Interview with Prof. John Ball

Interview Editorial Consultant: Tai-Ping Liu
Interviewers: Tai-Ping Liu(TPL), Fon-Che Liu(FCL)
Interviewee: John Ball(JB)
Date: October 17th, 2017
Venue: Institute of Mathematics, Academia Sinica

Prof. John M. Ball was born at Surrey, England on May 19, 1948. He graduated from the University of Cambridge in 1969 and earned his PhD at the University of Sussex in 1972. He used to be a professor of mathematics at Heriot-Watt University, and he currently is a professor at the University of Oxford. From 2003 to 2006, he served at the International Mathematical Union as the President. His research interests include elasticity, the calculus of variations, and infinite-dimensional dynamical systems. He was awarded the Theodore von Kármán Prize in 1999 and King Faisal International Prize in 2018. He was awarded the title 'Sir' in 2006.

TPL: I am still recovering from your talk on a very rich subject. You have been working on the calculus of variations for a while?

JB: Yes, since about 1973 maybe.

TPL: As I remember, at earlier times, you must have paid a lot of attention to the work of Morrey, is this so?

JB: Yes.

TPL: Did you ever meet him?

JB: No, unfortunately the year I spent in Berkeley, which was 1979-1980, he’d already begun to suffer from dementia. And in fact, it was kind of sad, because one day I went to the mail room in the math department and there were all Morrey’s reprints. I took one of his quasi-convexity paper. But I never met him.

TPL: Oh. Then many years later, you had a conference to celebrate this important work.

JB: In Princeton we had one, yes, that’s right, when I spent a year at the Institute for Advanced Study.

TPL: What had Morrey found that is so important?

JB: In some sense, from a historical perspective, I sort of rediscovered his work, which was in a paper he wrote in 1950 and also in his book that was written in 1966. And of course, the book contains fantastic material, and actually when you start reading it line by line it’s well written. But its organization is really all over the place. And actually in the introduction it says “this book is not written in its logical order”! And so, somehow there was this fantastic stuff that he had done. And I think the only paper that had referred to it was that of Norman Meyers at Minnesota, who extended what Morrey did to the higher order case. But in some sense it was a great theory without a good example. And I found that elasticity was a great example for his theory and that in some sense regenerated interest in it. So now there are hundreds of papers on quasi-convexity, but we still don’t understand it.

TPL: I see. So you rediscovered Morrey. That’s a great story. Once Nirenberg told a story in Taipei: Morrey was giving a talk in Italy and he said that you need to have this estimate, otherwise you wouldn’t even get to the second base. Morrey liked baseball, but nobody there seemed to understand what he meant by getting to the second base. Italians were not able to appreciate that. Okay, so let’s go back to the usual questions. So how did you get into elasticity?

JB: In Sussex, in the mathematics department. And Serrin and Robert Finn were there. But anyway, I realized I didn’t understand what these great people were doing, but I did understand they were doing something. Also it was something that was relevant to what I was trying to do. Robin Knops from Edinburgh in fact had given a seminar, and I was talking to him afterwards and saying that I didn’t really have a project. And he mentioned this to Stuart Antman, and there was a conference at which Stuart Antman was, and I remember that my friend was driving this mini car from the campus to the railway station, and in the back of the car were me and Stuart Antman and in the front was this friend and Avron Douglis. And Stuart Antman said to me, “ Robin Knops told me you are looking for a project; so this is what you can do, you can go and read this paper of Dickey, and this is what he does and this is what you could do to it.” Okay, so up to this point, I’d never met anybody who had said anything like this to me. I didn’t realize, actually, that there were people who could understand research so well, who could point to a particular paper and say “this is what it does and this is what you can do to it”. So this was an incredible revelation. So there I was with some kind of project which had come from Stuart Antman. And this student managed to get me to be informally supervised by David Edmunds. But I ended up with a DPhil in Mechanical Engineering. So this is how I first got interested in some kind of elasticity.

TPL: How did you then know to read the book of Morrey?

TPL: So, I guess that’s the key thing which cannot be explained. You simply had the insight.

FCL: It’s really remarkable that you say you rediscovered something Morrey has done. Actually when I was in America, I went to America in 1965 and Morrey’s book was published in ‘66 I think. So when it came out, I bought a copy because at that time I was wondering what I should do. I could understand the first four chapters. Those are about Sobolev spaces and certain easy elementary estimates and I think those parts are very interesting, but other than that it was very difficult for me. I could not understand it at all, so I stopped there. So when I heard that you rediscovered something Morrey had done I really feel that’s remarkable. And I want to ask you just one question: you were at Cambridge for some years. How do people there train the applied mathematician?

JB: Well, I was just an undergraduate there, and immediately after my undergraduate degree I went to Sussex. There was a course on the calculus of variations which I took when I was in Cambridge which was quite interesting. Subsequently I have discovered that, taken literally, most of the things I was taught in it were not true. But I enjoyed it and I learned things from it, but of course in Cambridge there was and still is a tremendous division between pure and applied mathematics. I once had a student who actually had a permanent academic job, but he left it. But before he left it he told a very good joke about Cambridge. He was in Bonn giving a seminar, and Stefan Müller, who was my student, was there. He understood the joke, though maybe nobody else did, and he related the joke to me anyway. So this guy was cleaning the board and he says “you know in Cambridge they have a bucket of water by the board. This is for two reasons, first, so they can clean the board, and second, to prove there exists a smooth solution to the Navier-Stokes equations!”

FCL: You know I spent one year in London as a student. From ‘67 to ‘68, and there I noticed that when people were talking about applied mathematics, they were really talking about mechanics. That was my impression.

JB: Mostly fluid mechanics. It was very dominant, but it came from G.I. Taylor, who was of course absolutely brilliant. So in some sense it was his legacy, and there were other big figures like Batchelor and Lighthill. Brooke Benjamin, he was my predecessor in Oxford, left Cambridge because of this attitude. He saw that in France there were things going on with PDE. And anything like that was not appreciated in Cambridge. Well, I had quite a hard time as a graduate student…I mean trying to do rigorous work in mechanics. You didn’t get easily appreciated in the UK at that time. It was viewed as not being interesting - the important thing was physical intuition and not dotting I’s and crossing T’s, as they would say.

TPL: I see, I begin to understand the joke. So proving the global smooth solution to Navier-Stokes equations is looking in the bucket of water…

JB: I doubt if anyone of this great school of fluid mechanics in Cambridge, even up until the 1970’s, knew anything about Leray’s work on the Navier-Stokes equations or would have cared. Brooke would have been probably the first to have thought of these things.

TPL: You were president of IMU, and I learned from you that you tried to do something for the third world. Let’s say Africa in particular and so on. So how is that?

JB: One of the first things that struck me on taking over as president was that the budget for the program for developing countries was so small, I can’t remember what it was, but it was ten thousand dollars a year or something. Absolutely nothing. And so I tried to increase it. Actually the Abel Foundation contributed to this and we organized it a bit better. But it still is a small operation compared to what it should be.

TPL: Suppose that you have your way, what would you do in Africa？That is a very difficult continent to work with.

JB: Well, the principles we tried to operate on, which really came from Herb Clemens who was the person responsible in IMU, was that the people on the ground should be involved in the planning of everything you do. So in other words, you shouldn’t take a kind of colonial attitude to foreign aid and say this is what is good for you. You need to develop it, in conjunction with them. So we were in Africa involved with AMMSI (the African Mathematics Millennium Science Initiative) which was funded by a grant of the Mellon Foundation which Philip Griffiths had obtained. And I am still involved in a resulting project with the London Mathematical Society which involves some kind of pairing up of researchers, mostly in the UK, but in some other countries too, who act as mentors to research groups in Africa. It’s mostly people going there and helping to supervise students and so on. It requires a lot of work. I think one important thing is to get young people involved. Lots of people, young and old, are only too happy to go to countries they’ve not been to and give a course and talk to people. So it really is a question of funding and organization.

TPL: But as you just said the organization should be initiated by the local people. That is a hard part, isn’t it?

JB: You have got to have this. Of course it helps if you’ve got some very good mathematicians in the country. I was recently in Vietnam, where there is Ngô Bảo Châu. He may not always be there but some months of the year he is there, and he is some kind of national hero, and runs a successful institute. So he can really develop things. But in some sense Vietnam was already quite a good mathematical country. In Africa the situation is much more difficult. The internet has given big opportunities for developing countries. In principle if you have a good internet connection then part of the problem of accessing information is solved. I say part of the problem. Certainly in our area, mathematics, you can get your hands on more or less anything on the internet if you take the trouble. It may not be as easy as if you’re in Harvard or someplace like this, but you can find somewhere where you can get hold of it. But it doesn’t mean you know how to use that material when you get it, and that’s part of the problem. So I think one of the things that still needs to be done is to get really good internet connections so that you can start. Traveling around in Africa is not so easy, but you can have research groups that meet by skype and so on and so forth. This can be done.

TPL: You are admired for thinking about this direction. When you went to Tibet you also tried to help mathematicians in Tibet. So you take this as your mission in a way.

JB: I don’t do so much right now unfortunately, but I would like to do more and perhaps now I have a bit more time maybe I will be able to.

TPL: What IMU does mostly is organizing the ICM?

JB: It organizes the ICM and it awards the Field Medals and other prizes of course. So it’s one of various International Scientific Unions. Something like IMU has to exist because you need some organization to represent mathematics.

FCL: You tried to improve the educational training of mathematicians in the third world when you became the president of IMU, or before that?

JB: Well, before I became president I was not on the executive board. Usually people are on the executive board for a year or so before they become the president. I sort of went straight in there somehow. And the first thing I did was to write a position paper on what I saw as things that could be done in the organization. One of them was this funding. And eventually of course the IMU got a permanent office. This was a big thing but this happened a bit after…

FCL: It’s a big job to get good people involved in your program. You also mentioned that the funding organization is also very important. Are those funding organizations basically public ones or private ones?

JB: You mean in the third world? That’s right, you have to scrabble around for grants from whoever you can find. For our programme with the London Mathematical Society we had some funding from the Leverhulme Foundation, for example, but then it ran out. And so it was picked up partly by the London Mathematical Society and partly by IMU from its limited funds. Then AMMSI, which supports mathematics in sub-saharan Africa, had funding from the Mellon Foundation and then that ran out. So you’re always looking around for money somehow.

TPL: Can you tell us some story about the prizes, while you were president?

JB: The great thing that happened when I was president was that Perelman, of course, got the Fields Medal, and well, that was really very, very interesting. It was clear that he was very likely to get the Fields Medal and it was also clear that he might not accept it. So even before the Fields Medal committee had met, we discussed on the executive committee what we would do if he was awarded the Fields medal and turned it down. And we very quickly came to the conclusion that we would award it come what may, and that if he chose not to collect it, then that was his problem. Because we just eliminated the other alternatives - I mean you couldn’t somehow award one less Fields Medal and not say who it was. That would be ridiculous and at the same time it was potentially very important mathematical work which should be recognized. So that was one of the first things that happened. And then another thing that was very important for me was that I began to realize that it could be a big news story, though I underestimated how big the news story would be. But I talked to Marcus du Sautoy, who was an expert in Oxford having lots of dealings with the media, and he taught me how to deal with the media. If you say something is off the record, nobody can quote you at all. If you said it’s un-attributable, they could quote you but they can’t say who said it. And otherwise they can write absolutely anything you said. So, I didn’t make this kind of mistake with IMU, I was very, very careful. And other people were not so careful. For example, when they talked to Sylvia Nasar who wrote that article in the New Yorker. So anyway that was exciting. And another thing was that at the time when we had to decide whether Perelman should get a Fields Medal or not, we could not be sure that he had proved the Poincaré conjecture. But we decided that in any case the work he had done deserved a Fields Medal. So actually if you look at the citation for him it did not mention the Poincaré conjecture because at the time we didn’t know.

TPL: So did you try to contact him?

JB: Yes, yes. I called him up and he said he wouldn’t accept it and so I said, “Well, could I come to St. Petersburg to talk to you, at least I could understand why you would not accept it.” And he said yes. So I went and I spent two days talking to him at St. Petersburg. He still didn’t accept it.

TPL: But you made a trip to see Ito.

JB: I went to see Ito, that was different though. That was different, yes. That was because Ito was not well enough to get the Gauss Prize himself.

TPL: But was he clear in his mind?

JB: Yes. It was wonderful. His daughter accepted the prize on his behalf at the Madrid Conference. And then I went to present it to him in Japan. The ceremony was going to be in the University. But he wasn’t well enough…he was in a nursing home, but he was not well enough to go. So I presented it to him in the nursing home. And it was actually a great occasion and all the doctors and nurses were there. He was very gracious and very pleased to get it somehow. His daughter Junko who accepted it in Spain and the other daughter looked after me very well. It was a very nice occasion actually.

FCL: I will continue my previous questions. In Britain, people in applied mathematics are usually more interested in mechanics, and is that also the reason why the calculus of variations has been always one of the main subjects in analysis in Britain.

JB: No it hasn’t. There was hardly anybody doing it. I mean I don’t think I knew anybody who was really doing rigorous calculus of variations when I was a graduate student in the UK, I don’t think so.There wasn’t anybody for some years doing any regularity theory for PDE. I mean in the whole UK. There was nothing. There was really nothing.

TPL: You know Morrey has done this great thing that’s a theory in search of an example, right? But how did Morrey get into this theory?

JB: It must have been purely on trying to extend what was already known. I guess he knew all the results on lower semi-continuity in the scalar case and he wanted to try to find out what the conditions were in the multidimensional calculus of variations. One reason why Morrey's stuff on quasi-convexity is difficult to understand is because quasi-convexity is difficult to understand. You know it's not the usual kind of condition that you see. It's non-verifiable, at least up to the present. What we know now is that it's not a local condition on the integrand. So the first time I read it I had been doing stuff with Jacobians and I saw in his book stuff about Jacobians, and I read what he wrote on quasi-convexity and I just didn't understand anything. But I just kept reading it again and again and I eventually began to realize what he was doing. And his actual theorems didn't quite apply to elasticity because there were growth conditions that weren’t applicable.

FCL: He seems to be more interested in geometry than analysis. Is that right?

JB: He wrote a lot in his book on geometry, that's right. Yes.

FCL: And this part to me is very difficult. And also of course all the quasi-convexity , that concept is also difficult for me.

JB: And it's really important for science. Because we already have these applications of quasi-convexity to microstructure of materials, and the fact that we can't characterize quasi-convex functions has direct implications for things we would like to predict about materials. So it's really an important problem and needs a breakthrough from somebody. The only wide class of quasi-convex functions that is known are polyconvex functions. There are other examples but you have to work very very hard to find them. And yet we know that polyconvex functions are far from being the general case. At least we think they are far from being the general case. But we don't have any larger class of functions which we know are quasi-convex that you can actually calculate, so it's really strange.

TPL: What do you see as the possible future research direction?

JB: Well, there is this whole area in which we are trying to understand quasi-convexity better. I suppose the first big breakthrough was the counterexample of Sverák. When he did that work, he was actually my post-doc. And now there's another set of advances by Székelyhidi, Faraco and Kirchheim in two dimensions, and also Iwaniec and collaborators are trying to develop the connections with quasiregular maps and geometric function theory. We still don't know whether rank one convexity is the same as quasiconvexity in the $2\times 2$ case. It's remarkable. Another thing that would be really useful is to be able to quasi-convexify functions and sets. Because that's the sort of thing that relates to microstructure and so on. And of course you can't do that if you don't know what quasiconvex functions are. Also in elasticity, essentially nothing is known about the regularity of minimizers. Actually there is not a single energy function for which you can prove for general boundary value problems that the minimizer is smooth. The only case when you can do that is when you have boundary data that is very close to a stress free state, where you can use the implicit function theorem to prove that there's a smooth solution to the Euler-Lagrange equations and then another subsidiary argument to show that in that case it’s the minimizer under some other conditions. But there's not a single energy function which respects the conditions you want to have in elasticity for which you can prove that the minimizer is smooth. And yet you don’t expect singularities. I mean there are of course singularities that you see. You see them in microstructure of alloys but that's related to the fact that the energy is not quasiconvex, and then there are things like cavitation, but having cavitation and fracture is related to the function space and growth of the energy in the deformation gradient. But there should be good conditions under which you can prove that every minimizer is smooth.

TPL: Good conditions correspond to some real physical materials.

JB: Yes, yes.

TPL: We're talking about the calculus of variations and so on. But overall the British science continues to be excellent.

JB: Yes, pretty strong. And I think British mathematics has changed a lot in the last 20 years or so. And now analysis has become more prominent and theorems in applied mathematics have become much more the norm. Whereas previously it was really the exception.

TPL: British Empire is basically gone, right? But British science continues to do well. Is there anything one can be attributed to?

JB: How good is British mathematics? For example it's not as good as France I would say, right? France must surely be the best country per population in terms of quality. So we're not doing everything right. Well, I mean we're in the top few countries, say, for mathematics probably, but we're not the best. I think it all depends on people, really. You have different subjects, in different phases of development, with key people and they attract other key people.

TPL: You are one of the key people.

JB: I doubt that, but G. I. Taylor would be a good example of a key person who developed a big school around him.

TPL: You have met quite a number of excellent mathematicians throughout your career so far. You want to talk about some of them that come to your mind right now?

JB: Well. Among my collaborators, I think Dick James is a very interesting case. He was a student of Ericksen. And I think one thing that he learned from Ericksen, and which in some sense partly rubbed off on me, was to have faith in mathematics. I think a lot of mathematicians tend to think when they are doing applied mathematics in a particular situation that they make all these assumptions and it doesn’t really mean anything. But there is a striking case that came from stuff that Dick and I did together. We derived some sort of compatibility conditions and there was an exceptional case which was when the middle eigenvalue of the transformation strain equals one, and another when some further conditions held. And when we wrote up papers originally, we mentioned that there might be very exceptional materials for which this middle eigenvalue was one but usually it was not. So I just thought that was an uninteresting case. But some 25 years later maybe, the group of Dick James constructed materials that satisfy all these conditions and these are amazing materials. Usually there is hysteresis, so you have a phase change that might be for example cubic to orthorhombic that occurs at a certain temperature. Now usually you have to go down below that temperature a bit in order for the material to transform and then on the way back you have to go above it. And so usually you have a hysteresis, which is about 30 degrees C. So for these materials, when you tune them exactly for these special conditions to be satisfied, the hysteresis is just 2 degrees C! And you get microstructures that nobody has ever seen before. And it's really comes from believing in mathematics somehow and that it really does apply, and not ignoring cases that you might think of as being pathological.

TPL: You like (Jerald L.) Ericksen.

JB: Yes, but he's a great hero of mine, Ericksen.

TPL: Tell us something about him. Where is he now?

JB: We went to his 90th birthday meeting just over a year ago. And so he retired quite early from Minnesota. He went from Johns Hopkins to Minnesota. But he must have retired maybe at age 60 or something, because he didn't like university, the way universities were going. And so he went out to Oregon, and he continued to do research there. I think he is, to me, very clearly the greatest figure in mechanics of the 20th century. He did so many fundamental things and he interacted very well with mathematicians. I think his work is really amazing.

TPL: He knows physics.

JB: Oh yes, he knows physics, but he is very very original. And you would ask him a question and he'd always ask you a question in reply. It's difficult to understand him sometimes, because he thinks differently. So I encountered his work in elasticity on the one hand, elastic crystals and then liquid crystals. So these are absolute central things.

TPL: Is he is similar to G.I Taylor?

JB: Well, no, because G.I Taylor was a great experimentalist, and Jerry didn't really do experiments. I mean he was very close to people who did experiments and he was very interested in experiments, but he is a theorist.

TPL: What is your plan, continue the liquid crystal, elasticity, this general field? Any specific direction you are thinking of now?

FCL: Is the phenomenon of defects in the liquid crystal, is that a general phenomenon or…

JB: Yes , it’s hugely important. Of course in many applications you're trying to avoid defects. But then there's lots of interesting situations where defects are actually important and you want to be able to understand them. For example there are bi-stable displays. In a standard liquid crystal display the image will disappear when you turn off the current. But suppose you want the image to stay there, like an electronic newspaper or something, when you get on the train to commute to New York, so you plug in your whatever it is and you want the image to remain. Now maybe the idea of electronic newspapers is being superseded by people using Kindles and stuff like that. But certainly there are situations where it's interesting to be able to create an image with the liquid crystal which stays there, so that when you turn off the power it remains. So that's one situation. And that depends on defects that stabilize things. And then there are interesting defects around particles. So people are interested in little particles embedded in liquid crystals, and think that you can even store information in them by some interesting devices that are connected with defects. There are line defects that somehow can wrap around and even become knotted and so on. So a fair amount of it is curiosity driven research to see what amazing things can happen with these defects I suppose. But there's also people who are really trying to apply the stuff.

TPL: I guess that Morrey and your discovery of Morrey, one part of that is curiosity.

JB: Yes. In some sense he was being driven by mathematical structure. So I think that's the interesting thing about this interaction between pure and applied mathematics. The applied mathematics suggests something, or some science or some phenomenon suggests something, and then some bit of mathematics gets invented, maybe not by mathematicians. And then somehow the professional mathematicians take it over and generalize it, and they sort of develop it into something in an intrinsic way, following things like beauty and simplicity and generality and so on. And then you get it turned it into something which can be applied to something else.

TPL: Your experience is very unique and very real. Next week you will have a meeting, an intense one.

JB: Yes. It’s the proposed merger which will probably take place between the International Council for Science and the International Social Science Council. So, we first have to discuss it separately and then together.

TPL: With the hard work ahead of you next week, let’s stop here. I hope that you will come back in the near future.

JB: That would be great.

• Tai-Ping Liu and Fon-Che Liu are faculty members at the Institute of Mathematics, Academia Sinica.