Prof. Paul H. Rabinowitz was born at Newark City, New Jersey in 1939. He attended New York University both as an undergraduate and a graduate and received a Ph.D. in 1966. Between 1966 and 1969, he was an Assistant Professor at Stanford University. Since 1969, he has been a faculty at the University of Wisconsin, Madison. He has worked in the field of variational methods of partial differential equations, and introduced basic theorems such as global bifurcation theorem and mountain pass theorem. He was elected as a member of National Academy of Science and awarded George David Birkhoff Prize in 1998 and Julius Schauder Medal in 2014.

TPL: Thank you for coming all the way for this meeting. I would like to ask you first about your famous theorem of bifurcation. Of course bifurcation phenomena are well-known. So people were aware of bifurcation phenomena before you proved this global theorem. How did this result come about to you?

PR: Since it was so long ago, I don’t remember all of the details. The story begins with my thesis. As part of the thesis, I looked at a nonstandard bifurcation problem where you are bifurcating from an infinite dimensional null space. Then I went to Stanford where Dave Gilbarg ran a PDE seminar. Dave received a preprint by a German mathematician named Welte who proved the existence of Taylor vortices using a rather novel argument from degree theory. Gilbarg asked one of the young faculty members to speak on this in the seminar. After agreeing to do so, this person had second thoughts because of a lack of knowledge of degree theory. I didn’t know much about degree theory either but I agreed to talk on it. After doing so, I realized that in fact you didn’t have to use degree theory but instead could employ a simpler variant of an argument that I used in my thesis. That was a key step towards my later work with Mike Crandall on the local question of bifurcation from a simple eigenvalue. The global result was an outgrowth of that. Motivated by the work of several people who had obtained existence results for nonlinear Sturm-Liouville problems, Mike and I used a mixture of the local bifurcation results and continuation via degree theory to get such global results for the Sturm-Liouville setting while we were both in Stanford together. Then Mike left for UCLA and a year later I left for Wisconsin. There I learned some useful topological tools from my new colleague, Charlie Conley. So finally combining this new information with some of the earlier work led to the global bifurcation theorem. In the end, the theorem resulted from a mixture of several influences and in having the right machinery.

TPL: That’s a beautiful story. I never heard it before. You have mentioned a number of names. I spent some time in Wisconsin at MRC and Charles Conley is a good friend of Joel Smoller, my dear thesis advisor. So you had contacts with Charles Conley.

PR: Right, he was my good friend and long time colleague in Wisconsin.

TPL: What kind of things did you talk about with him?

PR: We would talk about various mathematical problems. I would often have lunch with him. He had a very geometrical way of looking at problems. If you met him you know that. He liked to say the picture is the proof. I enjoyed very much his way of looking at things. He knew quite a lot about topology, all facets of topology, from point set through algebraic. He was a very interesting person to interact with.

TPL: I remember one time he told me he was studying algebra. And I asked ”Is that helpful to you?” And he said ”It will never be helpful to me if I don’t know it.” Tracing a little bit further back, were you a student of Jurgen Moser?

PR: Yes.

TPL: What kind of person was he? I met him several times but you must have a more intimate impression of him.

PR: Well, as his student, I have a different impression than somebody who met him in another way. He was very good to me and to his other students. I felt like I was a member of his family. He was very warm, extremely knowledgeable, and had wonderful insight and taste. No matter what question you asked him about mathematics, he got to the essence of it and had something helpful and useful to say.

TPL: Charles Conley was also his student. Were you two students at NYU around the same time?

PR: Charlie Conley was actually his student at MIT. Moser was at MIT for a while before he went to NYU. I believe Charley came with him as a postdoc as part of that move. Moser first came to the US and NYU as a Fulbright Fellow. Then he returned to Germany for a year before coming back to NYU for another year after which he went to MIT for three years. Then he accepted a position at NYU where he remained until he left for Switzerland. He was in NYU for 17 or 18 years. At MIT, I think he had a couple of students, most notably Charlie Conley. At NYU he had many students. When I became his student, I was the fifth one at the same time, which was quite a lot.

TPL: Where were you from? How did you come to mathematics?

PR: I came from New Jersey. I was born in Newark and I went to school there through high school . In deciding on universities and major fields, I considered a few possibilities. One field I thought about was chemical engineering. That was because I had an excellent high school chemistry course. If I had studied chemical engineering, I would have done so in Newark at what was then called Newark College of Engineering. Another area in which I had some interests was biochemistry. But in the end I opted for mathematics so I went to NYU as an undergraduate. I went through their undergraduate program and at the same time, I took some additional advanced courses. Then I decided to continue at NYU as a graduate student because it was really the best place at that time for the kind of mathematics that I was interested in, namely a mixture of analysis and applied mathematics. So I was both an undergraduate and a graduate at NYU and by the time I left, I had tenure as a student!

TPL: A very senior one! What was the period?

PR: I started as an undergraduate in 1957 and received a B.A., in 1961 when I began graduate school. I finished my Ph.D. work at the end of the calendar year 1965 and received it officially in 1966.

TPL: So all these people like Lax, Nirenberg, Joe Keller, Harold Grad and so on, they were all there at that period?

PR: They were all there.

TPL: So that was a great place for PDE.

PR: It was a very special place in analysis, especially PDE, and in applied math. Moser was the only person who did research on dynamical systems there. Actually I was the exception among his students at that time because I worked on a PDE problem.

TPL: But Moser in the latter half of his career moved more to PDE although dynamical systems was always sort of his primary interest.

PR: Already in the late 1950’s, Moser became interested in regularity problems, first for elliptic and then for parabolic PDEs. He found a very useful technique people sometimes call the Moser iteration method or Moser boot strapping method to help prove regularity. His method was rather different from the breakthrough work done earlier by De Giorgi. Moser mainly did this research in the sixties. He had very broad interests in analysis. He worked on many questions in analysis arising in areas like complex geometry, spectral theory for quasiperiodic potentials, and completely integrable systems such as KdV. (16:14)

TPL: The Courant Institute has been evolving quite a bit since then and also PDE has been evolving quite a bit. So what do you think of PDE in the past and in the future?

PR: When I started, there had been a significant development of the linear theory for PDEs. For linear elliptic theory, this was due to the work of people like Agmon- Douglas-Nirenberg and many, many others. What was quite open then were many basic questions in nonlinear elliptic theory. In particular quasilinear theory was undergoing a great development in those years due to the efforts of many people. This research was mainly in an analytical direction. I became interested in something between the analytical and topological directions. For example, one influence on me while I was a graduate student was a course taught by Jack Schwartz on nonlinear functional analysis. That subject was just beginning as a unified field of study. It dealt with the development of tools to treat nonlinear problems. This course led to a set of lecture notes that became one of the early books in that direction.

TPL: Dunford-Schwartz?

PR: Dunford-Schwartz was a bit earlier and became the definitive treatise on linear functional analysis. Nonlinear functional analysis (which has morphed into what is called nonlinear analysis these days) was the development that would be useful in treating nonlinear PDEs. I became interested in it. Bifurcation theory and variational methods are part of that field. It was a good time to work on such questions.

TPL: So what is the hope for the future?

PR: A lot of PDE areas have been extensively worked over. But in PDE, there is always an interplay between theory and applications. When the theory has been developed extensively, one has to look more closely at specific applications. This, in turn, leads to the further development of the theory. Thus PDE has moved strongly in two directions: to geometry and to applications in several areas of science. The central area is not worked on so heavily anymore. Of course there are other very active directions of research such as the intersection of probability with PDE. These new developments are good for the field and will lead again to more general theory.

CNC: Just to follow up on what you have said, you and Tai-ping are editors of certain very good journals so you have a chance to see a lot of good papers. So I would like to ask from your experience, can you see what would be the trend for PDE from what you see from the results in recent years done by many people?

PR: I don’t follow geometry very much, but in the spirit of what I just said, you see lots of applications to problems in continuum mechanics in various directions such as to water waves and kinetic theory. Likewise there have been extensive developments in the direction of applications to mathematical biology. There are new kinds of problems that one encounters here and eventually, with enough experience with specific problems, they will undoubtedly lead to new theories.

CNC: Tai-ping also has some thoughts?

TPL: Today we are only giving Paul a hard time.

CNC: When I went to Madison, I didn’t know what to do for my graduate study. I talked to some first year graduate students. They already had their thesis advisors in mind, so I thought they had earlier plans to go to Madison. I was not in that category. As you said, you had been in Courant since you were an undergraduate, so you knew many good people in the graduate school and probably had a plan to pick someone as your thesis advisor.

PR: It was more complicated than that. As a graduate student, the first thing you focus on is that you have to pass your Ph.D. qualifying exams. After clearing that hurdle, some people seemed to think they were going to work with X but it is not that simple because X might not take you on as his student unless you have performed sufficiently well. I didn’t have any preconceived notions about with whom I was going to work. After one passed the qualifying exam, you talked to your friends about what research the various faculty members were doing and you talked to some of your teachers to get some further advice. Then you shopped around a bit. I followed this course of action. I remember that I talked to Friedrichs, Moser and a couple of other faculty members. The person who I felt would be the most interesting advisor was Moser and he was willing to take on one more student on top of the four he already had. That’s how the process worked in my case. It was definitely a shopping- around process, with no commitment with anybody beforehand.

TPL: So your problem of choosing advisors was not unique.

PR: In Madison, if somebody came up to me and said I want to work with you, I would definitely not accept them before at least they had some courses with me or had taken a reading class with me because you have to get an impression of the person first hand and to estimate if they will be able to succeed.

TPL: You mentioned Friedrichs. What kind of person was Friedrichs? I talked to him only once on N-waves for hyperbolic conservation laws.

PR: I didn’t know him very well. He was a very senior member of the faculty at that time. One semester I was a grader for him. It was a course in real analysis. I recall that he was a very precise lecturer. For my tastes, he was a little too precise because excess precision takes some of the geometric insight away. But he was certainly a great mathematician and he did excellent work in many different areas. He was an extremely organized person. I recall a good anecdote about him. If you wanted to see him, you had to make an appointment; you couldn’t just go to his office and talk to him. In fact even at home his family members had to make appointments to talk to him during ”working” hours. Friedrich’s mother-in-law lived with the family. One day she discovered that a fire had started somewhere in the house. She came rushing to him calling out, ”Friedrichs! Friedrichs! I have to see you.” to which he replied, ”Tuesday at 3 pm.” I don’t know how true it is, but it makes for a colorful story!

TPL: One time I talked to Joe Keller and said that people always say Friedrichs is slow. But Friedrichs is a great mathematician, he can’t be slow. Joe Keller said that’s right and one time at a seminar, Friedrichs asked a question and Joe Keller knew the answer and immediately answered the question. After the seminar Friedrichs said to Joe Keller, ”I knew the answer also but maybe the young people didn’t know the answer. ”

PR: That’s nice.

TPL: You did your mountain pass work with Ambrosetti after the bifurcation results?

PR: Yes. I became somewhat interested in the calculus of variations when I was completing my thesis. There was a young postdoc, Melvin Berger, who arrived at Courant a few months before I finished and who was very enthusiastic about the subject. Berger later wrote a couple of books in that direction. He had lots of interesting things to say and got me interested in the field. Sometime after I did the bifurcation work I looked a bit more closely at some variational arguments. It’s another example of serendipity at work in that I was looking at some such results when I went on sabbatical for a year. I spent three months of the sabbatical in Pisa where I was invited by Giovanni Prodi. In Pisa, I shared an office with a young guy named Ambrosetti. He had been looking at some similar questions from a slightly different point of view. So we started to talk about these things together and the mountain pass theorem was an outgrowth of that interaction. There were some earlier results in simpler settings that were not so far from the mountain pass theorem and one can say that it was something that could have been done earlier because the tools were all there. I guess nobody encountered the right set of applications, such as the particular problems that we were looking at and that led us to the theorem.

TPL: Beautiful, I see. Because of the application, you felt that this really should be done.

PR: Yes, one could see there was a common feature in some of the problems we were studying. We explored the setting both in problems without symmetry where you had the mountain pass theorem per se as well as symmetric analogues where you could get results on multiplicity of solutions for various kinds of variational problems. So things came together and in our joint paper you will find theoretical results and applications of both types, the mountain pass theorem and its symmetric version. So again, like the work on bifurcation, it was a matter of being in the right place at the right time.

TPL: Plus the right people.

PR: It certainly could have been done earlier. The tools were all there.

CNC: Concerning the mountain path theorem, I think many people know it and they have experience working on that kind of problem, using the mountain path theorem to work on certain applications. But then also people know that you have more results on what is called critical point theory. But when you have more experience working on such kinds of critical point theory, then you realize one important ingredient is the Deformation Theorem. Sometimes you don’t even have the Palais-Smale condition but if you have a good enough version of the Deformation Theorem, then you can pick out a good Palais- Smale sequence to get the solutions. So the Deformation Theorem is also part of the idea involved in your work with Ambrosetti.

PR: Yes, if you look at the work published earlier than ours, it was mainly in settings where you had symmetry so there were results using the notion of Ljusternik- Schnirelmann category. They all involve some kind of abstract deformation theorem where you push down level sets using the associated negative gradient flow. If there were no obstructions, i.e. no critical points in the way, then you can deform one level set down to lower levels. That was a tool that we used. I don’t think there was anything especially new at the technical level that we did, rather the novelty in our work was just to show there was a simple, useful set of geometrical hypothesis which gives you the mountain pass theorem and which can be verified in many applications. Stated somewhat succinctly, if you have a local minimum which is not a global minimum and you have some kind of compactness, then you have a critical point.

TPL: You have a long career so far and it’s still going strong. What are the things that you particularly like to talk about concerning your research?

PR: In what sense?

TPL: Whatever.

PR: You always enjoy what you are doing locally in time, but of course there are highlights. You have already touched on some such highlights. Here is another one. Part of my thesis involved a bifurcation problem for a nonlinear wave equation. There were natural global questions associated with it. I tried my hand at them without success at that time. I never let go of the problem in the sense that every once in a while after I learned some new techniques or tools, I would go back and think about the old problem again. About some ten years later, I had gathered enough tools so that I could make another significant step. Using a combination of topological ideas and estimates of critical values and critical points, I could show a highly indefinite variational problem associated with a nonlinear wave operator had a nontrivial critical point corresponding to a time periodic solution of the wave equation. Naturally it is gratifying when something you have tried hard to do bears fruit. I remember soon thereafter, Jurgen Moser made an extended visit to Madison. He heard me talk on this research. Coincidentally he had just received a preprint of some very nice work by Alan Weinstein on finding periodic solutions of prescribed energy of Hamiltonian systems. Moser asked me whether I could apply my new arguments to this sort of problem. I thought about it and when I saw him next, I said it looks to me like one can indeed handle such questions of the existence of periodic solutions of Hamiltonian systems via variational arguments. As far as Moser was concerned, a statement like that was not too valuable. The key thing is whether you can really carry out the details. So Moser said to me ”That would be a very nice application. I will give you a quarter if you can do it”. Several months later I wrote to Moser that indeed, I had all the details.

TPL: Did he give you the quarter?

PR: Of course! He mailed me a quarter, I believe from Sweden. I still have it somewhere.

TPL: That’s a hard earned quarter! (47:47)

PR: So that is how I began working on Hamiltonian systems.

TPL: So you really have had a long association with that problem, starting from your dissertation, accumulating all kinds of knowledge and thinking about it. This story from beginning to the end took over 10 years.

PR: Yes. The thesis was completed late in 1965 and officially accepted in 1966. The papers on nonlinear wave equations and Hamiltonian systems appeared in 1978.

TPL: You give me the impression that you really enjoy in what you are doing.

PR: If it is not fun, why should you do it? That doesn’t mean everything is fun. But you should get considerable satisfaction from it.

TPL: Sure, sure. Do you go back to Courant sometimes?

PR: Very rarely. Visiting Moser was the main attraction for me, so after Moser left, I didn’t visit very often. There was nobody remaining to whom I was especially close, except for Louis Nirenberg, and I would see Louis somewhat regularly at conferences.

TPL: You are an outdoor guy.

PR: In some sense.

TPL: Madison is really beautiful. So you walk around a lot.

PR: I like to walk. I usually walk home from the office.

TPL: How long does it take?

PR: About 50 minutes to an hour.

TPL: One way?

PR: One way, I just go one way. I take the bus in and I walk home.

TPL: How long have you been in Wisconsin?

PR: I went there in 1969. Since I’m still there, it just shows what a lack of imagination I have! Madison is a pleasant place. It’s not too big. It’s not too small. It’s a comfortable place to live and I have always had good colleagues.

TPL: You got to know Ambrosetti well, I suppose. I met him a couple of times. Once at a trilateral meeting between Italy, Taiwan and Australia. And once at a meeting in Rome. Ambrosetti looked to me like a king there and he gave a talk. After his talk people applauded a lot and then I asked, ”Are there any questions?”. Nobody asked questions, unlike your talk this morning when there were lots of questions. So I didn’t know what to do. So I searched for the right word and said to him, ”Everybody is overwhelmed.” And he was obviously pleased.

PR: Very good! That was an excellent choice of words.

TPL: What kind of guy is he? Is he really a tough guy?

PR: Well, Antonio has high standards and his own way of looking at things.. But he has many interests and he has had many many good students. I did some research with one of his students, Vittorio Coti Zelati. Ambrosetti has been very influential in the development of nonlinear analysis in Italy. He may be the most influential person in that direction, certainly in his generation. They have great people in nonlinear PDE like De Giorgi and so forth. But if you look at what you might call nonlinear analysis rather than nonlinear PDE, he has been the main figure there.

TPL: Are there some particular people in Courant or elsewhere among the mathematicians you met that you have a good impression of and about whom you can tell us some stories?

PR: Well Courant is full of exceptional mathematicians. I didn’t know most of them well personally. The ones I knew best were Moser and Nirenberg. Peter Lax was another person who I admired but I wasn’t especially close to him.

TPL: How about Nirenberg? He always has a different kind of humor.

PR: Louis is one of my role models. He’s a wonderful person, warm and generous. He has a great sense of humor, with a story for every occasion. In addition, if you want to know what restaurant to go to or what book to read or what piece of music to listen to, he is the person to ask.

TPL: We have interviewed Peter Lax and Lax told us a story about Nirenberg. When Nirenberg applied to NYU, Courant looked at the undergraduate transcript of Nirenberg and it was straight A’s. And Courant’s immediate reaction was, ”What’s wrong with this guy?”

PR: When I was a graduate student, it was said that at least in Courant’s early days at NYU, you could always get accepted there and get an assistantship if you played an unusual instrument because Courant liked music so much. That would be definitely an advantage. I did meet Courant once and I heard him lecture twice. Once he gave a talk to the undergraduate math club about Hilbert and the other time he substituted for the teacher, Lazer Bromberg, in my course on methods of mathematical physics. Courant gave a talk on the Dirichlet Principle. He made a very favorable impression. In the math club, I remember he told us two nice stories about Hilbert. Periodically Hilbert would meet his assistants and they would go for a walk in the woods. One day somebody noticed that Hilbert had a slight tear in his trousers. As the weeks passed, Hilbert would be wearing the same pair of trousers for the walk and the tear was getting bigger and bigger. The assistants were getting concerned about it and finally they thought it was reaching dangerous proportions and somebody would have to say something. So one bravely said, ”Herr Professor Doctor Hilbert, you just tore your trousers” to which Hilbert replied, ”Oh that, I had it for weeks!” The second story was about a dinner invitation the Hilberts had received. They had just left their house when Mrs. Hilbert noticed that Hilbert’s collar wasn’t in perfect condition. (In those days collars were separate from shirts). She indicated that he should go back inside and change it. So she waited and waited and waited, but no Hilbert. Finally she went back in the house and up to their bedroom. There was Hilbert in bed, fast asleep. What had happened was a typical case of the absent minded professor. He took off the collar. When you took off your collar, what did you do next? You took off your shirt. So proceeding by analytic continuation, he just undressed and went to bed!

TPL: So the change of collar was step one in the iteration of preparing to go to sleep.

PR: As an absent-minded mathematician, he was probably thinking about something else. He just did what he did routinely.

TPL: So soon you are going back to rake the leaves in your backyard?

PR: That’s right. In a few days I will be raking leaves in my backyard unless there is snow; then I won’t have to!

TPL: I remember some years ago when Louis Nirenberg was at Stanford and I asked him what will you do after you retire? He said he planned to read all these great books that he had not had time for. And I took it that it’s not the great books in mathematics but rather in literature. Then several years later, I met him again. He had already retired and I asked him the same question again. He said, ”It did not work out so well. I continue to do mathematics.” I guess that’s what you do most of the time.

PR: My wife says ”You are supposed to be retired. Why are you working?” As I mentioned earlier, it still remains fun. I still enjoy it and as long as I feel that way, I will continue to do it.

TPL: Let’s quit at this point. And you come back and we can get together again in the not too distant future.

PR: I hope so.

• Tai-Ping Liu is a faculty member at the Institute of Mathematics, Academia Sinica.
• Chao-Nien Chen is a faculty member at the Department of Mathematics, National Tsing Hua University.