1990), pp. 147–150. MR 1065880 Zbl 0707.46039 article

[44] V. Jones: “Knot theory and statistical mechanics,” Sci. Am. 263 : 5 (November 1990), pp. 98–103. MR 1079724 article

[45] F. Araki and S. Iitaka: “Profiles of the ICM-90 Fields Medal prizewinner,” Sūgaku 42 : 4 (1990), pp. 361–366. MR 1083945 article

Fujihiro Araki	Related
Shigeru Iitaka	Related

[46] V. F. R. Jones: “Knots, braids and statistical mechanics,” pp. 149–184 in Advances in differential geometry and topology. World Scientific (Singapore), 1990. A later version of this was published in Integrable systems and quantum groups (1992). MR 1095535 Zbl 0772.57008 incollection

[47] V. F. R. Jones: “Baxterization,” pp. 5–11 in Differential geometric methods in theoretical physics: Physics and geometry (Davis, CA, 2–8 July 1988). Edited by L.-L. Chau and W. Nahm. NATO ASI Series. Series B. Physics 245. Plenum Press (New York), 1990. Other versions of this were published in Int. J. Mod. Phys. B 4:5 (1990) and Int. J. Mod. Phys. A 6:12 (1991). MR 1169469 Zbl 1210.82022 incollection

[48] G. Skandalis: “Médaille Fields de Vaughan Jones” [The Fields Medal of Vaughan Jones], Gaz. Math. 47 (1991), pp. 17–19. MR 1091904 article

[49] V. F. R. Jones: “Baxterization,” Int. J. Mod. Phys. A 6 : 12 (1991), pp. 2035–2043. Other versions of this were published in Int. J. Mod. Phys. B 4:5 (1990) and Differential geometric methods in theoretical physics (1990). MR 1100623 Zbl 0744.57005 article

[50] Y. Kawahigashi: “Vaughan F. R. Jones’ achievements, I,” Sūgaku 43 : 1 (1991), pp. 29–34. MR 1111238 Zbl 0778.57001 article

[51] J. Murakami: “Vaughan F. R. Jones’ achievements, II,” Sūgaku 43 : 1 (1991), pp. 35–40. MR 1111239 article

[52] V. F. R. Jones: Subfactors and knots (Annapolis, MD, 5–11 June 1988). CBMS Regional Conference Series in Mathematics 80. American Mathematical Society (Providence, RI), 1991. MR 1134131 Zbl 0743.46058 book

[53] J. S. Birman: “The work of Vaughan F. R. Jones,” pp. 9–18 in Proceedings of the International Congress of Mathematicians (Kyoto, 21–29 August 1990), vol. 1. Edited by I. Satake. Mathematical Society of Japan (Tokyo), 1991. Reprinted in Fields Medallists’ lectures (1997). MR 1159199 Zbl 0743.01023 incollection

Ichirō Satake	Related
Joan S. Birman	Related	Volume

[54] V. F. R. Jones: “Von Neumann algebras in mathematics and physics,” pp. 121–138 in Proceedings of the International Congress of Mathematicians (Kyoto, Japan, 21–29 August 1990), vol. 1. Edited by I. Satake. Springer (Tokyo), 1991. This may have seen an expanded republication in Introduction to modern mathematics (2015). MR 1159209 Zbl 0781.46046 incollection

[55] V. F. R. Jones: “Commuting transfer matrices and link polynomials,” Int. J. Math. 3 : 2 (1992), pp. 205–212. An expanded version of this was published in Differential geometric methods in theoretical physics (1992). MR 1146812 Zbl 0774.57005 article

[56] V. Jones: “Knots, braids, statistical mechanics and von Neumann algebras,” pp. 1–16 in New Zealand mathematics colloquium (Dunedin, New Zealand, 19–23 May 1991), published as New Zealand J. Math. 21. University of Auckland, April 1992. MR 1167459 Zbl 0758.57002 incollection

[57] V. F. R. Jones: “Knots, braids and statistical mechanics,” pp. 1–36 in Integrable systems and quantum groups (Pavia, Italy, 1–2 March 1990). Edited by M. Carfora, M. Martellini, and A. Marzuoli. World Scientific (Singapore), 1992. An earlier version of this was published in Advances in differential geometry and topology (1990). MR 1178540 Zbl 0922.57007 incollection

[58] R. Schmid: “Strings, knots, and quantum groups: A glimpse at three 1990 Fields medalists,” SIAM Rev. 34 : 3 (September 1992), pp. 406–425. MR 1180070 Zbl 0757.01029 article

Vladimir Gershonovich Drinfel’d	Related
Edward Witten	Related
Rudolf Schmid	Related

[59] V. F. R. Jones: “From quantum theory to knot theory and back: A von Neumann algebra excursion,” pp. 321–336 in Mathematics into the twenty-first century: Proceedings of the AMS centennial symposium (Providence, RI, 8–12 August 1988), vol. 2. Edited by F. E. Browder. American Mathematical Society (Providence, RI), 1992. MR 1184618 Zbl 0965.46042 incollection

[60] “Eulogy of Vaughan Jones,” NZMS Newslett. 55 (August 1992), pp. 9–10. Eulogy delivered by P. T. Tarling. MR 1185251 article

[61] V. F. R. Jones: “Coincident link polynomials from commuting transfer matrices,” pp. 137–151 in Differential geometric methods in theoretical physics: Proceedings of the 20th international conference (New York, 3–7 June 1991), vol. 1. Edited by S. Catto and A. Rocha. World Scientific (Singapore), 1992. This is an expanded version of an article published in Int. J. Math. 3:2 (1992). MR 1225110 Zbl 0813.57004 incollection

Alvany M. P. Rocha Caridi	Related
Sultan Catto	Related

[62] A. Connes, G. Faltings, V. Jones, S. Smale, R. Thom, and J. Wagensberg: “Round-table discussion,” pp. 87–108 in Mathematical research today and tomorrow (Barcelona, 1991), vol. 1525. Lecture Notes in Math. Springer, Berlin, 1992. MR 1247058 incollection

Alain Connes	Related
Gerd Faltings	Related
René Thom	Related
Jorge Wagensberg	Related
Stephen Smale	Related

[63] V. Jones: Spin models, subfactors and link invariants, 1993. Kyoto University Infinite Analysis Lecture Notes No. 3. Notes by T. Kato. misc

[64] P. de la Harpe and V. F. R. Jones: “Graph invariants related to statistical mechanical models: Examples and problems,” J. Comb. Theory, Ser. B 57 : 2 (1993), pp. 207–227. MR 1207488 Zbl 0729.57003 article

[65] M. Rosso and V. Jones: “On the invariants of torus knots derived from quantum groups,” J. Knot Theory Ramifications 2 : 1 (1993), pp. 97–112. MR 1209320 Zbl 0787.57006 article

[66] V. F. R. Jones: “Milnor’s work and knot polynomials,” pp. 195–202 in Topological methods in modern mathematics: Proceedings of a symposium in honor of John Milnor’s sixtieth birthday (Stony Brook, NY, 14–21 June 1991). Edited by L. R. Goldberg and A. V. Phillips. Publish or Perish (Houston, TX), 1993. MR 1215965 Zbl 0833.57001 incollection

Lisa Robin Goldberg	Related
Anthony Valiant Phillips	Related
John W. Milnor	Related

[67] A. Connes, M. Flato, H. Hironaka, A. Jaffe, and V. Jones: “Editorial (dedication of this issue to Huzihiro Araki),” Comm. Math. Phys. 155 : 1 (1993), pp. 1–2. MR 1228522 article

Heisuke Hironaka	Related
Alain Connes	Related
Arthur Jaffe	Related
Moshé Flato	Related
Huzihiro Araki	Related

[68] V. Jones: “Spin model on knot projections,” pp. 195–200 in Operator algebras, mathematical physics, and low dimensional topology (Istanbul, 1–5 July 1991). Edited by R. Herman and B. Tanbay. Research Notes in Mathematics 5. A. K. Peters (Wellesley, MA), 1993. MR 1259065 Zbl 0853.46079 incollection

Richard Howard Herman	Related
Betül Tanbay	Related

[69] V. F. R. Jones: “On a family of almost commuting endomorphisms,” J. Funct. Anal. 122 : 1 (1994), pp. 84–90. MR 1274584 Zbl 0821.46076 article

[70] M. G. V. Bogle, J. E. Hearst, V. F. R. Jones, and L. Stoilov: “Lissajous knots,” J. Knot Theory Ramif. 3 : 2 (1994), pp. 121–140. MR 1279916 Zbl 0834.57005 article

M. G. V. Bogle	Related
L. Stoilov	Related
John E. Hearst	Related

[71] V. F. R. Jones: “A quotient of the affine Hecke algebra in the Brauer algebra,” Enseign. Math. (2) 40 : 3–4 (1994), pp. 313–344. MR 1309131 Zbl 0852.20035 article

[72] V. F. R. Jones and D. P. O. Rolfsen: “A theorem regarding 4-braids and the \( V = 1 \) problem,” pp. 127–135 in Proceedings of the conference on quantum topology (Manhattan, KS, 24–28 March 1993). Edited by D. N. Yetter. World Scientific (Singapore), 1994. MR 1309931 Zbl 0900.20066 incollection

Dale P. O. Rolfsen	Related
David Nelson Yetter	Related

[73] V. F. R. Jones: “The Potts model and the symmetric group,” pp. 259–267 in Subfactors: Proceedings of the Taniguchi symposium on operator algebras (Kyuzeso, Japan, 6–10 July 1993). Edited by H. Araki, H. Kosaki, and Y. Kawahigashi. World Scientific (Singapore), 1994. MR 1317365 Zbl 0938.20505 incollection

The symmetric group \( S_k \) acts on a vector space \( V \) of dimension \( k \) by permuting the basis elements \( v_1,v_2 \), \( \dots,v_k \). The groups \( S_n \) acts on \( \bigotimes^n V \) by the tensor product factors. We show that the algebra of all matrices on \( \bigotimes^n V \) commuting with \( S_k \) is generated by \( S_n \) and the operators \( e_1 \) and \( e_2 \) where \begin{align*} e_1(v_{p_1}\otimes v_{p_2}\otimes \cdots \otimes v_{p_n}) &= \frac{1}{k}\sum_{i=1}^k v_i\otimes v_{p_2} \otimes\cdots\otimes v_{p_n} \\ e_2(v_{p_1} \otimes v_{p_2} \otimes \cdots \otimes v_{p_n}) &= \delta_{p_1,p_2}v_{p_1}\otimes v_{p_2}\otimes \cdots \otimes v_{p_n} \end{align*} The matrices \( e_1 \) and \( e_2 \) give the vertical and horizontal transfer matrices adding one site in the square lattice Potts model.

Yasuyuki Kawahigashi	Related
Huzihiro Araki	Related
Hideki Kosaki	Related

[74] V. F. R. Jones: “Notes on subfactors and statistical mechanics,” pp. 177–201 in Braid group, knot theory and statistical mechanics, II. Edited by C. N. Yang and M. L. Ge. Advanced Series in Mathematical Physics 17. World Scientific (River Edge, NJ), 1994. Also published in Braid group, knot theory and statistical mechanics, II (1989) and Int. J. Mod. Phys. A 5:3 (1990). MR 1338603 Zbl 0822.46074 incollection

A lot has been made in the last few years of connections between knot theory, statistical mechanics, field theory and von Neumann algebras. Because of their more technical nature, the von Neumann algebras have tended to be neglected in surveys. This is not an accurate reflection of their fundamental role in the subject, both as a continuing inspiration and as the vehicle of the discovery of the original ties between statistical mechanics and knot theory. In this largely expository article, we attempt to redress this balance by talking almost entirely about Neumann algebras and prove they occur as algebras of transfer matrices in statistical mechanical models. We shall focus mostly on the Potts model and the model of Fateev and Zamolodchikov, with a brief exposition of how vertex models are related to type \( \mathrm{III} \) factors.

Mo Lin Ge	Related
Chen Ning Yang	Related

[75] R. Bacher, P. de la Harpe, and V. Jones: “Carrés commutatifs et invariants de structures combinatoires” [Commuting squares and invariants of combinatorial structures], C. R. Acad. Sci. Paris Sér. I Math. 320 : 9 (1995), pp. 1049–1054. MR 1332609 Zbl 0826.16024 article

Pierre de la Harpe	Related
Roland Bacher	Related

[76] V. F. R. Jones: “Three lectures on knots and von Neumann algebras,” pp. 96–113 in Infinite-dimensional geometry, noncommutative geometry, operator algebras, fundamental interactions (Saint-François, Guadeloupe, 30 May–13 June 1993). Edited by R. Coquereaux, M. Dubois-Violette, and P. Flad. World Scientific (River Edge, NJ), 1995. MR 1382136 incollection

Robert Coquereaux	Related
P. Flad	Related
M. Dubois-Violette	Related

[77] V. Jones: “Fusion en algèbres de von Neumann et groupes de lacets (d’après A. Wassermann)” [Fusion in von Neumann algebras and loop groups (after A. Wassermann)], pp. 251–273 in Séminaire Bourbaki, vol. 1994/95. Astérisque 237. 1996. Exposé no. 800. MR 1423627 Zbl 0921.46066 incollection

[78] D. Bisch and V. Jones: “A note on free composition of subfactors,” pp. 339–361 in Geometry and physics (Aarhus, Denmark, 18–27 July 1995). Edited by J. E. Andersen, J. Dupont, H. Pedersen, and A. Swann. Lecture Notes in Pure and Applied Mathematics 184. Dekker (New York), 1997. MR 1423180 Zbl 0968.46045 incollection

Johan L. Dupont	Related
Jørgen Ellegaard Andersen	Related
Henrik Pedersen	Related
Dietmar Herbert Bisch	Related
Andrew Swann	Related

[79] D. Bisch and V. Jones: “Algebras associated to intermediate subfactors,” Invent. Math. 128 : 1 (1997), pp. 89–157. MR 1437496 Zbl 0891.46035 article

The Temperley–Lieb algebras are the fundamental symmetry associated to any inclusion of \( \mathrm{II}_1 \) factors \( N\subset M \) with finite index. We analyze in this paper the situation when there is an intermediate subfactor \( P \) of \( N\subset M \). The additional symmetry is captured by a tower of certain algebras \( \mathrm{IA}_n \) associated to \( N\subset \) \( P\subset M \). These algebras form a Popa system (or standard lattice) and thus, by a theorem of Popa, arise as higher relative commutants of a subfactor. This subfactor gives a free composition (or minimal product) of an \( A_n \) and an \( A_m \) subfactor. We determine the Bratteli diagram describing their inclusions. This is done by studying a hierarchy \[ (FC_{m,n})_{n\in \mathbb{N}} \] of colored generalizations of the Temperley–Lieb algebras, using a diagrammatic approach, à la Kauffman, that is independent of the subfactor context. The Fuss–Catalan numbers \[ \frac{1}{(m+1)n+1}{(m+2)n\choose n} \] appear as the dimensions of our algebras. We give a presentation of the \( FC_{1,n} \) and calculate their structure in the semisimple case employing a diagrammatic method. The principal part of the Bratteli diagram describing the inclusions of the algebras \( FC_{1,n} \) is the Fibonacci graph. Our algebras have a natural trace and we compute the trace weights explicitly as products of Temperley–Lieb traces. If all indices are \( \geq 4 \), we prove that the algebras \( \mathrm{IA}_n \) and \( FC_{1,n} \) coincide. If one of the indices is \( < 4 \), \( \mathrm{IA}_n \) is a quotient of \( FC_{1,n} \) and we compute the Bratteli diagram of the tower \( (\mathrm{IA}_k)_{k\in\mathbb{N}} \). Our results generalize to a chain of \( m \) intermediate subfactors.

[80] V. Jones and H. Moscovici: “Book review: Alain Connes, ‘Noncommutative geometry’,” Notices Am. Math. Soc. 44 : 7 (1997), pp. 792–799. MR 1460207 Zbl 0908.46041 article

Alain Connes	Related
Henri Moscovici	Related

[81] V. Jones and V. S. Sunder: Introduction to subfactors. London Mathematical Society Lecture Note Series 234. Cambridge University Press, 1997. MR 1473221 Zbl 0903.46062 book

[82] J. S. Birman: “The work of Vaughan F. R. Jones,” pp. 435–445 in Fields Medallists’ lectures. Edited by M. F. Atiyah and D. Iagolnitzer. World Scientific Series in 20th Century Mathematics 5. World Scientific (River Edge, NJ), 1997. Reprinted from Proceedings of the International Congress of Mathematicians (1991). MR 1622915 incollection

M. F. Atiyah	Related	Volume
Daniel Iagolnitzer	Related
Joan S. Birman	Related	Volume

[83] V. F. R. Jones: “A polynomial invariant for knots via von Neumann algebras,” pp. 448–458 in Fields Medallists’ lectures. Edited by M. Atiyah and D. Iagolnitzer. World Scientific Series in 20th Century Mathematics 5. World Scientific (River Edge, NJ), 1997. Lecture based on an article published in Bull. Am. Math. Soc. 12:1 (1985). MR 1622916 incollection

M. F. Atiyah	Related	Volume
Daniel Iagolnitzer	Related

[84] V. F. R. Jones: “Index for subfactors,” pp. 459–486 in Fields Medallists’ lectures. Edited by M. Atiyah and D. Iagolnitzer. World Scientific Series in 20th Century Mathematics 5. World Scientific (River Edge, NJ), 1997. This lecture was based on an article published in Invent. Math. 72:1 (1983). MR 1622917 incollection

M. F. Atiyah	Related	Volume
Daniel Iagolnitzer	Related

[85] V. Jones: “Book review: D. Evans and Y. Kawahigashi, ‘Quantum symmetries on operator algebras’,” Bull. Am. Math. Soc. 38 : 3 (1998), pp. 369–377. article

David E. Evans	Related
Yasuyuki Kawahigashi	Related

[86] Knot theory (Warsaw, 13 July–17 August 1995). Edited by V. F. R. Jones, J. Kania-Bartoszyńska, J. H. Przytycki, and V. G. Traczyk, Pawełand Turaev. Banach Center Publications 42. Polish Academy of Sciences, Institute of Mathematics (Warsaw), 1998. MR 1634442 Zbl 0890.00048 book

Józef Henryk Przytycki	Related
Joanna Kania-Bartoszyńska	Related
Paweł Traczyk	Related
Vladimir G. Turaev	Related

[87] V. F. R. Jones and J. H. Przytycki: “Lissajous knots and billiard knots,” pp. 145–163 in Knot theory (Warsaw, 13 July–17 August 1995). Edited by V. F. R. Jones, J. Kania-Bartoszyńska, J. H. Przytycki, P. Traczyk, and V. G. Turaev. Banach Center Publications 42. Polish Academy of Sciences, Institute of Mathematics (Warsaw), 1998. MR 1634454 Zbl 0901.57012 incollection

Józef Henryk Przytycki	Related
Joanna Kania-Bartoszyńska	Related
Paweł Traczyk	Related
Vladimir G. Turaev	Related

[88] V. F. R. Jones: “A credo of sorts,” pp. 203–214 in Truth in mathematics (Mussomeli, Italy, 13–20 September 1995). Edited by H. G. Dales and G. Oliveri. Oxford University Press (New York), 1998. MR 1688343 Zbl 1041.57500 incollection

H. Garth Dales	Related
Gianluigi Oliveri	Related

[89] V. F. R. Jones: Planar algebras, I. Preprint, September 1999. Zbl 1328.46049 ArXiv math/9909027 techreport

[90] D. Bisch and V. Jones: “Singly generated planar algebras of small dimension,” Duke Math. J. 101 : 1 (2000), pp. 41–75. Part II was published in Adv. Math. 175:2 (2003). Part III was published in Trans. Am. Math. Soc. 369:14 (2017). MR 1733737 Zbl 1075.46053 article

[91] V. F. R. Jones: “Ten problems,” pp. 79–91 in Mathematics: Frontiers and perspectives. Edited by V. Arnold, M. Atiyah, P. Lax, and B. Mazur. American Mathematical Society (Providence, RI), 2000. MR 1754769 Zbl 0969.57001 incollection

M. F. Atiyah	Related	Volume
Vladimir Igorevich Arnold	Related
Barry C. Mazur	Related
Peter D. Lax	Related

[92] V. Jones: “Matematik för ett nytt årtusende” [Mathematics for a new millennium], Normat 48 : 4 (2000), pp. 145–152. MR 1829722 Zbl 1031.01014 article

[93] Knots in Hellas ’98: Proceedings of the international conference on knot theory an its ramifications (Delphi, Greece, 7–15 August 1998), vol. 1. Edited by C. M. Gordon, V. F. R. Jones, L. H. Kauffman, S. Lambropoulou, and J. H. Przytycki. Series on Knots and Everything 24. World Scientific (River Edge, NJ), 2000. Volume 2 was published as J. Knot Theory Ramif. 10:2 (2001). Volume 3 was published as J. Knot Theory Ramif. 10:5 (2001). MR 1865695 Zbl 0959.00034 book

Józef Henryk Przytycki	Related
Cameron McAllan Gordon	Related	Volume
Louis H. Kauffman	Related
Sofia S. F. Lambropoulou	Related

[94] V. F. R. Jones: “The planar algebra of a bipartite graph,” pp. 94–117 in Knots in Hellas ’98: Proceedings of the international conference on knot theory an its ramifications (Delphi, Greece, 7–15 August 1998), vol. 1. Edited by C. M. Gordon, V. F. R. Jones, L. H. Kauffman, S. Lambropoulou, and J. H. Przytycki. Series on Knots and Everything 24. World Scientific (River Edge, NJ), 2000. MR 1865703 Zbl 1021.46047 incollection

Józef Henryk Przytycki	Related
Cameron McAllan Gordon	Related	Volume
Louis H. Kauffman	Related
Sofia S. F. Lambropoulou	Related

[95] Knots in Hellas ’98: Proceedings of the international conference on knot theory an its ramifications, volume 2 (Delphi, Greece, 7–15 August 1998), published as J. Knot Theory Ramif. 10 : 2. Issue edited by C. M. Gordon, V. F. R. Jones, L. H. Kauffman, S. Lambropoulou, and J. H. Przytycki. World Scientific (Singapore), 2001. Volume 1 was published as a standalone book in 2001. MR 1822488 book

Józef Henryk Przytycki	Related
Cameron McAllan Gordon	Related	Volume
Louis H. Kauffman	Related
Sofia S. F. Lambropoulou	Related

[96] Knots in Hellas ’98: Proceedings of the international conference on knot theory an its ramifications, volume 3 (Delphi, Greece, 7–15 August 1998), published as J. Knot Theory Ramif. 10 : 5. Issue edited by C. M. Gordon, V. F. R. Jones, L. H. Kauffman, S. Lambropoulou, and J. H. Przytycki. World Scientific (Singapore), 2001. Volume 1 was published as a standalone book in 2001. MR 1839692 book

Józef Henryk Przytycki	Related
Cameron McAllan Gordon	Related	Volume
Louis H. Kauffman	Related
Sofia S. F. Lambropoulou	Related

[97] V. F. R. Jones: “The annular structure of subfactors,” pp. 401–463 in Essays on geometry and related topics: Mémoires dédiés à André Haefliger [Essays on geometry and related topics: Memoirs dedicated to André Haefliger], vol. 2. Edited by É. Ghys, P. de la Harpe, V. F. R. Jones, V. Sergiescu, and T. Tsuboi. Monographies de l’Enseignement Mathématique 38. Enseignement Mathématique (Geneva), 2001. MR 1929335 Zbl 1019.46036 ArXiv math/0105071 incollection

Vlad Sergiescu	Related
Takashi Tsuboi	Related
Étienne Ghys	Related
Pierre de la Harpe	Related
André Haefliger	Related

[98] Essays on geometry and related topics: Mémoires dédiés à André Haefliger [Essays on geometry and related topics: Memoirs dedicated to André Haefliger], vol. 1. Edited by É. Ghys, P. de la Harpe, V. F. R. Jones, V. Sergiescu, and T. Tsuboi. Monographies de l’Enseignement Mathématique 38. Enseignement Mathématique (Geneva), 2001. MR 1929318 Zbl 0988.00114 book

Vlad Sergiescu	Related
Takashi Tsuboi	Related
Étienne Ghys	Related
Pierre de la Harpe	Related
André Haefliger	Related

[99] Essays on geometry and related topics: Mémoires dédiés à André Haefliger [Essays on geometry and related topics: Memoirs dedicated to André Haefliger], vol. 2. Edited by É. Ghys, P. de la Harpe, V. F. R. Jones, V. Sergiescu, and T. Tsuboi. Monographies de l’Enseignement Mathématique 38. Enseignement Mathématique (Geneva), 2001. Zbl 0988.00115 book

Vlad Sergiescu	Related
Takashi Tsuboi	Related
Étienne Ghys	Related
Pierre de la Harpe	Related
André Haefliger	Related

[100] D. Bisch and V. Jones: “Singly generated planar algebras of small dimension, II,” Adv. Math. 175 : 2 (May 2003), pp. 297–318. Part I was published in Duke Math. J. 101:1 (2000). Part III was published in Trans. Am. Math. Soc. 369:14 (2017). MR 1972635 Zbl 1041.46048 article

We classified in Bisch and Jones (Duke Math. J. 101 (2000) 41) all spherical \( C^* \)-planar algebras generated by a non-trivial 2-box subject to the condition that the dimension of \( N^{\prime}\cap M_2 \) is \( \leq 12 \). We showed that they are given by the Fuss–Catalan systems discovered in Bisch and Jones (Invent. Math. 128 (1997) 89) and one exceptional planar algebra. In the present paper, we extend these results and show that there is only one spherical \( C^* \)-planar algebra generated by a single non-trivial 2-box if the dimension of \( N^{\prime}\cap M_2 \) is 13. It is given by the standard invariant of the crossed product subfactor \[ R \rtimes \mathbb{Z}_2 \subset R \rtimes D_5 ,\] where \( D_5 \) denotes the dihedral group with 10 elements.

[101] V. F. R. Jones and F. Xu: “Intersections of finite families of finite index subfactors,” Int. J. Math. 15 : 7 (2004), pp. 717–733. MR 2085101 Zbl 1059.46043 ArXiv math/0406331 article

[102]Special issue based on the 2nd international conference “Knots in Poland” (Warsaw, 7–13 July 2003 and Będlewo, 14–27 July 2003), published as Fundam. Math. 184. Issue edited by V. F. R. Jones, V. Tuarev, B. Wajnryb, and J. H. Przytycki. Polish Academy of Sciences, Institute of Mathematics (Warsaw), 2004. Special volumes of Fundamenta Mathematicae based on this conference were also published in 2005 (188) and 2006 (190). MR 2128038 Zbl 1065.57500 book

Józef Henryk Przytycki	Related
Dov Bronislaw Wajnryb	Related

[103]Special issue based on the 2nd international conference “Knots in Poland” (Warsaw, 7–13 July 2003 and Będlewo, 14–27 July 2003), published as Fundam. Math. 188. Issue edited by V. F. R. Jones, V. Tuarev, B. Wajnryb, and J. H. Przytycki. Polish Academy of Sciences, Institute of Mathematics (Warsaw), 2005. Special volumes of Fundamenta Mathematicae based on this conference were also published in 2004 (184) and 2006 (190). MR 2193063 Zbl 1087.57500 book

Józef Henryk Przytycki	Related
Dov Bronislaw Wajnryb	Related

[104] V. Jones: “The Jones polynomial,” pp. 179–181 in Encyclopedia of mathematical physics, vol. 3. Edited by J.-P. Françoise, G. L. Naber, and S. T. Tsou. Elsevier (Amsterdam), 2006. incollection

Gregory L. Naber	Related
Tsou Sheung Tsun	Related
Jean-Pierre Marcel Françoise	Related

[105] M. Conder and V. Jones: “Highly transitive imprimitivities,” J. Algebra 300 : 1 (2006), pp. 44–56. MR 2228633 Zbl 1111.20001 article

Some new observations are made about imprimitive permutation groups associated with subfactors of von Neuman algebras. Of particular interest are examples of a group \( G \) containing two maximal subgroups \( H \) and \( K \) such that \( G\neq HK \), and such that the action of \( G \) on the space of cosets of \( H\cap K \) has small rank (few suborbits). The rank 6 case turns out to correspond to the action of the collineation group on flags of a Desarguesian projective plane, and a special case of interest for rank 7 corresponds to the action of a 4-transitive group on ordered pairs of distinct points. Some other new (and unexpected) fundamental properties of groups are described along the way.

[106] V. F. R. Jones and S. A. Reznikoff: “Hilbert space representations of the annular Temperley–Lieb algebra,” Pac. J. Math. 228 : 2 (2006), pp. 219–249. MR 2274519 Zbl 1131.46042 article

[107] D. Aharonov, V. Jones, and Z. Landau: “A polynomial quantum algorithm for approximating the Jones polynomial,” pp. 427–436 in STOC’06: Proceedings of the 38th annual ACM symposium on theory of computing (Seattle, WA, 21–23 May 2006). Edited by J. M. Kleinberg. ACM Press (New York), 2006. MR 2277168 Zbl 1301.68129 incollection

The Jones polynomial, discovered in 1984 [Jones 1984], is an important knot invariant in topology. Among its many connections to various mathematical and physical areas, it is known (due to Witten [1989]) to be intimately connected to Topological Quantum Field Theory (TQFT). The works of Freedman, Kitaev, Larsen and Wang [2002a, 2002b] provide an efficient simulation of TQFT by a quantum computer, and vice versa. These results implicitly imply the existence of an efficient quantum algorithm that provides a certain additive approximation of the Jones polynomial at the fifth root of unity, \( e^{2\pi i/5} \), and moreover, that this problem is BQP-complete. Unfortunately, this important algorithm was never explicitly formulated. Moreover, the results in [Freedman et al. 2002a, 2002b] are heavily based on TQFT, which makes the algorithm essentially inaccessible to computer scientists.

We provide an explicit and simple polynomial quantum algorithm to approximate the Jones polynomial of an \( n \) strands braid with \( m \) crossings at any primitive root of unity \( e^{2\pi i/k} \), where the running time of the algorithm is polynomial in \( m \), \( n \) and \( k \). Our algorithm is based, rather than on TQFT, on well known mathematical results (specifically, the path model representation of the braid group and the uniqueness of the Markov trace for the Temperley–Lieb algebra). By the results of [Freedman et al. 2002b], our algorithm solves a BQP complete problem.

The algorithm we provide exhibits a structure which we hope is generalizable to other quantum algorithmic problems. Candidates of particular interest are the approximations of other downwards self-reducible \( \#\mathrm{P} \)-hard problems, most notably, the Potts model.

Zeph A. Landau	Related
Dorit Aharonov	Related
Jon Michael Kleinberg	Related

[108]Special issue based on the 2nd international conference “Knots in Poland” (Warsaw, 7–13 July 2003 and Będlewo, 14–27 July 2003), published as Fundam. Math. 190. Issue edited by V. F. R. Jones, V. Turev, B. Wajnryb, and J. H. Przytycki. Polish Academy of Sciences, Institute of Mathematics (Warsaw), 2006. Special volumes of Fundamenta Mathematicae based on this conference were also published in 2004 (184) and 2005 (188). Zbl 1101.57300 book

Józef Henryk Przytycki	Related
Dov Bronislaw Wajnryb	Related

[109] P. Grossman and V. F. R. Jones: “Intermediate subfactors with no extra structure,” J. Am. Math. Soc. 20 : 1 (2007), pp. 219–265. MR 2257402 Zbl 1131.46041 ArXiv math/0412423 article

If \( N\subseteq P \), \( Q\subseteq M \) are type \( \mathrm{II}_1 \) factors with \[ N^{\prime}\cap M =\mathbb{C}_{\mathrm{id}} \quad\text{and}\quad [M:N] < \infty \] we show that restrictions on the standard invariants of the elementary inclusions \[ N\subseteq P,\quad N\subseteq Q,\quad P\subseteq M \quad\text{and}\quad Q\subseteq M \] imply drastic restrictions on the indices and angles between the subfactors. In particular we show that if these standard invariants are trivial and the conditional expectations onto \( P \) and \( Q \) do not commute, then \( [M:N] \) is 6 or \( 6+4\sqrt{2} \). In the former case \( N \) is the fixed point algebra for an outer action of \( S_3 \) on \( M \) and the angle is \( \pi/3 \), and in the latter case the angle is \( \cos^{-1}(\sqrt{2}-1) \) and an example may be found in the GHJ subfactor family. The techniques of proof rely heavily on planar algebras.

[110] V. F. R. Jones: “In and around the origin of quantum groups,” pp. 101–126 in Prospects in mathematical physics: Young researchers symposium of the 14th international congress on mathematical physics (Lisbon, 25–26 July 2003). Edited by J. C. Mourão, J. P. Nunes, R. Picken, and J.-C. Zambrini. Contemporary Mathematics 437. American Mathematical Society (Providence, RI), 2007. MR 2354658 Zbl 1129.81046 ArXiv math/0309199 incollection

Roger F. Picken	Related
José Manuel Cidade Mourão	Related
João Luís Pimentel Nunes	Related
Jean-Claude Zambrini	Related

[111] V. F. R. Jones: “Two subfactors and the algebraic decomposition of bimodules over \( \mathrm{II}_1 \) factors,” Acta Math. Vietnam. 33 : 3 (2008), pp. 209–218. MR 2501843 Zbl 1182.46049 article

[112] D. Bisch, D. Gaboriau, V. F. R. Jones, and S. Popa: “Von Neumann algebras and ergodic theory of group actions,” Oberwolfach Rep. 5 : 4 (2008), pp. 2763–2814. MR 2568656 Zbl 1178.46003 article

This workshop was the first Oberwolfach meeting on von Neumann algebras and orbit equivalence ergodic theory. The organizers took special care to invite many young mathematicians and more than half of the 28 talks were given by them. The meeting was very well attended by over 40 participants, leading senior researchers and junior mathematicians in the field alike. Participants came from about a dozen different countries including Belgium, Canada, Denmark, France, Germany, Great Britain, Japan, Poland, Switzerland and the USA.

The first day of the workshop featured beautiful introductory talks to orbit equivalence and von Neumann algebras (Gaboriau), Popa’s deformation/rigidity techniques and applications to rigidity in \( \mathrm{II}_1 \) factors (Vaes), subfactors and planar algebras (Bisch), random matrices, free probability and subfactors (Shlyakhtenko), subfactor lattices and conformal field theory (Xu) and an open problem session (Popa). There were many excellent lectures during the subsequent days of the conference and many new results were presented, some for the first time during this meeting. A few of the highlights of the workshop were Vaes’ report on a new cocycle superrigidity result for non-singular actions of lattices in \( SL(n,\mathbb{R}) \) on \( \mathbb{R}^n \) and on other homogeneous spaces (joint with Popa), Ioana’s result showing that every sub-equivalence relation of the equivalence relation arising from the standard \( SL(2,\mathbb{Z}) \)-action on the 2-torus \( \mathbb{T}^2 \) is either hyperfinite, or has relative property (T), and Epstein’s report on her result that every countable, non-amenable group admits continuum many non-orbit equivalent, free, measure preserving, ergodic actions on a standard probability space. Other talks discussed new results on fundamental groups of \( \mathrm{II}_1 \) factors, \( L^2 \)-rigidity in von Neumann algebras, \( \mathrm{II}_1 \) factors with at most one Cartan subalgebra, subfactors from Hadamard matrices, a new construction of subfactors from a planar algebra and new results on topological rigidity and the Atiyah conjecture. Many interactions and stimulating discussions took place at this workshop, which is of course exactly what the organizers had intended.

Sorin Teodor Popa	Related
Dietmar Herbert Bisch	Related
Damien Gaboriau	Related

[113] V. Jones: “On the origin and development of subfactors and quantum topology,” Bull. Am. Math. Soc. (N.S.) 46 : 2 (2009), pp. 309–326. MR 2476415 Zbl 1169.46030 article

[114] D. Aharonov, V. Jones, and Z. Landau: “A polynomial quantum algorithm for approximating the Jones polynomial,” Algorithmica 55 : 3 (2009), pp. 395–421. MR 2512029 Zbl 1191.68313 ArXiv quant-ph/0511096 article

The Jones polynomial, discovered in 1984, is an important knot invariant in topology. Among its many connections to various mathematical and physical areas, it is known (due to Witten) to be intimately connected to Topological Quantum Field Theory (TQFT). The works of Freedman, Kitaev, Larsen and Wang provide an efficient simulation of TQFT by a quantum computer, and vice versa. These results implicitly imply the existence of an efficient (namely, polynomial) quantum algorithm that provides a certain additive approximation of the Jones polynomial at the fifth root of unity, \( e^{2\pi i/5} \), and moreover, that this problem is BQP-complete. Unfortunately, this important algorithm was never explicitly formulated. Moreover, the results of Freedman et al. are heavily based on TQFT, which makes the algorithm essentially inaccessible to computer scientists.

We provide an explicit and simple polynomial quantum algorithm to approximate the Jones polynomial of an \( n \) strands braid with \( m \) crossings at any primitive root of unity \( e^{2\pi i/k} \), where the running time of the algorithm is polynomial in \( m \), \( n \) and \( k \). Our algorithm is based, rather than on TQFT, on well known mathematical results (specifically, the path model representation of the braid group and the uniqueness of the Markov trace for the Temperley–Lieb algebra). By the results of Freedman et al., our algorithm solves a BQP complete problem.

Our algorithm works by encoding the local structure of the problem into the local unitary gates which are applied by the circuit. This structure is significantly different from previous quantum algorithms, which are mostly based on the Quantum Fourier transform. Since the results of the current paper were presented in their preliminary form, these ideas have been extended and generalized in several interesting directions. Most notably, Aharonov, Arad, Eban and Landau give a simplification and extension of these results that provides additive approximations for all points of the Tutte polynomial, including the Jones polynomial at any point, and the Potts model partition function at any temperature and any set of coupling strengths. We hope and believe that the ideas presented in this work will have other extensions and generalizations.

Zeph A. Landau	Related
Dorit Aharonov	Related

[115] V. Jones, D. Shlyakhtenko, and K. Walker: “An orthogonal approach to the subfactor of a planar algebra,” Pac. J. Math. 246 : 1 (2010), pp. 187–197. MR 2645882 Zbl 1195.46067 ArXiv 0807.4146 article

Kevin Walker	Related
Dimitri Y. Shlyakhtenko	Related

[116] A. Guionnet, V. F. R. Jones, and D. Shlyakhtenko: “Random matrices, free probability, planar algebras and subfactors,” pp. 201–239 in Quanta of maths: Conference on non commutative geometry in honor of Alain Connes (Paris, 29 March–6 April 2007). Edited by E. Blanchard, D. Ellwood, M. Khalkhali, M. Marcolli, H. Moscovici, and S. Popa. Clay Mathematics Proceedings 11. American Mathematical Society (Providence, RI), 2010. MR 2732052 Zbl 1219.46057 ArXiv 0712.2904 incollection

Sorin Teodor Popa	Related
Étienne Blanchard	Related
Alain Connes	Related
Masoud Khalkhali	Related
Dimitri Y. Shlyakhtenko	Related
Alice Guionnet	Related
David Alexandre Ellwood	Related
Matilde Marcolli	Related
Henri Moscovici	Related

[117] A. Guionnet, V. Jones, and D. Shlyakhtenko: “A semi-finite algebra associated to a subfactor planar algebra,” J. Funct. Anal. 261 : 5 (September 2011), pp. 1345–1360. MR 2807103 Zbl 1230.46054 article

We canonically associate to any planar algebra two type \( \mathrm{II}_{\infty} \) factors \( \mathfrak{M}_{\pm} \). The subfactors constructed previously by the authors in [Guionnet et al. 2010] are isomorphic to compressions of \( \mathfrak{M}_{\pm} \) to finite projections. We show that each \( \mathfrak{M}_{\pm} \) is isomorphic to an amalgamated free product of type \( \mathrm{I} \) von Neumann algebras with amalgamation over a fixed discrete type \( \mathrm{I} \) von Neumann subalgebra. In the finite-depth case, existing results in the literature imply that \[ \mathfrak{M}_{+} \cong \mathfrak{M}_{-} \] is the amplification a free group factor on a finite number of generators. As an application, we show that the factors \( M_j \) constructed in [Guionnet et al. 2010] are isomorphic to interpolated free group factors \[ L(\mathbb{F}(r_j)),\quad r_j = 1 + 2\delta^{-2j}(\delta - 1)I , \] where \( \delta^2 \) is the index of the planar algebra and \( I \) is its global index. Other applications include computations of laws of Jones–Wenzl projections.

Dimitri Y. Shlyakhtenko	Related
Alice Guionnet	Related

[118] V. F. R. Jones and D. Penneys: “The embedding theorem for finite depth subfactor planar algebras,” Quantum Topol. 2 : 3 (2011), pp. 301–337. MR 2812459 Zbl 1230.46055 ArXiv 1007.3173 article

[119] E. Artal Bartolo and M. T. Lozano Imízcoz: “Sir Vaughan Frederick Randal Jones,” Gac. R. Soc. Mat. Esp. 14 : 3 (2011), pp. 579–591. MR 2868302 article

María Teresa Lozano Imízcoz	Related
Enrique Artal Bartolo	Related

[120] V. F. R. Jones: “Quadratic tangles in planar algebras,” Duke Math. J. 161 : 12 (2012), pp. 2257–2295. MR 2972458 Zbl 1257.46033 ArXiv 1007.1158 article

[121] A. Guionnet, V. F. R. Jones, D. Shlyakhtenko, and P. Zinn-Justin: “Loop models, random matrices and planar algebras,” Comm. Math. Phys. 316 : 1 (2012), pp. 45–97. MR 2989453 Zbl 1277.82013 ArXiv 1012.0609 article

Jean Zinn-Justin	Related
Dimitri Y. Shlyakhtenko	Related
Alice Guionnet	Related

[122] M. Izumi, V. F. R. Jones, S. Morrison, and N. Snyder: “Subfactors of index less than 5, III: Quadruple points,” Comm. Math. Phys. 316 : 2 (2012), pp. 531–554. MR 2993924 Zbl 1272.46051 ArXiv 1109.3190 article

One major obstacle in extending the classification of small index subfactors beyond \( 3+\sqrt{3} \) is the appearance of infinite families of candidate principal graphs with 4-valent vertices (in particular, the “weeds” \( \mathcal{Q} \) and \( \mathcal{Q}^{\prime} \) from [Morrison and Snyder 2012]. Thus instead of using triple point obstructions to eliminate candidate graphs, we need to develop new quadruple point obstructions. In this paper we prove two quadruple point obstructions. The first uses quadratic tangles techniques and eliminates the weed \( \mathcal{Q}^{\prime} \) immediately. The second uses connections, and when combined with an additional number theoretic argument it eliminates both weeds \( \mathcal{Q} \) and \( \mathcal{Q}^{\prime} \). Finally, we prove the uniqueness (up to taking duals) of the 3311 Goodman–de la Harpe–Jones subfactor using a combination of planar algebra techniques and connections.

Noah Snyder	Related
Scott Morrison	Related
Masaki Izumi	Related

[123] S. Curran, V. F. R. Jones, and D. Shlyakhtenko: “On the symmetric enveloping algebra of planar algebra subfactors,” Trans. Am. Math. Soc. 366 : 1 (2014), pp. 113–133. MR 3118393 Zbl 1296.46054 ArXiv 1105.1721 article

Stephen Robert Curran	Related
Dimitri Y. Shlyakhtenko	Related

[124] 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

[125] É. Ghys, R. Grigorchuk, V. Jones, and T. Nagnibeda: “Pierre a septante ans” [Pierre is seventy], pp. 599–603 in Special issue on the occasion of Pierre de la Harpe’s 70th birthday, published as Groups Geom. Dyn. 8 : 3. European Publishing House (Zürich), 2014. MR 3267516 incollection

Rostislav Ivanovich Grigorchuk	Related
Étienne Ghys	Related
Pierre de la Harpe	Related
Tatiana Smirnova-Nagnibeda	Related

[126] V. F. R. Jones and D. Penneys: “Infinite index subfactors and the GICAR categories,” Comm. Math. Phys. 339 : 2 (2015), pp. 729–768. MR 3370617 Zbl 1338.46067 ArXiv 1410.0856 article

Given a \( \mathrm{II}_1 \)-subfactor \( A\subset B \) of arbitrary index, we show that the rectangular GICAR category, also called the rectangular planar rook category, faithfully embeds as \( A - A \) bimodule maps among the bimodules \( \bigotimes_A^n L^2(B) \). As a corollary, we get a lower bound on the dimension of the centralizer algebras \( A^{\prime}_0\cap A_{2n} \) for infinite index subfactors, and we also get that \( A^{\prime}_0\cap A_{2n} \) is nonabelian for \( n\geq 2 \), where \( (A_n)_{n\geq 0} \) is the Jones tower for \( A_0 = A \subset B = A_1 \). We also show that the annular GICAR/planar rook category acts as maps amongst the \( A \)-central vectors in \( \bigotimes_A^n L^2(B) \), although this action may be degenerate. We prove these results in more generality using bimodules. The embedding of the GICAR category builds on work of Connes and Evans, who originally found GICAR algebras inside Temperley–Lieb algebras with finite modulus.

[127] A. Brothier and V. F. R. Jones: “Hilbert modules over a planar algebra and the Haagerup property,” J. Funct. Anal. 269 : 11 (December 2015), pp. 3634–3644. MR 3406863 Zbl 1339.46058 ArXiv 1503.02708 article

[128] D. Bisch, T. J. Gannon, V. F. R. Jones, and Y. Kawahigashi: “Subfactors and conformal field theory,” Oberwolfach Rep. 12 : 2 (2015), pp. 849–926. MR 3443131 Zbl 1380.00034 article

Dietmar Herbert Bisch	Related
Yasuyuki Kawahigashi	Related
Terry J. Gannon	Related

[129] V. Jones: “Von Neumann algebras in mathematics and physics,” pp. 285–321 in Introduction to modern mathematics. Edited by S.-Y. Cheng, L. Ji, Y.-S. Poon, J. Xiao, L. Yang, and S.-T. Yau. Advanced Lectures in Mathematics 33. International Press (Somerville, MA), 2015. This may have been an expanded republication of the author’s 1990 ICM lecture. MR 3445454 Zbl 1356.46001 incollection

Shiu-Yuen Cheng	Related
Yat Sun Poon	Related
Lizhen Ji	Related
Jie Xiao	Related
Le Yang	Related
Shing-Tung Yau	Related

[130] A. Connes, V. Jones, M. Musat, and M. Rørdam: “Uffe Haagerup — in memoriam,” Notices Am. Math. Soc. 63 : 1 (January 2016), pp. 48–49. MR 3410853 Zbl 1338.46005 article

Alain Connes	Related
Uffe Haagerup	Related
Mikael Rørdam	Related
Magdalena Elena Muşat	Related

[131] V. F. R. Jones: “Knots, groups, subfactors and physics,” Jpn. J. Math. (3) 11 : 1 (2016), pp. 69–111. MR 3510680 Zbl 1361.46048 article

[132] 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

[133] D. Bisch, V. F. R. Jones, and Z. Liu: “Singly generated planar algebras of small dimension, III,” Trans. Am. Math. Soc. 369 : 4 (2017), pp. 2461–2476. Part I was published in Duke Math. J. 101:1 (2000). Part II was published in Adv. Math. 175:2 (2003). MR 3592517 Zbl 1370.46036 ArXiv 1410.2876 article

Zhengwei Liu	Related
Dietmar Herbert Bisch	Related

[134] The 2014 Maui and 2015 Qinhuangdao conferences in honour of Vaughan F. R. Jones’ 60th birthday (Kihei, HI, 14–18 July 2014 and Qinhuangdao, China 13–17 July 2015). Edited by S. Morrison and D. Penneys. Proceedings of the Centre for Mathematics and its Applications 46. Australian National University (Canberra), 2017. MR 3635666 Zbl 1386.46002 book

Scott Morrison	Related
David Penneys	Related

[135] 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

[136] 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.

[137] 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

[138] Knots, low-dimensional topology and applications — Part I, published as J. Knot Theory Ramif. 28 : 11. Issue edited by C. C. Adams, C. M. Gordon, V. F. R. Jones, L. H. Kauffman, S. Lambropoulou, K. Millett, J. H. Przytycki, R. Ricca, and R. Sazdanovic. World Scientific (Singapore), 2019. book

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

[139] Knots, low-dimensional topology and applications — Part II, published as J. Knot Theory Ramif. 28 : 13. Issue edited by C. C. Adams, C. M. Gordon, V. F. R. Jones, L. H. Kauffman, S. Lambropoulou, K. Millett, J. H. Przytycki, R. Ricca, and R. Sazdanovic. World Scientific (Singapore), 2019. book

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

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

[141] 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 3986037 Zbl 1419.57001 book

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

[142] 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},
}

[143] 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

[144] 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.

[145] C. C. Adams, C. M. Gordon, V. F. R. Jones, L. H. Kauffman, S. Lambropoulou, K. Millett, J. H. Przytycki, R. Ricca, and R. Sazdanovic: “Preface” in Knots, low-dimensional topology and applications — Part I, published as J. Knot Theory Ramif. 28 : 11. Issue edited by C. C. Adams, C. M. Gordon, V. F. R. Jones, L. H. Kauffman, S. Lambropoulou, K. Millett, J. H. Przytycki, R. Ricca, and R. Sazdanovic. World Scientific (Singapore), October 2019. article no. 1902001, 7 pages. MR 4038325 Zbl 07139437 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

@article {key4038325m,
	AUTHOR = {Adams, Colin C. and Gordon, Cameron
		 McA. and Jones, Vaughan F. R. and Kauffman,
		 Louis H. and Lambropoulou, Sofia and
		 Millett, Kenneth and Przytycki, Jozef
		 H. and Ricca, Renzo and Sazdanovic,
		 Radmila},
	TITLE = {Preface},
	JOURNAL = {J. Knot Theory Ramif.},
	FJOURNAL = {Journal of Knot Theory and its Ramifications},
	VOLUME = {28},
	NUMBER = {11},
	MONTH = {October},
	YEAR = {2019},
	DOI = {10.1142/S0218216519020012},
	NOTE = {\textit{Knots, low-dimensional topology
		 and applications --- {P}art {I}}. Issue
		 edited by C. C. Adams, C. M. Gordon,
		 V. F. R. Jones, L. H. Kauffman,
		 S. Lambropoulou, K. Millett,
		 J. H. Przytycki, R. Ricca,
		 and R. Sazdanovic. article
		 no. 1902001, 7 pages. MR:4038325. Zbl:07139437.},
	ISSN = {0218-2165},
}

[146] C. C. Adams, C. M. Gordon, V. F. R. Jones, L. H. Kauffman, S. Lambropoulou, K. Millett, J. H. Przytycki, R. Ricca, and R. Sazdanovic: “Preface” in Knots, low-dimensional topology and applications — Part II, published as J. Knot Theory Ramif. 28 : 13. Issue edited by C. C. Adams, C. M. Gordon, V. F. R. Jones, L. H. Kauffman, S. Lambropoulou, K. Millett, J. H. Przytycki, R. Ricca, and R. Sazdanovic. World Scientific (Singapore), November 2019. article no. 1903001, 15 pages. MR 4070944 Zbl 07177773 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

@article {key4070944m,
	AUTHOR = {Adams, Colin C. and Gordon, Cameron
		 McA. and Jones, Vaughan F. R. and Kauffman,
		 Louis H. and Lambropoulou, Sofia and
		 Millett, Kenneth and Przytycki, Jozef
		 H. and Ricca, Renzo and Sazdanovic,
		 Radmila},
	TITLE = {Preface},
	JOURNAL = {J. Knot Theory Ramif.},
	FJOURNAL = {Journal of Knot Theory and its Ramifications},
	VOLUME = {28},
	NUMBER = {13},
	MONTH = {November},
	YEAR = {2019},
	DOI = {10.1142/S0218216519030019},
	NOTE = {\textit{Knots, low-dimensional topology
		 and applications --- {P}art {II}}. Issue
		 edited by C. C. Adams, C. M. Gordon,
		 V. F. R. Jones, L. H. Kauffman,
		 S. Lambropoulou, K. Millett,
		 J. H. Przytycki, R. Ricca,
		 and R. Sazdanovic. article
		 no. 1903001, 15 pages. MR:4070944. Zbl:07177773.},
	ISSN = {0218-2165},
}

[147] V. F. R. Jones: Bergman space zero sets, modular forms, von Neumann algebras and ordered groups. Preprint, June 2020. ArXiv 2006.16419 techreport

\( A^2_{\alpha} \) will denote the weighted \( L^2 \) Bergman space. Given a subset \( S \) of the open unit disc we define \( \Omega(S) \) to be the infimum of \[ \{s \mid \exists f \in A^2_{s-2},\,f\neq 0, \textrm{ having } S \textrm{ as its zero set}\} .\] By classical results on Hardy space there are sets \( S \) for which \( \Omega(S)=1 \). Using von Neumann dimension techniques and cusp forms we give examples of \( S \) where \[ 1 < \Omega(S) < \infty .\] By using a left order on certain Fuchsian groups we are able to calculate \( \Omega(S) \) exactly if \( \Omega(S) \) is the orbit of a Fuchsian group. This technique also allows us to derive in a new way well known results on zeros of cusp forms and indeed calculate the whole algebra of modular forms for \( PSL_2(\mathbb{Z}) \).

[148] V. F. R. Jones and J. Yang: Motzkin algebras and the \( A_n \) tensor categories of bimodules. Preprint, August 2020. ArXiv 2008.04487 techreport

[149] V. F. R. Jones: “On spectral measures for certain unitary representations of R. Thompson’s group \( F \),” J. Funct. Anal. 280 : 1 (January 2020). article no. 108777, 27 pages. MR 4156130 ArXiv 1905.05806 article

The Hilbert space \( \mathcal{H} \) of backward renormalisation of an anyonic quantum spin chain affords a unitary representation of Thompson’s group \( F \) via local scale transformations. Given a vector in the canonical dense subspace of \( \mathcal{H} \) we show how to calculate the corresponding spectral measure for any element of \( F \) and illustrate with some examples. Introducing the “essential part” of an element we show that the spectral measure of any vector in \( \mathcal{H} \) is, apart from possibly finitely many eigenvalues, absolutely continuous with respect to Lebesgue measure. The same considerations and results hold for the Brown–Thompson groups \( F_n \) (for which \( F=F_2 \)).

[150] D. Bisch, D. E. Evans, R. Kirby, and S. Popa: “Memories of Vaughan Jones,” Notices Amer. Math. Soc. 68 : 9 (2021), pp. 1540–1563. MR 4323829 Zbl 1480.01028 article

David E. Evans	Related
Dietmar Herbert Bisch	Related
Robion C. Kirby	Related	Volume

Complete Bibliography