Department of Mathematics

I learned from Jerry how to treat my PhD students. He was generous above all. After spending lots of time reading his beautiful papers on surgery and knot theory, I decided to work on a problem in singularity theory given to me by Dennis Sullivan. Jerry was OK with that. He encouraged me to go my own way, and his support and gentle criticism were essential to my success.

Jerry was an exact contemporary of mine, and we shared many common mathematical interests. Although I have not had personal contact with him for probably two decades, I remember him vividly as a sweet and gentle person, with a deep engagement with mathematics.

We were among the generation of topologists which managed to exploit powerful methods developed by John Milnor and C.T.C. Wall to attack previously inaccessible problems in high dimensional geometric topology. Jerry was preeminent among us in developing high dimensional knot theory.

While I interacted with him only for brief periods, I learned a lot from personal discussions with him, and from studying his work. He enriched the mathematical life of many others as well, and for this he will be remembered with great fondness and respect.

The following obituary notice will be published in the June 2006 edition of the London Mathematical Society Newsletter:

Professor J.P. Levine (1937-2006)

The distinguished American topologist Jerry Levine died on 8th April, 2006. He was a student of Norman Steenrod at Princeton University, receiving his Ph.D. in 1961. Following appointments at MIT and Berkeley, from 1966 on he was at Brandeis University. In the 1960's and 1970's he was one of the pioneers of the surgery method of the classification of high-dimensional manifolds, their automorphisms and their embeddings, particularly of spheres and projective spaces. The classification of highly-connected high-dimensional codimension 2 knots and the computation of the cobordism groups of high-dimensional codimension 2 knots were particular highlights; he spoke at the 1970 Nice ICM on "The role of the Seifert matrix in knot theory". More recently, he made important contributions to the algebraic and geometric topology of low-dimensional knots and links. Professor Levine was a frequent visitor to the United Kingdom, spending the academic year 1963-64 in Cambridge, and 1972-1973 in Oxford. He had been invited to visit Durham, Edinburgh and Warwick in the current academic year on a Scheme 2 visit of the Society, but his terminal illness prevented him from taking up the invitation.

A.Ranicki

Andrew also adds:

I particularly appreciated the help and support offered by Jerry to my late student Des Sheiham. Indeed, Des's Ph.d. on the computation of the high-dimensional boundary link cobordism group solved one of the problems posed by Jerry and Kent Orr in their survey of knots and links in the Wall 60th birthday volumes. I last met Jerry at the 2004 BIRS meeting on knot theory where he was usual wonderfully lowkey but influential self.

I met Jerry in 1977-8 when he visited Geneva where I was a student. He shared an office with Vaughan Jones opposite mine and was lecturing on higher dimensional knot theory in our 3'ieme cycle. I wasn't aware at the time, but our interests (and physical contact) in math would cross each other in many ways over the following years. I lectured twice in his topology seminar at Brandeis, once during my 83-85 assistant professorship at Yale, and later during my year 96-97 of research with the CNRS. My last physical contact was at UCSD in 2000. Of the mathematicians with whom I ran in contact over the years, Jerry is among those who were closest.

I do not know how to express my sadness with Professor Levine's death. He was one of my heroes in mathematics. Since I started studying knot theory, his papers have been my favorite textbooks, and like many other knot theorists, my results could not have been obtained without his work. Visiting Brandeis and talking with him in person was one of my greatest memories from visiting the U.S. He was always very nice and supportive of me and I could feel kindness and warmth whenever I saw him.

I am not now, nor have I ever been, Jerry Levine's student. Thus began, in a feeble attempt at humor, my short tribute to Jerry at his 60th birthday conference banquet in Tel Aviv, Israel. Countless times I've been asked if I was Jerry's student. Ironically, upon arriving in Tel Aviv the day before, Michael Farber asked over dinner, "You were Jerry's student, weren't you Kent?"

For the record, and with much gratitude, I acknowledge Julius L. Shaneson for that important role in my life. Yet as I ponder Jerry's far-reaching influence on knot theory, and on me, I realize that I might have asked the same question of Michael - or of Cameron Gordon, or Tim Cochran, or Jonathan Hillman, or Stavros Garoufalidas, or De Witt Sumners, or Xiao-Song Lin, or Nathan Habegger, or Pat Gilmer, or Cherry Kearton, or of many others who have been so influenced by Jerry's striking breakthroughs in knot theory that the above question seems entirely reasonable. Jerry set the stage for all of us, and I do not hesitate to state my conviction that he is knot theory's most fundamental and profound influence.

Jerry founded a new era of knot theory during the 1960's, and at each new direction since, he leaped into the field making significant contributions. At the end of his career, Jerry's contributions were still fresh and modern, still deep and penetrating. At age 68, cruel illness cut short Jerry's career and cheated mathematics of his enormous insight and talent.

But like so many others, I feel this loss far more personally.

My respect for Jerry increased steadily with the length of time I knew him. He was a good teacher, a great thesis advisor, and an outstanding colleague. It took me a long time to understand the depth of his sense of humor and humanity. His generosity and gentleness, however, were obvious from the moment I met him.

The road trips to the Cornell Topology Festival stand out in my mind. Typically it involved Jerry deftly moderating the conversations of three obnoxious graduate students, and by the end of those road trips we always knew much more than when we started. More topology usually, but also more about the world, as happened one year when we tuned the radio to the Iran-Contra hearings for most of the trip.

Oh, and running into Jerry and Sandy in the middle of a crosswalk in downtown Seattle at 10 pm one night 5 years ago provided me with the most striking coincidence of my life, and gives me an story I'll be able to use for a long time.

So long, Jerry, you were a major influence on me.

The sad news of Jerry's passing brought my memory back to 20 years ago, in 1986, when I was reading Jerry's paper "An approach to homotopy classification of links" (published in 1988 in Trans. AMS). It was this paper which gave me the inspiration to complete the homotopy classification of links with Nathan Habegger.

In March of 1988, with my family, I drove from Princeton to Boston to visit Jerry. Jerry arranged a joint topology seminar of Brandeis and MIT for me. In Brandeis, I met two of Jerry's students, Gyo Taek Jin and Paul Kirk for the first time. We were all new Ph.D.'s then. What a nice trip! A lot of memories.

As many other people, my career owes a lot to Jerry's generous support. Jerry's example will always reminds me of how we should support the younger generations.

Thank you so much, Jerry.

The first word which comes to my mind when I think of Jerry is 'generosity'. Jerry was extremely generous, especially with his time. As his student I would easily talk for two hours a week with him, and this often right after talking to another student for two hours. How he managed to find the time and energy for this is a mystery to me. I hope that if I ever have students I will remember his generosity and pass it on to my students.

In my meetings with him and in his courses he frequently showed his wit and very dry humour. It took me often a few seconds that Jerry had just cracked a great joke. I am sure Jerry would have been a great poker player since he rarely showed what he was thinking. Only when Jerry was really surprised he would slightly raise his eyebrow, which could then mean anything from `well--done!' to `hmmm, I didn't expect such a mistake'.

As I said, Jerry was very generous with his time and I met him often even after I graduated. I spent several weeks in June 2005 at Brandeis and met him about twice a week. Just before our last meeting I got an email from Jerry saying that he had to cancel the meeting because he didn't feel well. I never saw him again.

I was shocked and saddened to get the e-mail from Kent Orr and learn of Jerry's death. Jerry was a great friend and mathematical inspiration for me. I first met Jerry in Cambridge, England in 1964 when I was a first-year graduate student and he was visiting Cambridge as a guest of Chris Zeeman. Andre Haefliger gave a series of lectures on Jerry's work—I attended these lectures, and appreciated that it was very interesting and important, but it took me a few more years to understand the details in the terrific mathematics in Jerry's work. When I began my career as a research mathematician at FSU after graduate school, it was Jerry's work on knot cobordism that had a major influence on me.

I fondly remember a visit to Brandeis with my family when I was at the Institute for Advanced Study in 1974, and the terrific party Sandy and Jerry gave for us at 39 Dexter Road. Sandy and Jerry were great hosts, and my wife Neddy and I enjoyed hosting them on two visits they made over the years to Tallahassee. Jerry, we will miss you!

I will remember Jerry as a very good friend, always kind and helpful and very inspiring to talk to. After he received the prestigious Alexander von Humboldt Prize in 1989 Jerry was repeatedly a guest of honor at Siegen University. His presence never failed to attract the world elite in knot theory and to create an enormously fruitful and enthusiastic scientific atmosphere.

Jerry has been a mentor, collaborator, and longtime friend during the beginning of my career in Boston. I have very fond memories of him explaining to me topology and geometry, in our frequent meetings in his office at Brandeis. He was very accessible, and would always listen and explain the many things I did not know of. During the six years of my stay in Boston, I would meet with Jerry regularly, twice a week, to discuss mathematics, month after month. Little by little, I became familiar with the breadth and wealth of his research: from surgery theory, to high dimensional knot theory, to low dimensional topology, to problems in slice knots and links.

We would often chat about things, history, politics and people. I have good memory for his sharp sense of humor. I traveled with Jerry to numerous conferences, and visited Israel, Japan, and Korea among other places.

Most of all, what struck me in Jerry was his integrity, and his good nature. Somehow Jerry had a good word to say just about everyone, and would get along with everyone.

Jerry was a good friend. I miss our math and life conversations. I find it hard to believe that he has passed away.

Aiwnia i mnimi (May our memory of him be eternal)

Memorial For A Dear Friend

We first met Jerry when he and Sandy became our next door neighbours, albeit, for a short period, in Great Shelford, Cambridgeshire, U.K. But an instant friendship was formed which has lasted for over forty years.

Jerry had a brilliant mind, a gentle soul and a very ready wit. He will be most sorely missed.

I was devastated by the sad news about Jerry's death. Before 1987 I lived in the Soviet Union and knew only Jerry's papers. I admired Jerry's mathematics and thought of him as of God: his papers were so beautiful and perfect, his style was so elegant and precise.

After I moved to Israel in 1987 Jerry invited me to visit Brandeis in 1988. That was my first ever visit to the US. When I met Jerry I was surprised to see a very "human" person, very intelligent, a little shy, charming, very friendly, sincere, very supportive, humorous and fun to be with. We became friends and later collaborators.

I am not sure if people loved more Jerry's mathematics or Jerry's personality. Jerry was the leader (the word "father" perhaps suits better) of a large area of topology which united many people attracted by his amazing mathematics and by his personal charm. He was clearly the best of us all, the strongest, deepest and most experienced.

Last time I met Jerry at a conference at the Courant Institute in New York in March 2005. He was not yet ill. We had good time together, discussed mathematics, laughed and attended a party. He chaired the session when I gave my talk.

Luckily, names of mathematicians remain in memory of generations. Jerry's name will be honored and praised by many students studying his theorems and enjoying his articles. Mathematical discoveries of Jerry will be alive and will carry his name forever.

I suggest publishing a book of mathematical essays in geometric topology dedicated to Jerry's memory. It could be a durable tribute to him and a useful publication by itself.

When I heard the sad new via email, thousands of feelings, memories, and thoughts passed through my head, making it difficult to know where my story about Jerry should begin. In fact, I have not written anything so personal for anyone else, except perhaps for my wife, whether in English or Korean. The chronological order would be easier to mathematicians.

It was 1979 when I decided to go abroad to pursuit a higher degree and to enter the Ph.D. program at Brandeis after I received the bachelor's degree from Seoul National University in Korea. I recall going to see Jerry on one day toward the end of the first year to get some help in preparing a talk to be given by every second-year graduate student. He suggested that I read the article An Interpretation of G. Whitehead's Generalization of H. Hopf's Invariant by Michel Kervaire, where I learned about the Thom-Pontryagin construction and transversality. They were eventually used as important tools in my thesis, and I now realize that Jerry had a master plan for how to guide me from the beginning. Jerry had me read several papers and books on surgery, including the article Groups of homotopy spheres by Kervaire and Milnor and the book Surgery on compact manifolds by C.T.C. Wall before he gave me the thesis problem of understanding the Cappell-Shaneson work on boundary link concordance via Seifert matrices, which Jerry had applied to knot concordance with great success. I saw him about an hour every week to report my progress and ask questions, sometimes non-mathematical. Even though Jerry had many students at that time, he always tried to be helpful and open-minded. During some semesters, I met him around 11 a.m. and after meeting, we often went to the Sherman Student Center together to have a bowl of chili and crackers as lunch.

I still remember Jerry's delighted face when I told him that I received a job offer from UT Austin and I left Brandeis. I met Jerry on several occasions since then. I invited him twice to Korea in 1988 and 2000, and I recall that he liked Korean food very much even if he enjoyed almost all ethnic food. Jerry kindly invited me to Brandeis twice in 1990 and 1996 when I was visiting the U.S. for my sabbatical years. The most memorable event was a workshop held in Tel Aviv University in August 2001 to pay tribute to Jerry for his contribution as a researcher and a teacher. Many of Jerry's students attended. I am attaching several photos from Tel Aviv, 2001 and Yongpyong, Korea, 2000.

I was fortunate to become a student of Jerry's when he was in his 40s, the most productive period in his career. I think that Jerry was one of the most sophisticated users of algebra in topology. He left a profound influence not only as a distinguished mathematician, but also as a considerate teacher, as a beloved husband, and as an affable father. To me, Jerry will be remembered forever as a professor who taught me how to love the things that I do and the people that I live with.

I was a second year graduate student in Geneva when my PhD advisor, Claude Weber, had the brilliant idea to send me to Brandeis for several months. I will always keep warm memories of this fall semester 2000. Jerry took care of me with great kindness and generosity: he organized my stay and treated me as one of his own students. He soon became a model for me: in his teaching, in his huge knowledge, and in his gentleness.

As so many other people, I owe a lot to Jerry. I feel very fortunate to have met him.

Jerry was my thesis advisor. I will always be grateful to him for allowing me to be his student. Much has already been said about Jerry's remarkable combination of a deeply insightful intellect with a gentle, welcoming spirit. To these beautiful tributes I will only add a small personal recollection. Once, while driving back to Waltham from the M.I.T. topology seminar, the conversation turned to graduate student attrition (a subject in which I was keenly interested, since I was still a graduate student at the time, and definitely not one of the stronger ones). I cannot recall Jerry's exact words, but the gist of it was something like, "I've learned that it's very difficult to tell in advance who will be able to write a Ph.D. thesis." In that conversation, Jerry revealed that his uncommon ability to draw out the best in others was not a mere accident of personality, but the result of a deliberate choice. With his level of accomplishment, Jerry could have chosen to reject all but the most obviously promising graduate students. He chose instead a more subtle path, one that recognized the potential contribution of many, and accepted the responsibility of helping them realize that potential. I found his words encouraging enough at the time, but I am only now, seventeen years beyond my Ph.D., beginning to appreciate the rare depth of the lessons he taught on how to live life as an academic. I miss Jerry terribly, but it is heartening to note, as I reflect on his life and legacy, that I am still learning from him.

When I joined Brandeis, I took up a rigorous study of singularities of algebraic varieties and knot theory with Jerome Levine, one of the finest mathematicians and human beings that I have ever come across, a rather compelling combination in the sociology of mathematics!

My reading courses in Topology with Jerry lasted for 3 years (besides my earlier studies in Tata with Gurjar.) The courses with Jerry were a concentrated study dealing with knot theory, surgery, cobordism, and singularities of algebraic varieties. He would spend great amount of time with me; quite frequently, we would start at 4 pm, and would go on until 7 pm. What started as a topic for the minor exam--which lasts for one semester--continued for 3 years. In fact during this time I had studied more research articles in topology than in algebraic geometry. Two people in the faculty thought I was working with Levine! In fact once Jerry asked me before he was going to a faculty meeting, "what should I tell the faculty?" I was perplexed at this and asked to clarify, then he told me "should I say you are working with me"!! I told him I always was working with Eisenbud and that I will be continuing to do so. In fact I had come to Brandeis because of Eisenbud, but Jerry was a fantastic bonus.

But even after this incident, Jerry never said "sorry I cannot spend time with you anymore as I have to spend time with my students" he went on spending time with me teaching me topology and imparting his unique way of looking at things; it was so geometric, that I felt like an abstract algebraist whenever I was with him. Considering the fact that I was considered a geometer, this was quite something! But he was also powerful in algebraic methods, in fact many of his papers introduces some deep invariant in the algebraic realm that would translate back deep geometric content. But it was his geometric intuition that was spectacular. In all of my years in academics, I have never come across someone who had this geometric intuition. It felt invigorating to be around him. Sometimes when he was explaining the geometric intuition behind the surgery operations, I felt goose bumps, I am not kidding. Usually these kinds of things happen when you listen to great music, but with Jerry this was possible with Topology! Perhaps his insights, always highly imaginative, conveyed to me the common primordial elements also present in music.

He was considered "Mr Knot" and a world class topologist. It was truly a great privilege to have been associated with him. A very giving man, who cared deeply for his students, he would patiently go through many papers in topology, and teach me ways of looking at things. It is a sad fact that some of that is getting rusted because of lack of people in those areas to talk to in Kansas.

I feel deeply for this man, for his great knowledge, for his humanity and for his selflessness that is rare in the rather brutal world of academics. Always utterly calm and collected, he conveyed a deep love for mathematics and for doing mathematics for its own sake. There was one occasion I remember when he got quite excited; I was dreamily musing about applications of knot theory to DNA and chemical bonding and so on, he immediately got excited and got up from his chair and took out a book from his shelf on topological stereochemistry. It was Eisenbud who suggested to me that I minor with Levine, knowing my readings in Tata earlier. To both of them my heartfelt thanks!

Jerry Levine passed away on April 8, 2006 at the age of 68. I wept then and I have tears in my eyes when I think of him now.

I came to Brandeis in 1989 as a fresh post-doc from Germany for a year. Jerry became my post-doc advisor and has been the

most-influential figure in my mathematical life. It seems just like yesterday sitting with him in the Thai restaurant in Boston after he had picked me up from the airport, he liked Thai food a lot. I recall the moment when I told him a proof, which later got published in joint work with Nathan Habegger, and he would think a moment and say: This sounds right but there is still a gap. It took me two days to find the gap and fill it. Or the many times John Klein, who was Kiyoshi Igusa's PhD student at the time, would explain his ideas to Jerry, who would listen carefully and then comment. Or the times when I could sit and listen to James Hughes, Jerry's own PhD student at that time, discuss with his advisor. Or another time when we were sitting together how to call a specific construction, which became strong fusion. Or another time when he would look at me and say: You got to learn to think functorial. Jerry gave me the feeling to be part of research and made me love it, even though I knew and still know my limits very well. But he accepted my limits the way they were and he appreciated my modest capabilities and ideas. He definitely was unique in this respect.

I would have stopped working in research early if there had not been this year with him, if there had not been the invitation for Thanksgiving on that snowy November night 1989, and the seminars at MIT he took me along in his Volkswagen. I recall the many times he would take me to my place in the evenings, or drop me at the supermarket. I still recall the lunch times where he would pay for me in the Faculty club because I was no member. Or the weekend we would go sight-seeing in Boston and have some Greek fast food in Faneuil Hall. Jerry was a great mentor and teacher and human being.

Thank you Jerry for all you did, I will never forget you as long as I live.

Like many people who came in contact with Jerry, my life was affected by him on many levels.  Our relationship was built over many years of talking, usually late in the afternoon with teaching done and graduate students advised. Jerry was endlessly patient and generous with his students, and I felt that I got the same treatment.

As a graduate student, I knew of his famous and influential papers, but I had actually read only one--his magisterial `Knot Modules, I' and really only the portion of that paper related to my thesis work. I was amazed by the depth and sophistication of the algebraic ideas that Jerry brought to bear on what I'd previously thought was the low-tech world of knot theory. It was not to be the first time that Jerry opened my eyes to a much broader vista than I'd previously understood.

 I can’t begin to count how many times, when struggling with some problem (mathematical or otherwise) I thought—“I’ll just go see what Jerry makes of this”. The habit is so deeply engrained that I still occasionally start down the hall to his office.  Jerry had a keen sense of irony, and I’m sure he would have gotten a laugh out of me looking to him for advice on how to organize his memorial conference.

 It’s hard to truly capture the feel of our many conversations as they ranged over mathematics, family, life, and—to be honest—plenty of gossip.  Maybe one story can give something of the flavor of the mathematical side. Once I was trying to compute something in algebraic topology (classifying bundles over projective spaces, if you must know). I was busy making a hash of this in my office and popped down to see if Jerry could help make some order out of my calculations. We drew all over his board for an hour or so, and were ready to give up and go home when he said, “I think I might have done something about this”.  He dug around in his file cabinet, and pulled out an old paper from 1963. When I looked at it later, two things jumped out at me: the elegant and clear solution to my problem, and the thanks to his advisor—Norman Steenrod—for help on what was clearly his PhD thesis. Things like this just seemed to happen in that office, and many visitors to Brandeis over the years came away with fresh insight and new ideas.

As much mathematics as I learned from Jerry, there was even more about life as a mathematician. Jerry was an excellent judge of people, and while he often kept his own counsel, he would sometimes let slip an opinion about colleagues or students as we shared some digestive biscuits (a holdover from his Oxford days) late in the afternoon. I learned a great deal about how to be an advisor from watching him--when to push and when to let someone find their own way.  It wasn’t just students who benefited from his gentle way of dealing with people. I recall one day early in my career when I expressed some doubts about the quality of the work I was doing. Jerry said something understated like `not everybody feels that way’, and in that simple way conveyed something very meaningful.

I can’t resist one other little tale, from my early days at Brandeis.  Jerry and I went one day to the student center for a quick lunch.  When we got to the cashier, she looked at two heads of frizzy hair, and said to me, “It must be so nice to have your father visiting”.  Jerry blushed and was embarrassed, but this remains one of my favorite memories. Jerry was a special colleague, teacher, and friend, and is terribly missed.