Celebratio Mathematica

Sir Vaughan Fre­d­er­ick Ran­dal Jones, who died at age 67 on Septem­ber 6, 2020, was one of the most in­flu­en­tial and in­spir­a­tion­al math­em­aticians of the last four dec­ades. His ori­gin­al and pen­et­rat­ing ana­lys­is of in­clu­sions of von Neu­mann al­geb­ras led to the cre­ation of new fields of re­search, while re­in­vig­or­at­ing old ones, thereby set­ting off an ex­traordin­ary in­ter­play between dis­par­ate areas of math­em­at­ics, from ana­lys­is of op­er­at­or al­geb­ras, to low-di­men­sion­al to­po­logy, stat­ist­ic­al mech­an­ics, quantum com­put­ing, and quantum field the­ory. Vaughan’s work had a ma­jor im­pact with un­ex­pec­ted, stun­ning ap­plic­a­tions, even out­side of math­em­at­ics, for ex­ample to the study of knot­ted DNA strands and pro­tein fold­ing in bio­logy. A cru­cial idea lead­ing to these strik­ing con­nec­tions was his ground­break­ing dis­cov­ery in the early 1980s that the sym­met­ries of a factor (a von Neu­mann al­gebra with trivi­al cen­ter), as en­coded by its sub­factors, are quant­ized. They gen­er­ate “quant­ized groups,” a com­pletely new type of struc­ture, en­dowed with a di­men­sion func­tion giv­en by a trace and an in­dex that can be non­in­teg­ral.

This art­icle gives a pan­or­amic view of the sci­entif­ic im­pact and en­dur­ing leg­acy of Vaughan’s work, as well as his per­son­al­ity and style of work­ing through the con­tri­bu­tions of col­leagues and friends across math­em­at­ics and phys­ics. Over the years, Vaughan’s count­less math­em­at­ic­al in­ter­ac­tions forged nu­mer­ous lifelong friend­ships, and he will be sorely missed by all.

Vaughan was born on Decem­ber 31, 1952, in Gis­borne on the North Is­land of New Zea­l­and to par­ents Jim Jones and Joan Jones (née Collins) and grew up in Auck­land. Between the ages of eight and twelve he was edu­cated at the board­ing school St Peter’s School in Cam­bridge in rur­al North Is­land. Vaughan at­ten­ded Auck­land Gram­mar School un­til the age of six­teen and then stud­ied Math­em­at­ics at Auck­land Uni­versity from 1969 to 1973. He left New Zea­l­and in 1974 for gradu­ate study at the Uni­versity of Geneva with the in­ten­tion of writ­ing a thes­is in Phys­ics, but gradu­ally moved in 1974–76 to work un­der the su­per­vi­sion of An­dré Hae­fli­ger in Math­em­at­ics. It was in Switzer­land where Vaughan met Martha (Wendy), who held a schol­ar­ship to study at the Uni­versity of Fri­bourg and sub­sequently worked at the United Na­tions in Geneva. They mar­ried in 1979 and raised three chil­dren to­geth­er, Beth­any, Ian, and Alice.

Vaughan had sev­er­al ap­point­ments in the USA un­til his death. However, the friend­ships he made dur­ing these form­at­ive years in New Zea­l­and re­mained with him throughout his life. His love and loy­alty to New Zea­l­and would bring him back later, at least an­nu­ally from 1994 on, to in­vig­or­ate math­em­at­ics in his nat­ive land, of­ten with sum­mer schools, us­ing his net­work of col­leagues and friends world­wide and his sci­entif­ic stand­ing to at­tract oth­er world-renowned stars to New Zea­l­and.

In the fall of 1975, when Vaughan was switch­ing from phys­ics to math­em­at­ics, he met Alain Connes at a con­fer­ence in Stras­bourg and was very im­pressed. Connes had just fin­ished his sem­in­al work on \( \mathrm{II}_1 \) factors, a class of von Neu­mann factors that have a trace with range \( [0, 1] \) on the lat­tice of their pro­jec­tions. In one of his pa­pers, Connes gave a clas­si­fic­a­tion of peri­od­ic auto­morph­isms of the hy­per­fin­ite factor \( R \), an im­port­ant \( \mathrm{II}_1 \) factor that can be seen as the quant­ized ver­sion of the unit in­ter­val. Vaughan was struck by the beauty of these math­em­at­ic­al ob­jects and the con­tinu­ous di­men­sion phe­nomen­on, which has the re­mark­able fea­ture that one can take the \( t \times t \) mat­rix al­gebra \( M^t := \mathbb{M}_t (M) \) over a \( \mathrm{II}_1 \) factor \( M \) for any real num­ber \( t > 0 \). He avidly stud­ied all of the pa­pers in this sub­ject, from the pi­on­eer­ing 1936–1943 work of Mur­ray and von Neu­mann, who dis­covered these ob­jects, all the way to Connes’s re­cent pre­prints. He gathered a list of ten pos­sible thes­is top­ics and trav­elled to Par­is to show them to Connes, who went rap­idly down the list, “No, no, no, maybe, no, …, good, …,” and the “good” one be­came Vaughan’s thes­is. That top­ic was to gen­er­al­ize Connes’s res­ult on peri­od­ic auto­morph­isms to ar­bit­rary fi­nite groups, which Vaughan did in [1]. Vaughan con­tin­ued to vis­it Connes in Par­is and then at the In­sti­tute for Ad­vanced Study where Connes was a mem­ber in 1978–1979, with Hae­fli­ger as his form­al ad­visor. Vaughan re­ceived his Docteur ès Sci­ences from the Uni­versity of Geneva in 1979, and his thes­is was awar­ded the Vacher­on Con­stantin Prize.

Masami­chi Take­saki was im­pressed by Vaughan’s thes­is and brought him to UCLA on a Hedrick as­sist­ant pro­fess­or­ship in 1980. But after one year at UCLA, Vaughan re­turned to the East Coast to join his wife Wendy who was study­ing at Prin­ceton. UPenn seized the op­por­tun­ity and made him an of­fer. So dur­ing 1981–1985, Vaughan was at UPenn, first as a ju­ni­or fac­ulty mem­ber then as an as­so­ci­ate pro­fess­or, with 1984–1985 ac­tu­ally spent at MSRI. In 1985, he was ap­poin­ted full pro­fess­or at UC, Berke­ley, where he re­mained un­til he re­tired in 2013 with the title Pro­fess­or Emer­it­us. From 2011 on he held the Steven­son Dis­tin­guished Chair at Vander­bilt Uni­versity. Vaughan was also a Dis­tin­guished Alumni Pro­fess­or at the Uni­versity of Auck­land and Found­ing Dir­ect­or of the New Zea­l­and Math­em­at­ics Re­search In­sti­tute from 1994 on. He kept in con­tact with Europe in­clud­ing spend­ing one-year sab­bat­ic­als at the IHES dur­ing 1986–1987 and 1989–1990 and at the Uni­versity of Geneva in 1993–1994 and 1998–99.

In his thes­is, Vaughan de­veloped a nov­el al­geb­ra­ic ap­proach to the clas­si­fic­a­tion of ac­tions of fi­nite groups on \( \mathrm{II}_1 \) factors, in which the ac­tion of the fi­nite group was en­coded by the iso­morph­ism class of an in­clu­sion of \( \mathrm{II}_1 \) factors, via a crossed product con­struc­tion. Soon after his thes­is, this led him to con­sider ab­stract in­clu­sions of \( \mathrm{II}_1 \) factors, \( N \subset M \), or what he later called sub­factors, to­geth­er with a nat­ur­al no­tion of di­men­sion of \( M \) as an \( N \)-mod­ule, that he called the in­dex and de­noted \( [M : N] \). He no­ticed right away that the hy­per­fin­ite \( \mathrm{II}_1 \) factor \( R \) con­tains sub­factors of any in­dex \( \geq 4 \). This fol­lows from the fact that \( R^t\approx R \) for any \( t > 0 \), a res­ult that is due to Mur­ray and von Neu­mann. He also noted that for sub­factors \( N\subset M \) arising from in­clu­sions of groups \( H\subset G \), the sub­factor in­dex was equal to the in­dex \( [G : H] \) of the sub­group. By early 1980, he was able to prove that the in­dex of a sub­factor \( N \subset M \) can only take the val­ues 1 and 2 when \( [M : N] < 1 + \sqrt{2} \). He cir­cu­lated a pre­print and gave talks at con­fer­ences about these find­ings. The gen­er­al re­ac­tion of col­leagues in the field was that most cer­tainly only the val­ues 1, 2, 3 could oc­cur un­der 4.

But by Novem­ber 1981, Vaughan made the amaz­ing dis­cov­ery that the in­dex of a sub­factor can take ex­actly the val­ues \[ \{4 \cos^2 (\pi/n) \mid n \geq 3\}= \{1, 2, (3 + \sqrt{5})/2, 3, \dots \}, \] when less than 4. Most im­port­antly, he showed that all these val­ues can oc­cur as in­dices of sub­factors of the hy­per­fin­ite factor \( R \) ([2]). The proof of the re­stric­tions on the in­dex, which is of stun­ning beauty, in­volves the con­struc­tion of an in­creas­ing se­quence of factors (a tower), ob­tained by “adding” it­er­at­ively pro­jec­tions (i.e., idem­potents) sat­is­fy­ing a set of ax­ioms which, to­geth­er with the ex­ist­ence of the trace and its prop­er­ties, provide the re­stric­tions.

In the sum­mer of 1982, Vaughan real­ized that, be­cause of the al­geb­ra­ic re­la­tions they sat­is­fy, the pro­jec­tions in the tower of factors provide an un­ex­pec­ted fam­ily of semisimple quo­tients of the Hecke al­geb­ras of type \( A_n \) and com­pletely new rep­res­ent­a­tions of Artin’s braid groups, in­dexed by a para­met­er \( \lambda \in \mathbb{R} \), which are unit­ary ex­actly at val­ues cor­res­pond­ing to the in­dices in the dis­crete range. Dur­ing the fol­low­ing two years, Vaughan gradu­ally learned of the im­port­ance of braid groups to the the­ory of knots, due to Al­ex­an­der’s the­or­em that any knot is a closed braid and Markov’s the­or­em show­ing when two braids give rise to the same knot via “two moves.” While he real­ized that one of the Markov moves was auto­mat­ic­ally in­vari­ant when ap­ply­ing the trace to the braid ele­ment in this rep­res­ent­a­tion in the tower of factors, it was in May 1984 that he dealt with the second Markov move, through a stroke of geni­us renor­mal­iz­a­tion idea, that al­to­geth­er gave rise to a poly­no­mi­al in­vari­ant for knots and links — the Jones poly­no­mi­al, \( V_K(q) \) for an ori­ented link \( K \) ([3], [4]). Once Vaughan had defined his poly­no­mi­al, it was easy to see that it was not the clas­sic­al Al­ex­an­der poly­no­mi­al, and that it could dis­tin­guish a knot from its mir­ror im­age, and then, with more work, that it solved three Tait Con­jec­tures, cen­tury-old con­jec­tures that con­cerned pro­jec­tions of a knot on the plane and their sim­pli­fic­a­tions. Next, \( V_K \) was quickly gen­er­al­ized to a 2-vari­able poly­no­mi­al, the HOM­FLYPT poly­no­mi­al, named after the ini­tials of five groups who in­de­pend­ently dis­covered it. Bio­lo­gists im­me­di­ately used the Jones poly­no­mi­al to ana­lyze knots ap­pear­ing in strands of DNA.

The in­vari­ant \( V_K \) was re­mark­able in gen­er­at­ing new dir­ec­tions for re­search. Most in­ter­est­ing for to­po­lo­gists was Khovan­ov ho­mo­logy, a cat­egor­i­fic­a­tion of \( V_K \) us­ing the Kauff­man brack­et (it­self a way of de­scrib­ing \( V_K \)). Khovan­ov ho­mo­logy, whose Euler char­ac­ter­ist­ic is \( V_K \), de­term­ines the un­knot which \( V_K \) is not known to do and is re­lated by a spec­tral se­quence to knot Flo­er ho­mo­logy.

In 1988, Wit­ten gave a phys­ic­al in­ter­pret­a­tion for \( V_K \) for links (Wilson loops) in terms of Chern–Si­mons the­ory at level \( l \) that cor­res­ponds not just to \( V_K \), but also to the HOM­FLYPT poly­no­mi­al. This point of view led to nu­mer­ous 3-man­i­fold quantum in­vari­ants at roots of unity us­ing the colored \( V_K \). Vaughan ori­gin­ated these spec­tac­u­lar de­vel­op­ments which now form a new branch of math­em­at­ics called Quantum To­po­logy.

In a par­al­lel de­vel­op­ment which star­ted in 1983, a con­nec­tion was made with cal­cu­la­tions by Tem­per­ley and Lieb in solv­able stat­ist­ic­al mech­an­ics. This triggered yet an­oth­er series of in­ter­ac­tions with phys­ics, via stat­ist­ic­al mech­an­ics and con­form­al quantum field the­ory. In the lat­ter, a sim­il­ar di­cho­tomy of dis­crete and con­tinu­ous parts oc­curs for the cent­ral charge in the rep­res­ent­a­tions of the Vi­ra­s­oro al­gebra which de­scribes cer­tain pro­ject­ive rep­res­ent­a­tions of the dif­feo­morph­ism group of the circle. Sub­factors provide a nat­ur­al frame­work for study­ing two-di­men­sion­al con­form­al quantum field the­or­ies. In­deed the dis­crete series of the cent­ral charge in the rep­res­ent­a­tion the­ory of the Vi­ra­s­oro al­gebra can be un­der­stood via con­form­al nets of factors, as cosets of \( \operatorname{SU}(2) \) the­or­ies. However, the power of the quantum sym­metry sub­factor for­mu­la­tion is that it per­mits the won­drous pos­sib­il­ity of con­struct­ing new exot­ic con­form­al field the­or­ies bey­ond the known well-stud­ied ones arising from loop groups, doubles of fi­nite groups, or nat­ur­al con­struc­tions such as cosets, with in­tense on­go­ing work.

Per­haps the deep­est and most en­dur­ing of Vaughan’s re­volu­tion­ary work is with­in the the­ory of \( \mathrm{II}_1 \) factors and more gen­er­ally in al­geb­ras of op­er­at­ors on Hil­bert space. \( \mathrm{II}_1 \) factors arise nat­ur­ally from groups, their ac­tions on spaces, and unit­ary rep­res­ent­a­tions of groups. Un­til Vaughan’s work, sym­met­ries of a \( \mathrm{II}_1 \) factor \( M \) were thought to be its al­gebra auto­morph­isms, which un­der mul­ti­plic­a­tion gen­er­ate a group of auto­morph­isms of \( M \), with pos­sible tor­sion, like in the case of sym­met­ries of clas­sic­al spaces. But Vaughan’s work showed that sym­met­ries of a factor \( M \) may be “quant­ized.” Moreover, it also showed that the prop­er way to view a sym­metry in this frame­work is to en­code it as a sub­factor \( N \subset M \), or equi­val­ently as the Hil­bert \( N \)-\( M \) bimod­ule \( {}_N L^2 (M)_M \) and the Jones in­dex \( [M : N] \) as the codi­men­sion of \( N \) in \( M \). Such quant­ized sym­met­ries gen­er­ate a quant­ized group (tensor cat­egory) \( \mathcal{G}_{N\subset M} \) un­der tak­ing ad­joints and mul­ti­plic­a­tion (fu­sion un­der re­l­at­ive tensor product), called the stand­ard in­vari­ant of \( N\subset M \), with a Cay­ley type bi­part­ite graph \( \Gamma_{N\subset M} \). When the num­ber of ir­re­du­cible ele­ments of \( \mathcal{G}_{N\subset M} \) is fi­nite, something that Vaughan proved to be auto­mat­ic if \( [M : N] < 4 \), then the square norm of the graph equals the in­dex, \[ \|\Gamma_{N\subset M}\| 2 = [M : N]. \] The ADE clas­si­fic­a­tion of graphs of norm less than 2 as \[ \{2\cos(\pi/n) \mid n \geq 3\} \] of­fers yet an­oth­er way of de­riv­ing the re­stric­tions on the in­dex \( < 4 \).

One can hardly over­state the im­port­ance and depth of these dis­cov­er­ies. This led right away to a huge num­ber of beau­ti­ful and ex­cit­ing prob­lems, such as the clas­si­fic­a­tion of sub­factor in­clu­sions \( N \subset M \) when \( M \) is hy­per­fin­ite, the prob­lem of ax­io­mat­iz­ing the ob­jects \( \mathcal{G}_{N\subset M} \) and char­ac­ter­iz­ing the bi­part­ite graphs \( \Gamma_{N\subset M} \) that can oc­cur as graphs of sub­factors, and the prob­lem of in­vest­ig­at­ing what kind of quantum sym­met­ries can “act” on a spe­cif­ic factor and what val­ues of the in­dex can oc­cur, etc.

Many out­stand­ing res­ults by a large num­ber of people have fol­lowed. Vaughan was much in­volved in these de­vel­op­ments, not­ably find­ing the best way to char­ac­ter­ize the ob­jects \( \mathcal{G}_{N\subset M} \) arising as stand­ard in­vari­ants of sub­factors as a two-di­men­sion­al dia­gram­mat­ic struc­ture of tangles called a planar al­gebra (1999). Vaughan de­veloped planar al­geb­ras as a tool to ef­fi­ciently carry out in­tric­ate com­pu­ta­tions with the stand­ard in­vari­ant of a sub­factor. It al­lowed for to­po­lo­gic­al ar­gu­ments in the ana­lys­is of sub­factors and led to re­mark­able res­ults in the clas­si­fic­a­tion pro­gramme of sub­factors, in­clud­ing the con­struc­tion of stun­ning “exot­ic” quant­ized sym­met­ries, cap­tured as planar al­geb­ras. These power­ful tools were suc­cess­fully used by Vaughan and some of his former stu­dents to clas­si­fy all such sym­met­ries up to in­dex 5 (1995–2014) ([5]), which was then pushed fur­ther up to \( 5.25 \). While tra­di­tion­al clas­si­fic­a­tion at­tempts fo­cused on sub­factors with small in­dices, the planar al­gebra ap­proach shif­ted the point of view to a gen­er­at­ors and re­la­tions ap­proach. Thus, singly gen­er­ated planar al­geb­ras and then Yang–Bax­ter re­la­tion planar al­geb­ras played a key role, through which im­port­ant colored vari­ants of the Tem­per­ley–Lieb al­geb­ras were dis­covered as the fun­da­ment­al quant­ized sym­met­ries as­so­ci­ated to in­ter­me­di­ate sub­factors.

Planar al­geb­ras, to­geth­er with a quest to pro­duce a con­form­al the­ory from sub­factors, led Vaughan to a study of the Thompson groups as dis­crete ap­prox­im­a­tions to the dif­feo­morph­ism group of the circle, and again to un­ex­pec­ted spin-offs for the the­ory of knots and links (2015–2020).

More de­tails of all these math­em­at­ic­al de­vel­op­ments will be found in a forth­com­ing is­sue of the Bul­let­in of the AMS which is ded­ic­ated to Vaughan.

Vaughan was awar­ded the Fields Medal in Kyoto in 1990, and was elec­ted Fel­low of the Roy­al So­ci­ety in the same year, Hon­or­ary Fel­low of the Roy­al So­ci­ety of New Zea­l­and Te Apārangi in 1991, mem­ber of the Amer­ic­an Academy of Arts and Sci­ences in 1993 and of the US Na­tion­al Academy of Sci­ences in 1999, and for­eign mem­ber of na­tion­al learned academies in Aus­tralia, Den­mark, Nor­way, and Wales. He re­ceived the On­sager Medal in 2000 from the Nor­we­gi­an Uni­versity of Sci­ence and Tech­no­logy. In 2002, he was made a Dis­tin­guished Com­pan­ion of the NZ Or­der of Mer­it DCNZM, later re­des­ig­nated Knight Com­pan­ion KNZM. The same year, he be­came an hon­or­ary mem­ber of the Lon­don Math­em­at­ic­al So­ci­ety. The Jones Medal of the Roy­al So­ci­ety of New Zea­l­and Te Apārangi is named in his hon­or.

Vaughan had a strong com­mit­ment of ser­vice to the com­munity. In 1994, he was the prin­cip­al founder and Dir­ect­or of the New Zea­l­and Math­em­at­ic­al Re­search In­sti­tute, lead­ing sum­mer schools and work­shops in New Zea­l­and each Janu­ary. He was Vice Pres­id­ent of the Amer­ic­an Math­em­at­ic­al So­ci­ety in 2004–2006, and Vice Pres­id­ent of the In­ter­na­tion­al Math­em­at­ic­al Uni­on in 2014–2018.

Vaughan had an un­usu­al and very per­son­al style of do­ing re­search. He would freely share ideas about a pro­ject and dis­cuss ini­tial spec­u­la­tions and pos­sible ap­plic­a­tions and con­crete steps for how one might ob­tain the fi­nal res­ult. Vaughan was a warm and gregari­ous in­di­vidu­al whose hu­mor and hu­mil­ity led to the gen­er­os­ity and open­ness from which the math­em­at­ic­al com­munity drew sub­stan­tial be­ne­fit. Vaughan had over 30 gradu­ate stu­dents and was a sought-after doc­tor­al ad­visor. His pres­ence at math­em­at­ic­al events was stim­u­lat­ing for all who came in con­tact with him. He will be dearly missed by his fam­ily and his many friends all over the world.

[Ed­it­or’s note: The text above is from the In­tro­duc­tion of “Memor­ies of Vaughan Jones” pub­lished in the No­tices in Oc­to­ber 2021. For the full art­icle, click on the PDF link at the up­per right of this page.]