Celebratio Mathematica

Vaughan F. R. Jones

Vaughan and the gift of friendship

by Sorin Popa

It has been al­most four months since Vaughan Jones un­timely death, yet I am still strug­gling to come to terms with the ter­rible real­ity that he is no longer with us, and to cope with the last­ing pain of los­ing one of my closest friends. He would have been 68 this Decem­ber 31st, the very last day of this tra­gic year.

Much has been writ­ten on Vaughan after he passed away Septem­ber 6th, and more will come. There will be an AMS No­tices me­mori­al art­icle on Vaughan, with con­tri­bu­tions from a large num­ber of col­leagues and friends, shar­ing per­son­al memor­ies and com­ment­ing on his math­em­at­ic­al leg­acy. Sev­er­al con­fer­ences and journ­al volumes in his memory are sched­uled for next year, and a de­tailed de­scrip­tion of his work will ap­pear in an is­sue of the AMS Bul­let­in. I will par­ti­cip­ate in some of this and thus have plenty of op­por­tun­ity to con­trib­ute my thoughts on why Jones in­dex and sub­factor the­ory was com­pletely re­volu­tion­ary, how this led Vaughan to the dis­cov­ery of the Jones poly­no­mi­al in­vari­ant for knots and links, the strik­ing con­nec­tions with oth­er areas that fol­lowed, and the re­mark­able im­pact his work has had on sev­er­al fields of math­em­at­ics ever since then.

The testi­mon­ies from fam­ily and friends show how much more than this Vaughan ac­tu­ally was. But while they re­late his pas­sion for sports, rugby and kite surf­ing fore­most, but also ski, ten­nis, golf, sail­ing, etc., about Vaughan as a barista, about his mu­sic­al tal­ent and great bari­tone voice, these stor­ies may say too little about how gen­er­ous and al­tru­ist­ic he was with all people around him.

Vaughan cher­ished friend­ship, and the num­ber of friends he had, all over the world and from all peri­ods of his life, start­ing with his early school years, is amaz­ing. Per­haps this came from his genu­ine care and in­terest in people, his deep em­pathy for oth­ers, and a joy for shar­ing.

Al­most every time I was at a con­fer­ence with Vaughan, in Europe, US or New Zea­l­and, he would dis­ap­pear for a day or so to vis­it friends who lived in the area. I of­ten wondered wheth­er he had friends in all cit­ies, or wheth­er he only at­ten­ded con­fer­ences in places where he had friends…. Or, he would take a few days after a math­em­at­ic­al meet­ing to vis­it a friend in an­oth­er city, by train or by car, be­fore fly­ing back home. Of­ten this was be­cause that friend was passing through a bad peri­od of life: “He is in shambles, needs some shoul­der­ing.” Some­times such a de­tour re­quired an ef­fort, but he would do it any­way.

Vaughan vis­it­ing us in Ro­mania, at the Math In­sti­tute in Bucharest, in June 1984, is quite telling. More than “a friend in need”, this was about a whole math­em­at­ic­al com­munity in need. People in our siz­able func­tion­al ana­lys­is group, op­er­at­or al­gebra and op­er­at­or the­ory com­bined, were no longer al­lowed to go to math­em­at­ic­al meet­ings out­side of Ro­mania after the fall of 1981. The com­mun­ist re­gime froze all per­mis­sion to travel to sci­entif­ic events. We were like be­hind an iron cur­tain walled up be­hind the usu­al iron cur­tain…. Luck­ily, cor­res­pond­ence by mail was pos­sible (there was of course no email at that time), and we did man­age to have some sci­entif­ic con­tact by or­gan­iz­ing an in­ter­na­tion­al con­fer­ence in the sub­ject al­most every year. But those were ex­cit­ing times for our sub­ject, there were fre­quent meet­ings all over the world. Not be­ing able to at­tend them, in a peri­od where this way of com­mu­nic­a­tion was cru­cial, was ex­tremely frus­trat­ing. We cer­tainly felt like we were be­ing “locked up”.

Vaughan and I kept a reg­u­lar cor­res­pond­ence from March 1981 through April 1987. This star­ted with a joint math pro­ject, but then it was mostly up­dat­ing each oth­er on our work and work of oth­ers around us, and dis­cuss­ing prob­lems. Once we, in Ro­mania, be­came isol­ated, that cor­res­pond­ence be­came very im­port­ant. Vaughan kept us this way in touch with his dis­cov­er­ies, even be­fore pre­prints were cir­cu­lated. But we really wanted him to vis­it, so we could talk at length. We or­gan­ized a Con­fer­ence in Busteni (a moun­tain re­sort in Ro­mania) in Septem­ber 1983 and he was cer­tain to come, but had to can­cel in the last mo­ment.

His vis­it was post­poned to June 18–25, 1984. This was planned for months, and we were wait­ing for his vis­it with a cer­tain amount of ex­cite­ment.

At this point I should re­call that Vaughan had his amaz­ing math­em­at­ic­al bout dur­ing pre­cisely those years, start­ing with the in­dex of sub­factors break­through in Oc­to­ber–Novem­ber 1981, fol­lowed by his dis­cov­ery of rep­res­ent­a­tions of the braid groups in the tower of factors, in June 1982. And then, as it hap­pens, by his dis­cov­ery of the poly­no­mi­al in­vari­ant for knots, some time at the end of May 1984! There was of course no way for us to know about this very last one, when wait­ing for him at the air­port just a couple of weeks later.

Vaughan wrote to me that he would take a flight from New York to Bucharest, with a two-day stop in Geneva to vis­it friends, ar­riv­ing in Bucharest June 18. I went with Mi­hai (Pims­ner) to pick him up at the air­port. Mi­hai bor­rowed his fath­er’s car, we parked, then went to the ar­rival gate look­ing for Vaughan to show up. After about one hour of wait­ing and all pas­sen­gers hav­ing got­ten out, we be­came quite wor­ried. We star­ted to run from one agent to an­oth­er say­ing there was a miss­ing pas­sen­ger, and if they could check if he took the plane in Geneva. But in the middle of all that, Vaughan showed up in the tun­nel walk­ing slowly head down while writ­ing on a note book. Get­ting closer, we no­ticed he was draw­ing knots, with scribbled cal­cu­la­tions on the side! He raised his head and in­stead of hello said calmly “Sor­in, Mi­hai, I got a really big res­ult”. That’s how we learned about the Jones poly­no­mi­al, at the Otopeni Air­port in Bucharest, then in the car driv­ing Vaughan to the hotel. Talk­ing math was only in­ter­rup­ted when he no­ticed the ter­rible de­moli­tions all over Bucharest “My God, this looks worse than Beirut”. It sum­mar­ized well the situ­ation of a city and people un­der siege….

Vaughan gave two great talks at the in­sti­tute, the first one on the in­dex for sub­factors, the second one on the poly­no­mi­al in­vari­ant for knots. Between the end of May, when he ob­tained the res­ult, and his ar­rival in Bucharest, June 18, Vaughan had already done tons of cal­cu­la­tions, de­riv­ing many strik­ing con­sequences. We were in awe, with this unique feel­ing of wit­ness­ing a ma­jor math­em­at­ic­al dis­cov­ery. It was just his second talk and pub­lic an­nounce­ment of this res­ult, with the only pre­vi­ous lec­ture giv­en in Geneva, a few days be­fore, when stop­ping on his way from NY to Bucharest!

Vaughan’s vis­it was a re­in­vig­or­at­ing, ma­jor event for us all. We dis­cussed math­em­at­ics frantic­ally all day long. Mi­hai and I had worked on vari­ous ana­lys­is as­pects of sub­factor the­ory since the fall of 1982; we ob­tained a num­ber of in­ter­est­ing res­ults which had their im­pact in that area, so we had things to show as well. There was lots of party­ing and laugh­ing in the even­ings, with a mem­or­able “col­loqui­um din­ner” at the only res­taur­ant in town where qual­ity meat was avail­able. We had beer then switched to wine. But then Vaughan ordered beer again. As I cau­tiously sug­ges­ted this may not be good, he replied “Oh, don’t worry, I do this all the time.” The second morn­ing when I went to the hotel to take Vaughan to the in­sti­tute, he was very very sick, blam­ing… mix­ing beer with wine! “But Vaughan, you said you do this all the time!” Des­pite be­ing really sick, he whispered with a laugh “Yeah, and all the time it hap­pens like this.”

We con­tin­ued our cor­res­pond­ence for sev­er­al more years after his vis­it. I would copy-xer­ox Vaughan’s let­ters upon re­ceiv­ing and cir­cu­late them in our group like art­icle pre­prints. Dur­ing the 1984–1985 MSRI year, which had two pro­grams, one called “Sub­factors” the oth­er “Knot The­ory”, Vaughan or­ches­trated an “of­fi­cial let­ter” to Ro­mani­an au­thor­it­ies re­quest­ing that Picu (Voicules­cu), Mi­hai and my­self be al­lowed to hon­or the in­vit­a­tions we re­ceived from MSRI. It was signed by Cal Moore, MSRI’s Deputy Dir­ect­or, and Joan Birman, one of the Pro­gram Dir­ect­ors, and ad­dressed to the Min­is­ter of Sci­ence and Tech­no­logy (Ceau­ses­cu’s wife, of all people!). To no ef­fect. None of us could in fact ever travel again, un­til we left Ro­mania for good.

My cor­res­pond­ence with Vaughan went on all the way un­til I man­aged to leave with my fam­ily, in May 1987. Picu had already left in the Sum­mer of 1986, and Mi­hai in the spring of 1989. Vaughan’s vis­it and our cor­res­pond­ence were def­in­itely im­port­ant on the math side. But in ret­ro­spect, I think that the friend­ship part in all this was way more im­port­ant. Dur­ing sev­er­al years, throughout a very in­tense re­search peri­od of his ca­reer and des­pite hav­ing many com­mit­ments and a very busy peri­od of his life, Vaughan took his time to write all those long let­ters and to come vis­it us for a week, be­hind the iron cur­tain. That was quint­es­sen­tial Vaughan Jones.

Sor­in Popa re­ceived his PhD in 1983 from the Uni­versity of Bucharest, Ro­mania, with Dan Voicules­cu as his ad­viser. He has been a pro­fess­or of math­em­at­ics at UCLA since 1987, where he now holds the Yuki, Ky­oko and Masami­chi Take­saki Chair in Op­er­at­or Al­geb­ras. He was an in­vited speak­er at ICM 1990 in Kyoto and a plen­ary speak­er at ICM 2006 in Mad­rid. He re­ceived a Gug­gen­heim Fel­low­ship in 1995, the Os­trowski Prize in 2009 and the E.H. Moore Prize of the AMS in 2010. In 2013 he was elec­ted to the Amer­ic­an Academy of Arts and Sci­ences. Popa is an ana­lyst work­ing in op­er­at­or al­geb­ras, sub­factor the­ory, group the­ory and er­god­ic the­ory, es­pe­cially on ri­gid­ity as­pects per­tain­ing to these areas.