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.