Celebratio Mathematica

Vaughan Jones was born in New Zea­l­and on Decem­ber 31, 1952. His lifelong in­terest in math and sci­ence began very early. As he liked to re­mem­ber from his first years at school, he dis­covered by him­self that

a hun­dred plus a hun­dred is two hun­dred. These were gi­gant­ic num­bers. I knew that one plus one is two and I de­duced that a hun­dred plus a hun­dred is two hun­dred. That was great. I was the only one in the classroom to have thoughts of this kind. (Trans­lated from [e17].)

He did his un­der­gradu­ate stud­ies at the Uni­versity of Auck­land in the years 1970–73, and at­ten­ded sev­er­al mas­ters-level courses there that he made good use of later. He heard of von Neu­mann al­geb­ras in a course by Mi­chael Len­non, who is still re­membered in Auck­land as an out­stand­ing teach­er. A course by Paul Hafn­er taught him Galois the­ory, which may have in­spired him later when he thought of a sub­factor \( N \subset M \) as an ana­logue of a field ex­ten­sion \( L / K \) [10]. And his first en­counter with knot the­ory may well have been a course by Dav­id Gauld, dur­ing which he and an­oth­er stu­dent cal­cu­lated the Al­ex­an­der poly­no­mi­als of the granny and square knots.

He was al­ways in­ter­ested both in math­em­at­ics and phys­ics. Paul Hafn­er re­mem­bers that

he was clutch­ing Jauch’s book1 on quantum mech­an­ics, point­ing out that this was a ma­jor at­trac­tion. I do not know how he had come across it, maybe Mike Len­non, or pos­sibly people from the Phys­ics De­part­ment?

Just after his MSc, he con­trib­uted a pa­per in the 1974 New Zea­l­and Math­em­at­ics Col­loqui­um: “Di­men­sion the­ory of or­tho­mod­u­lar lat­tices”; the ab­stract is re­pro­duced on page 148 of a his­tor­ic­al art­icle by Garry Tee [e14].

We quote Paul Hafn­er again:

I could tell him that the Swiss Gov­ern­ment sup­por­ted in­ter­na­tion­al aca­dem­ic col­lab­or­a­tion by way of fed­er­al schol­ar­ships (“Bundess­ti­pen­di­en”), with one avail­able to New Zeal­anders. Tra­di­tion­ally, these schol­ar­ships were ‘owned’ by the De­part­ment of Ger­man­ic Lan­guages and Lit­er­at­ures in Auck­land. At an oc­ca­sion where I met the Swiss Am­bas­sad­or to New Zea­l­and, maybe a year pri­or, the Am­bas­sad­or ex­pressed his dis­pleas­ure that there did not ap­pear to be any sci­ent­ists among the ap­plic­ants — now here was my chance to dis­rupt by point­ing Vaughan in the right dir­ec­tion. He ap­plied and was suc­cess­ful.

This is how he ar­rived in Geneva to work with Josef-Maria Jauch, whose main in­terests were the math­em­at­ic­al found­a­tions of quantum mech­an­ics.

In 1974, his first three months in Switzer­land were in Fri­bourg, for a lan­guage course to learn French. “It was the coolest three months in my life,” (les mois les plus sympa de ma vie) as he has said. The course was a real suc­cess: Vaughan be­came al­most bi­lin­gual. A good les­son for all stu­dents and col­leagues who be­lieve that Eng­lish is enough to settle down any­where in the aca­dem­ic world, for­get­ting that think­ing to­geth­er with oth­er people and in an­oth­er lan­guage is both pos­sible and worth­while. It is dur­ing this peri­od in Fri­bourg that he met his fu­ture wife Wendy. Later, in Geneva, they got mar­ried in 1979 (and they cel­eb­rated their forty years in 2019). Vaughan also vis­ited Jauch once, and work­ing in his group looked prom­ising and pleas­ant. But Jauch died sud­denly the next week, and Vaughan had to modi­fy his plans. He star­ted to work with Con­stantin Piron, a col­league and former stu­dent of Jauch, and also with Jean-Pierre Eck­mann. His first pub­lished art­icle re­cor­ded in Math­S­ciNet is from this peri­od (see [1]).

As he con­tin­ued to be in­ter­ested in both math­em­at­ics and phys­ics, he at­ten­ded lec­tures by An­dré Hae­fli­ger, prob­ably one on fo­li­ations and one on de Rham the­ory. Fla­vio da Sil­veira, an­oth­er PhD stu­dent of Hae­fli­ger, re­mem­bers that Vaughan of­ten asked in­ter­est­ing ques­tions and, in the cor­ridor be­fore or after the lec­tures, was of­ten to be seen in the cen­ter of a group of stu­dents act­ively dis­cuss­ing the course. At one point, Vaughan asked An­dré if he could pos­sibly be­come a teach­ing as­sist­ant in the math­em­at­ics de­part­ment. The an­swer was “Yes, I need some­body for my course, and you can start to­mor­row.” This was in Feb­ru­ary of 1976, if I re­mem­ber cor­rectly.

It happened that Mahnaze Ar­d­jomande, the wife of da Sil­veira, also an as­sist­ant in Geneva as a PhD stu­dent of Narasim­han, had de­cided to work with her two sis­ters on a pro­ject to found a sci­entif­ic mu­seum for chil­dren in Te­her­an (this was be­fore the 1979 Ir­a­ni­an Re­volu­tion). The da Sil­veira’s left Geneva for Te­her­an without much no­tice, and thus there were now two va­cant as­sist­ant po­s­i­tions. Vaughan took the one offered by Hae­fli­ger. As I was work­ing in Lausanne and vis­it­ing Geneva weekly, Claude Weber asked me if I knew some­body who could take the oth­er po­s­i­tion; I was in­ter­ested my­self, and star­ted the next week. (Ad­min­is­trat­ive de­tails would fol­low.)

This is how Vaughan Jones be­came the PhD stu­dent of An­dré Hae­fli­ger. Hae­fli­ger has a very strong and im­port­ant place in math­em­at­ics, and he had a strong in­flu­ence on Jones, but he is cer­tainly not a spe­cial­ist of von Neu­mann al­geb­ras; this shows the ori­gin­al­ity of the stu­dent and the open mind of the ad­visor. To­geth­er with the un­of­fi­cial and de­cis­ive co­dir­ec­tion of Alain Connes, the out­come was a very strong thes­is [2]. The main res­ult is a clas­si­fic­a­tion of the ac­tions of fi­nite groups on \( \mathbb{R} \), the hy­per­fin­ite factor of type II\( _1 \), in terms of three in­vari­ants. As Jones writes mod­estly, “the meth­od of proof that I use is a de­vel­op­ment of that of [e3]”, where Connes clas­si­fies the ac­tions of fi­nite cyc­lic groups on \( \mathbb{R} \). Sub­factors of \( \mathbb{R} \) ap­pear already here, as fixed-point al­geb­ras of ap­pro­pri­ate fi­nite group ac­tions ([2], end of Sec­tion 2.1).

I note in passing that Jones spent 6 years al­to­geth­er in Geneva as a PhD stu­dent; this shows that even tal­en­ted people like him may need time to fin­ish their work, des­pite what politi­cians some­times seem to sug­gest.

In Geneva, as in all places, Vaughan Jones was a mar­velous com­pan­ion. His in­form­al style of work­ing en­cour­aged ex­changes of all kinds, and he was al­ways ready to share his ideas. He was not only a first-class math­em­atician, he was also an ex­cel­lent bari­tone and for many years sang in the Uni­versity choir, and en­joyed prac­ti­cing cham­ber mu­sic with his ad­visor An­dré Hae­fli­ger and the choir’s dir­ect­or Chen Li­ang-Sheng. He shared his love of sports with his many friends, and al­ways con­duc­ted him­self with mod­esty, hu­mor, and re­spect for every­body. On one oc­ca­sion, however, he com­por­ted him­self with glor­i­ous pan­ache: dur­ing his thes­is de­fense, he dressed in a su­perb smoking, like a king ad­dress­ing his courtiers — with the jury mem­bers on the front bench: An­dré Hae­fli­ger, Alain Connes, Michel Ker­vaire, and me.

Shortly after his thes­is, Vaughan Jones es­tab­lished his re­mark­able res­ult con­cern­ing sub­factors in a factor of type II\( _1 \) and their in­dices, which are pos­it­ive num­bers in the uni­on of the dis­crete set \begin{equation}\label{eqon} \{ 4 \cos^2 (\pi/n) \}_{n \ge 3} = \{ 1, 2, (3 + \sqrt{5})/2, 3, \dots \} \end{equation} and the con­tinuum \( \mathopen[ 4, \infty \mathclose] \). His first res­ult for in­dices was the ex­ist­ence of a first gap, between 1 and 2, a real sur­prise since any­thing known so far re­lated to di­men­sions and factors of type II\( _1 \) was con­tinu­ous; very wisely, and fol­low­ing the ad­vice of Connes, Jones did not pub­lish this pre­lim­in­ary res­ult and worked out the com­plete pic­ture as it ap­pears in his spec­tac­u­lar art­icle [4], and already in the pre­vi­ous an­nounce­ment [3]. The start­ing step in the proof is to as­so­ci­ate to a pair \( N \subset M \) of a factor and a sub­factor a pro­jec­tion \( e_N \) from \( M \) onto \( N \), and the pair \( M \subset \langle M, e_N \rangle \); the al­gebra \( M_2 := \langle M, e_N \rangle \) gen­er­ated by \( M \) and \( e_N \) makes sense be­cause both \( M \) and \( e_N \) act on a Hil­bert space which is a nat­ur­al com­ple­tion of \( M \); it is shown that \( M_2 \) is a fi­nite factor and that the in­dex of \( M_1 := M \) in \( M_2 \) is equal to the in­dex of \( M_0 := N \) in \( M \). The next step is to it­er­ate this con­struc­tion to ob­tain a nes­ted se­quence of II\( _1 \) factors \[ M_0 = N \subset M_1 = M \subset M_2 = \langle M_1, e_N \rangle \subset M_3 = \langle M_2, e_2 \rangle \subset \cdots \] and a se­quence \( e_1 = e_N, e_2, e_3, \dots \) of pro­jec­tions (\( e_i^2 = e_i \), and \( e_i \) self-ad­joint) which among oth­er things sat­is­fy \[ e_i e_{i \pm 1} e_i = \tau e_i, \hskip.5cm e_i e_j = e_j e_i \quad\text{if} \hskip.2cm \vert i - j \vert \ge 2 , \] where \( \tau \) is the re­cip­roc­al of the in­dex of \( N \) in \( M \). The fi­nal step is an ana­lys­is of the al­gebra gen­er­ated by the \( e_i \)’s, which gives the pos­sible val­ues of \( \tau \).

I like one thing that he says in [11] about his work on the sub­ject of [4]:

In this re­search for in­dex of sub­factors, I think I spent at least 80% of my time on that try­ing to prove something that I thought was true and turned out to be wrong.

This shows a way in which some of his res­ults were in­deed a sur­prise, even for him.

One day, after lec­tur­ing on sub­factors in Geneva, he spent some time in dis­cus­sion with Didi­er Hatt-Arnold, at this time a PhD stu­dent of Ker­vaire and now a mu­si­cian in Ber­lin. Didi­er poin­ted out that some of his equa­tions sug­gest a re­la­tion with braid groups, and he star­ted to teach Vaughan about braids; I ad­ded a ref­er­ence to a sur­vey by Mag­nus [e2]. The point is that Al­ex­an­der had already shown in 1923 that any knot can be ob­tained as a closed braid and that, for each pos­it­ive \( n \), braids on \( n \) strands con­sti­tute a group \( B_n \). The fol­low­ing present­a­tion \begin{equation}\label{eqtw} B_n = \biggl\langle \sigma_1, \dots, \sigma_{n-1}\biggm| \begin{aligned} & \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} \text{ for} \hskip.2cm i \in \{1, \dots, n-2 \} \\ & \sigma_i \sigma_j = \sigma_j \sigma_i \text{ for} \hskip.2cm i, j \in \{1, \dots, n-1 \} , \hskip.1cm \vert i - j \vert \ge 2 \end{aligned} \biggr\rangle , \end{equation} ap­pears in an art­icle of 1925 by Emil Artin. Jones un­der­stood that what is form­ally com­mon between \eqref{eqon} and \eqref{eqtw} in­dic­ates an in­ter­est­ing re­la­tion between the al­gebra \( TL_n \) gen­er­ated by \( e_1, \dots, e_{n-1} \), which is called the Tem­per­ley–Lieb al­gebra, and the group \( B_n \). More pre­cisely, there is for each \( t \in \mathbf C^* \) a ho­mo­morph­ism \( r_t \) from \( B_n \) to the group of units of \( TL_n \), giv­en by \( r_t(\sigma_i) = \sqrt{t} ( te_i - (1 - e_i)) \). There is a nat­ur­al trace on \( TL_n \) and the Jones poly­no­mi­al \( V_L (t) \) of an ori­ented link \( L \), which is the clos­ure of a braid \( b \in B_n \), is defined by \( V_L (t) = (-(t+1)/ \sqrt{t} )^{n-1} tr \left( r_t(b) \right) \) [5]. This geni­us dis­cov­ery owes a lot to a cru­cial con­ver­sa­tion with Joan Birman at Columbia Uni­versity; see also [13]. There have been many ex­pos­i­tions of parts of Jones’ new ideas; let us quote one by his un­of­fi­cial thes­is co­dir­ect­or [e6] and one from sem­inars in Bern and Geneva [e7].

The Jones poly­no­mi­al marked a new peri­od in knot the­ory, and more gen­er­ally in low-di­men­sion­al to­po­logy. Some of the early con­sequences were the solu­tion by Kauff­man, Mur­as­ugi, Menasco, and Thistleth­waite of con­jec­tures on knots for­mu­lated by Peter Tait in art­icles pub­lished between 1876 and 1885; see [e11], [e9], [e10], [e13], [e15]. The ori­gin of these con­jec­tures was Tait’s at­tempt to tab­u­late all knots; for this it is use­ful to es­tab­lish that ap­pro­pri­ate pro­jec­tions of a knot on a plane show the min­im­al pos­sible num­ber of cross­ings. More re­cent tab­u­la­tions use, among oth­er things, much of Tait’s strategy, plus the Jones poly­no­mi­al to dis­tin­guish knots. “After over 25 years of la­bor­i­ous hand­work, Tait, Kirk­man, and Little had cre­ated a table of al­tern­at­ing knots through 11 cross­ings and non­al­tern­at­ing knots through 10 cross­ings”; the quo­ta­tion is from [e16], which de­scribes the work lead­ing to a table of all prime knots with 16 or few­er cross­ings.

The Jones poly­no­mi­al was very soon gen­er­al­ized. This gen­er­al­iz­a­tion was so nat­ur­al that it was dis­covered sim­ul­tan­eously and in­de­pend­ently by five sets of au­thors. Four re­search an­nounce­ments were sub­mit­ted to the Bul­let­in of the AMS with­in a few days in late Septem­ber and early Oc­to­ber 1984. They were pub­lished as one pa­per, with an in­tro­duct­ory sec­tion writ­ten “by a dis­in­ter­ested party” [e5]. The work of the fifth group was slow in reach­ing the United States, be­cause of poor mail ser­vice (Po­land was ruled un­der mar­tial law between 13 Decem­ber 1981 and 22 Ju­ly 1983), and was pub­lished later [e12]. Jones’ own ac­count ap­peared as [7]. The gen­er­al­iz­a­tion as­so­ci­ates to an ori­ented link \( L \) a Laurent poly­no­mi­al in two vari­ables \( P_L(\ell, m) \); the poly­no­mi­al \( P_L \) has spe­cial­iz­a­tions \( V_L \) and the Al­ex­an­der poly­no­mi­al of \( L \), but con­tains strictly more in­form­a­tion than these two [e8].

Knot poly­no­mi­als had also im­pact on bio­logy. As An­drzej Stas­iak re­mem­bers:

The fame of Jones’ and oth­ers’ poly­no­mi­als co­in­cided with the first bio­lo­gic­al stud­ies of DNA knots in which pre­cise knot types were de­term­ined; see for ex­ample [e4] […]. Bio­lo­gists and math­em­aticians star­ted to at­tend the same meet­ings, as math­em­aticians liked to see real life knots and wanted to un­der­stand how they form and what may be their func­tion. More re­cently though, when we can work with co­ordin­ates of thou­sands of pro­teins, we can search for lin­ear knots in them and char­ac­ter­ize knots res­ult­ing from their clos­ure. As this work is done with thou­sands or even mil­lions of con­fig­ur­a­tions (each sub­chain is ran­domly closed hun­dreds of time) all con­fig­ur­a­tions are ana­lyzed by com­puters and poly­no­mi­als are cal­cu­lated to de­term­ine the knot type. However in such a work we used HOM­FLYPT poly­no­mi­als; see for ex­ample [e18].

There have been fur­ther fas­cin­at­ing res­ults show­ing these in­ter­ac­tions between op­er­at­or al­geb­ras and low-di­men­sion­al to­po­logy. We will only al­lude to the found­a­tion of the the­ory of planar al­geb­ras [8] and an ana­logue of Al­ex­an­der the­or­em show­ing that “the Thompson group is in fact as good as the braid groups at pro­du­cing un­ori­ented knots and links” [12]. For more on the im­port­ance of these res­ults, see [e19] and [e20].

Vaughan Jones has had a deep in­flu­ence on people in Geneva. For me in par­tic­u­lar, after the time of his thes­is, I had the chance of be­ing near him on many oth­er oc­ca­sions: when he spent sab­bat­ic­al peri­ods in Geneva; in IHÉS; in large groups after lec­tures by Alain Connes in Par­is around Leffe beers; in Berke­ley and MSRI; cowrit­ing a book with Fred Good­man and him; or hik­ing around. Moreover, it is amaz­ing to real­ize that he was so in­flu­en­tial at the same time in so many oth­er places — New Zea­l­and and the US of course, but also France, Eng­land, Wales, and oth­er coun­tries. To all of us, his vis­its were gifts.

It was Vaughan Jones’ pleas­ure to draw a pic­ture of his be­loved type II\( _1 \) factor show­ing a big fat black point (see [6]). In Geneva and else­where, all his col­leagues and friends miss him, and will re­mem­ber him as a large, bright, friendly sun.

I am grate­ful to Jean-Pierre Eck­mann, Dav­id Gauld, Fred Good­man, An­dré Hae­fli­ger, Didi­er Hatt-Arnold, Paul Hafn­er, Fla­vio da Sil­veira, An­drzej Stas­iak, and Claude Weber for hav­ing shared their memor­ies with me, and checked mine.