Interview with Prof. Leon Simon

Interview Editorial Consultant: Tai-Ping Liu
Interviewer: Tai-Ping Liu(TPL)
Interviewee: Leon Simon(LS)
Date: May 4th, 2018
Venue: Institute of Mathematics, Academia Sinica

Prof. Leon Simon was born in Adelaide, Australia July 6, 1945. He received his BSc (1967) and PhD (1971) from the University of Adelaide and was employed from 1968 to 1971 as a Tutor in Mathematics by the University. Then, he held academic positions first at Flinders University, then at Australian National University, the University of Melbourne, the University of Minnesota and ETH Zurich. Since 1973, he has served at Stanford University. He has authored the textbook Lectures on Geometric Measure Theory, and the monograph Theorems on Regularity and Singularity of Energy Minimizing Maps. He was awarded Bôcher Memorial Prize in 1994 and the Leroy P. Steele Prize in 2017. He was elected as a Fellow of the Australian Academy of Science in 1983 and a fellow of the Royal Society in 2003.

TPL: When you were a teenager, how did you get interested in mathematics?

LS: Well, I didn’t really get into it until I was quite old. I wasn’t that interested in high school in mathematics, too busy playing sports and social activities. So I was moderately ok academically, but nothing special. And I just did enough in high school to get accepted into the University of Adelaide, right. And it wasn’t until that time, my first year in university that I started to get interested in math. And even in the first year, I thought that I was probably about in the middle of the class. It was about 100 people in the class.

TPL: But did you major in mathematics?

LS: No, in the first year, in those days, you typically took one mathematics course for the whole year, and three other science courses: usually physics, chemistry, and biology.

TPL: Oh so it’s general science.

LS: Yes a general science course, initially without any particular specialization. And I thought I was probably about in the middle of the mathematics class. When I arrived I had no idea that I would ever specialize in mathematics. It was completely out of the question as far as I was concerned. But at some point during those lectures, I realized that even though I found mathematics in high school fairly hard, and I hadn’t done particularly well in it, I found the lectures in my first university class, which were well presented, seemed suddenly relatively easily understandable. It was as though something happened in my brain between my high school experience and that first year, at least that is the way it felt.

TPL: Is it a new experience?

LS: A new experience, I could suddenly understand much of what was being presented in the lectures. Not entirely without effort, but pretty much without huge effort. And I could solve the problems that were set for homework and so on. So I thought well ok, that’s good. I’m probably about in the middle of the class. But I was pleasantly surprised that at the end of the year I ended up in about the top five or ten percent. We only had one exam at the very end of the year in those days—there was no continuous assessment then.

TPL: Oh this is a British . . .

LS: Yes, British system. It was in Adelaide, capital city of South Australia, with some British influences dating back to the colonial era of the 19th century. Anyway just one exam at the end of the year. So you know, I was very happy about doing well in that, and it made me think, well maybe I’m better at this than I thought. And, in the second year, I approached it somewhat differently. Instead of just deciding that I wanted to sit there and do well enough, I decided to see if I could really do well, like that. And it seemed to work. I somehow got more and more interested and enthused about the subject. Then I went on from there. In the third year, I had my future PhD adviser, Jim Michael, as one of the lecturers. He was lecturing two courses in that third year, real and complex analysis. That was my good fortune—it was really the turning point where I decided I wanted to do mathematics.

TPL: Why did he choose to stay there? Jim Michael.

LS: I think personal reasons of family—after all he was originally from a country farming area close to Adelaide. Of course for me it was extremely lucky he stayed in Adelaide. He was a very modest man, and he wasn’t flashy at all, quite the reverse. He was umm . . . I would say one of these people quite slow to pick up on ideas. Sometimes you feel you had to really work hard to get an idea though to him, but once he got it into his brain, he could do more with it than anybody else. A bit similar to Constant Reid’s description of Hilbert, she claimed that he was also like that. That’s probably an exaggeration in Hilbert’s case. She claimed that there were times in Göttingen when almost everyone in the room listening to a lecture could understand before Hilbert and would be trying to explain it to him. That might have been an exaggeration in Hilbert’s case, but Jim Michael was a little like that.

TPL: I see. So this university is in a quiet environment, suited him just well.

LS: It suited him very well. And he had the esteem of all the students and the faculty. We understood how good he was even if it might not have been apparent to people outside the mathematical community. But within the Australian mathematical community, even outside of his home state, he had a reputation for being an exceptional analyst.

TPL: When talking about Australian mathematics, the general impression is that it originally came from Britain, right?

LS: Yes. In fact there was a period in the 1960s when the universities expanded very rapidly. The government had a policy to increase the number of people who were taking degree courses. So in that case they were recruiting from everywhere, especially British nationals and Australians who had done graduate work in Britain. Some of the people hired during that period had only a very modest research record—it wasn’t until later, into the 1970s, that it became more difficult to get a permanent academic job.

TPL: How about 19th century?

LS: No, in the case of South Australia, I think the colony was only started in the mid 19th century, and I don’t think the university was founded until early in the 20th century, the University of Adelaide. So it was relatively new. The original universities in Melbourne and Sydney were quite a bit older than that.

TPL: But still not earlier than late 19th century.

LS: Right, well Captain Cook came in 1770 and planted a British flag there. And it wasn’t until quite a bit later than that that the colonies got going. They were penal colonies by the way, so we’re all descended from criminals, except I’m not because I come from South Australia, which was not a penal colony.

TPL: So actually the academics really come into prominence in Australia in the second half of 20th century.

LS: I think so. There was some very good work done, and Australia was always like that. There were these little pockets of real excellence with people that stood head and shoulders over everyone else and set a real tone of excellence. But then I think the average level was considerably less. But nevertheless, that was good enough that somehow people knew what good work was.

TPL: Why is that? Is that because the place is comfortable enough and people relax enough?

LS: Well there were various things. One is that somehow you think if it was relaxed enough, people would be self-satisfied and any mediocre work would be considered good, right? But it wasn’t like that with these people that stood out, and they were recognized, without malice, as being really good, and they didn’t need to be cut down to size and so on, at least in general. There’s a claim that Australia has a “tall poppy syndrome,” and anyone who stands out gets their head chopped off (figuratively speaking). I don’t think that was true at all in those times. I think there was always these pockets of real excellence, and people really looked up to those people. Jim Michael would’ve been an example of that I think.

TPL: In ideal world where people just do what their talent or what their instinct allows them, then people like Jim Michael just go back home and think about that and so on, and when he comes back, people, at least some of them like you who can recognize, and that is satisfying to him.

LS: I think so, but he also loved to teach, he loved to pass on mathematical ideas, even though he was rather, what’s the word . . . a little bit retiring and shy until he got in front of a class. I can remember going to social occasions at his house when he would invite a whole group of people. These were the most awkward and tense things you could imagine. He would sit people around the room and then there would be various attempts at a conversational gambit, you know, that would uncomfortably peter out. So he was a little bit awkward socially. A wonderful man on the personal level, but a bit awkward socially. But then when he got in front of a class, it was a whole different story. He spoke with authority and you could tell immediately that this man was deep and could pass on insights that would not be accessible in most other places. Even though he didn’t attempt really to impress, I don’t think, he just did what he thought would be effective for passing on the material.

TPL: Maybe Australia is a new society, and therefore such a person is allowed and basically anything goes in term of social manners?

LS: Possibly. He certainly was an interesting case. I don’t think I’ve actually seen that to the same extent with anybody else.

TPL: I see, so he is a singularity.

LS: He was sort of a singularity in that regard I think, but everyone knew that. I noticed that after he had died, when some of us got together, for the Australian Academy of Science, to write an obituary about his life and work. And I noticed that several people mentioned this, that he was a little bit, diffident is the word, diffident when it came to his interacting with people. But as soon as he got in front of a class, he spoke with authority and assurance. It was great to see actually. And even though I wasn’t like that, I thought that was admirable, that he could do that. And it was so refreshing because you could tell that he was never really trying to impress somehow. He somehow didn’t play the game, if you know what I mean—most of us don’t want to admit that we’re dumb idiots, and sometimes hide our ignorance, right? But he wouldn’t do that. He was brutally honest, not just in that way but in many other ways too. And you know, fortunately for him, people knew how clever he was, so it didn’t cost him very much I don’t think. But Peter Lax once said to me, when we were talking about Jim Michael, “Yes, Jim’s a wonderful character, but he should play the game a bit more.” Now I hope I’m quoting Peter correctly there, but I think I am.

TPL: It fits together for me. But now that we are talking about Jim Michael, what exactly he has achieved?

LS: Ok, I’m not, probably terribly well prepared to explain that, but . . .

TPL: No, just in general.

LS: Yes in general, well the first paper I ever read was a paper by him. And he didn’t make me read it, I just decided that I wanted to read it. It was a brief paper on the proof of the Cauchy Integral Theorem in complex analysis. And he had the first proof that was completely general, it didn’t make any assumptions of any kind about smoothness of the curve, or whether or not the degree could be unbounded, the degree of the curve, the number of times it winds around, could be unbounded in some places or whatever. And I remember that at one point in this paper there was an extraordinarily clever argument using the topological degree, where he gets some estimate. The whole paper depended on this. And I remember being absolutely so impressed by how clever it was. I think the first real idea in mathematics that I’d seen other than the classical ideas that you see in lectures. The first real idea in a research paper I’d seen, and somehow for me that was very important, because I realized to get a good paper, you don’t just write up some piece of nonsense that you can write up because you know how to do it. You do something that requires a real idea, and a really genuine idea, to solve, and that was a wonderful experience for me. First paper, I was very lucky to choose that. Second paper I ever read, by the way, was by Bob Finn. Also similarly, several brilliant ideas there. His paper on two-dimensional gradient estimates, gradient estimates for the minimal surface equation in the like in two dimensions.

TPL: Ok! Did you tell this to Jim Michael?

LS: I don’t remember telling him, and he probably would’ve been very embarrassed if I had told him it was brilliant. I think he knew how I felt—I was full of admiration for that little paper.

TPL: In general, how did you interact with him?

LS: Of course by the time I was his PhD student we had developed a comfortable and enjoyable relationship and spoke just about every day at morning and/or afternoon tea. But in the early stage when I was still an undergraduate I would always go to see him in his office and sometimes I was nervous. Often, when I had finished talking about what I’d come to talk about, it became a bit awkward. You sort of had to edge towards the door. He didn’t know how to terminate a session somehow. I was also a bit shy at that point, right? And I had trouble getting out of the room.

TPL: I remember in the book on Courant. Courant was very good at that. And if he wants to terminate the conversation, and he would stand up and said: Oh, you want to stay, don’t you? But then Courant gradually moved toward the door.

LS: See, that would be the antithesis of Jim Michael because he would consider that dishonest, you see. He would never say that.

TPL: But Courant had different role to play.

LS: Yes, of course. But he is the absolute antithesis of Jim Michael. He wouldn’t think about handling a situation in that sort of way. But I think he also felt it too. He knew the conversation had to terminate, but he didn’t quite know how to do it. And it was awkward at that undergraduate stage—I mean at that stage to me he was sort of way up there and I was down here. I do remember once though during that period when we were discussing something, he said, well, I think you know more than I do about that now. And I was astounded that I could know more about that than he did. I mean, I forget what the topic was. I think it was about topology and topological degree or something, applications, but it was one of the things he was a recognized expert in, so I was surprised when he said that, and it somehow helped me to see possibilities about what I might be able to achieve.

TPL: So honesty can be . . .

LS: It can be very helpful, and ever since then, I’ve always tried to emulate that with my own students, and not to give the impression that I think I know more than they do about everything, even in my main areas of interest.

TPL: Since Jim Michael is so honest, his word is believable and is really effective.

LS: Yes, yes, definitely, that’s the thing. You could always be confident that what he is saying was straight, that there was no twisting of the truth. Also, I could not imagine him publishing a paper that turned out to be wrong. I mean, I couldn’t imagine that because he was so careful and conscientious in checking things and so on. His output was not high by the way. He would only write a paper every year or two. I think we could look him up in math reviews, I’m not sure how much he published, but probably not a great many papers I would imagine.

TPL: Going back to your high school years, could I infer from what you just said that actually the high school education was not very suitable to you?

LS: No, no, it was. I remember in the last year, it’s just that I was the unsuitable part of that equation I think. I just wasn’t really engaged by the subject. In the last year, I had a teacher, who by the way was in this religious order, he was a monk, but he was a very decent human being in that religious order, which in general was troubled—a recent Australian Royal Commission established a horrifying pattern of abuse of students by a significant percentage of people in that particular religious order. But the man I am speaking of was a good man. He had a very unique way of looking at mathematics and teaching it. A typical lesson, which went for I think about an hour, would be that, instead of him teaching, he would be getting one student to write notes on the blackboard. He had these standard notes of the main things in the course. And we would copy them down. And then tackle assigned problems about it that night. And that went on for the first half of the year. And he would do very little teaching, but he did make sure that we did the homework—there was no getting out of that.

TPL: Ok, he set a standard.

LS: He set a standard and made sure we put in the hard hours at night. That was extremely important for me I think, because somehow my brain, I can’t really analyze it, but my brain wasn’t very flexible at that time. And I think I needed that consistent effort, more structure and that consistent effort every day. Of course I wasn’t consciously aware of it at the time, but I think that period and that consistent effort was important in developing whatever mathematical ability I would end up with.

TPL: We all know Euler went to see Johann Bernoulli and received a homework, did some reading and homework and then went back Bernoulli to report them. It sounds like that to me. I was talking to S. S. Chern, the geometer, and he said he went to see Elie Cartan every other week. Cartan gave him problems, and I’m quoting him, he said, “All easy problems, and I could solve them,” and he then he added: “Don’t give the students too hard problem.”

LS: He got that from Cartan, right? He learned that from Cartan. It’s better to give easier problems rather than hard ones.

TPL: Right

LS: Maybe something to that actually. I don’t believe in this business of only giving hard problems. Then they have to bang their head against the wall for hours and hours, and can sap the confidence and enthusiasm for the subject even in people who have outstanding mathematical potential. I do think the majority of the problems should be pretty straightforward, and maybe occasionally one harder problem that they could find more challenging and much more inspiring if they do manage to solve it.

TPL: I see, ok.

TPL: After you graduated, you went to various places.

LS: Well I went to Stanford first—no, actually that’s not right, I had a job for a year after my PhD at the other university in Adelaide, Flinders University. And in those days, when you got appointed to a lectureship, you could pretty well be sure that it was going to be permanent unless you didn’t work out, unless you were really hopeless or something, right? So I more or less had that feeling that it was going to be permanent, I was going to be there for the rest of my professional life. And one day I got a letter, a letter from Stanford (there was of course no email and certainly no inkling of the idea of an internet in those days, so it was an airmail letter which took about a week to arrive), from David Gilbarg in Stanford, saying would you like to apply for a three-year assistant professorship in Stanford?

TPL: That’s pretty impressive.

LS: I was just bowled over. I found out later that Neil Trudinger, who was one of the examiners of my thesis, had sent a copy of the thesis to Gilbarg. And I guess he said something like this person seems to be pretty good, or some such thing.

TPL: Actually I was saying that it’s impressive that Gilbarg recognized it.

LS: He did. Well, I don’t know if he got it right, but the fact is, he wrote this letter, which came out of the blue completely for me. I was totally astounded.

TPL: He found you in that part of Australia.

LS: Right, I would probably still be at Flinders University proving gradient estimates for the minimal surface equation or something if that hadn’t happened. Because I wouldn’t have thought of applying to a big university overseas. Somehow it wouldn’t have occurred to me that it was even possible somehow.

TPL: Let me jump to another subject. Ok, so this is geometric measure theory, which everybody agrees is the hard analysis, and it’s tough business.

LS: Right.

TPL: And it has produced a number of first-rate mathematicians.

LS: Oh yes, quite a few, especially the founders of the subject, Federer, Almgren, Fleming, Reifenberg, De Giorgi. I mean those, they were my heroes. CB Morrey wasn’t quite in geometric measure theory, but he was really into that stuff. If you look at his book, it’s not just PDE. There’s a lot of geometric analysis in the later chapters. For instance, he gives a treatment of Reifenberg’s work on the Plateau problem. And he talks about the d-bar Neumann problem. And there’s lots of stuff in that book. So he’s really in that mix actually, even though he would have said, most people would said, he is a PDE guy.

TPL: Have you met Morrey?

LS: I did. I only met him once. I went to Berkeley to give a talk, and he was a very gracious man. He took me out to dinner after, and it was really, he was a very nice man, very gracious, very gentlemanly man, and very obviously wanted to encourage me. So it wasn’t just a case of going and giving a talk and saying after all, thanks see you later. It wasn’t like that. He took me out to dinner, brought his wife along to the dinner actually, I remember. And it was very nice.

TPL: You mentioned these names. Maybe you can tell us just a little bit what they have done.

LS: Ok, yes, well Federer and Fleming wrote this amazing Annals paper in 1960 called “Normal and Integral Currents,” which sounds like a rather uninspiring name I must say. Probably a bit of a turn-off as far as the title is concerned. But what they proved there is quite amazing, something that I would never have never believed could sensibly be considered, let alone actually be true. I thought it was impossible that it could be true. They said that if you take. . . uhh I’ll paraphrase actually alright? Just a special case of what they proved. They said that if you take a sequence of compact manifolds in Euclidean space, some manifolds of Euclidean space, say a sequence of 2-dimensional surfaces without boundary for example. Of arbitrary topology, possibly extremely convoluted, but with bounded area, and they remain in a fixed compact set, they don’t run off to infinity. And then they’re all oriented so you can integrate forms over them, you can integrate a two forms over these surfaces. And then Federer and Fleming’s result says there’s a subsequence of this sequence of surfaces, which in a weak sense converges to what is basically a limiting surface, which sounds ridiculous; I mean, you’d feel that couldn’t possibly be true because you can just think, for example, of a sequence of very thin truncated cylinders, capped off at there ends, which are increasing in number and decreasing in cross-sectional diameter and becoming dense in say the unit cube in Euclidean space. And according to Federer-Fleminig, the only restriction you need to keep on them is that the total area of them is bounded, right? But for this example you can just get some multiple Lebesgue three-dimensional measure in the limit, if you take the limit in the appropriate measure-theoretic sense right? There’s no surface there—it just smooths out, it fills up the whole space. So you’d think Federer- Fleming couldn’t possibly be true. The point is that they take convergence in the sense of currents—in the sense of integrating forms in other words: you integrate an arbitrary smooth form (defined in the surrounding space) over each of the manifolds in this subsequence, and the claim is that there is a limiting surface (not smooth, but at least having a tangent space almost everywhere in the appropriate sense) and the subsequence of integrals of that fixed form converges to the integral of that same form over the limiting surface. Now the limiting surface is not smooth, that would be too much to believe. But it’s what you call a rectifiable set–it’s like a measure-theoretic version of a manifold. It’s got an approximate tangent space almost everywhere. It really does more or less look like a surface, if you squint a little anyway, it looks like a surface. And the reason that that can work against your apparent intuition is that if you took something like the thin cylinders in the example I was talking about above, you would get cancellation, because you’re integrating the forms over an oriented surface, and contributions from one side of each thin cylinder asymptotically cancel out with the contribution from the diametrically opposite side. So in that that case we would get zero limit. So in the particular case of the above example of the thin cylinders you get zero limit, but in general you’ll get some non-trivial limiting rectifiable surface, and you can imagine that it’s immensely important in general that you can do things like that. So as far as I’m aware, no one even had an inkling that that might be true before they published that paper.

TPL: That’s great.

LS: Fantastic result.

TPL: And then . . .

LS: Reifenberg proved something similar in the mod two case, the unoriented case.

TPL: This is a big area, so you are talking about De Giorgi and Morrery and Almgren, they all do this.

LS: Yes that’s right. Yes, they all had fundamental contributions that were so important in various areas of that big field.

TPL: Geometric measure theory, it has geometry, it has measure, it has PDE, so this is a geometric analysis.

LS: Yes it means different things to different people. There is an area of geometric measure theory where people work on what you would call more measure theoretical aspects of it. They’d be looking at fractional dimensional sets and the structure of those and things like that. And there’s another side of it, which is the side I like, that’s really in geometric analysis, where you’re solving problems in geometry us- ing these geometric measure theory techniques. So it’s a big spectrum of things, and it can mean different things to different people.

TPL: So how about De Giorgi falling into this . . .

LS: De Giorgi definitely is in the geometric analysis side of things. He was interested in high dimensional minimal surfaces, that was his main interest, at least at that time, in the 1950s and 60s. Later he did things like G-convergence and gamma-convergence. But at that time he was mainly interested in minimal surfaces. And he basically solved the co-dimensional one plateau problem, and got everyone well on the way to the required regularity theory. And then later worked with Bombieri and Miranda to prove the gradient estimate for the minimal surface equation, that was the topic of my thesis, gradient estimates, quasilinear and quasi-elliptic, and it had just come out, that paper, just around the time I was writing my thesis. It had only come out about a year before that, so I was very lucky. I got interested in that exactly at the right time, when those people were doing their main work. And Federer’s book came out about the same year. I remember going to the library, and I was just browsing in the library. I saw this thick book called geometric measure theory. I had no clue what it meant, but I said that looks pretty interesting.

TPL: Did you read it?

LS: I didn’t read the whole book, it was too hard for me at the time. But I did get the book out, and I started looking through it, and I realized that this was incredible, this book was incredible.

TPL: I know many people who had the book on their shelves.

LS: On their shelves, and they can’t bear to open it because it’s so difficult to read. Do I spy a copy over there in your bookshelf? Maybe not.

TPL: At Stanford, not here.

LS: Yes, anyway that’s a common thing because Federer’s writing style was very, very difficult for most people to comprehend. But if you really have the patience and determination to try to read it, all the details will be there. It’s just that they’re in this strange form. It’s like learning another language somehow. Bob Hardt referred to it as “Federese,” this language.

TPL: His particular way of writing.

LS: Yes, he would have very little in the direction of pointing out what were the key results, and what was the aim of the next lemma, relative to what he was trying to achieve. Rather, there would be a whole sequence of maybe ten or fifteen lemmas. And when you finally work out what they’re saying, some of them will be saying something pretty bone-headed and down to earth, which perhaps shouldn’t have even been stated as a separate lemma, but then others will be the crucial thing, with the real content. But he wouldn’t help you out very much by making that clear. Somehow it would’ve made a huge difference if someone simply had gone through as an executive editor (he probably never would’ve agreed to this), but if someone had the authority to go through and to add comments and explanations at various stages, it would’ve made things infinitely more easy to read. But he was also completely economical to the point of being a fault, I would say.

TPL: Can I ask you this question? What make you really happy? Which result of yours make you really happy?

LS: I don’t know, I suppose the one that really was the biggest struggle was the uniqueness of tangent cones paper. That seemed impenetrable, and I was working very hard for very many months, maybe almost a year actually. Every day and every night, I did most of my work at night in those days, and I wouldn’t get up until about 9, and then I’d go into the department at around 10 or something, but I would be up every night working on that. And somehow I had this feeling, you know how it is, you have this feeling that maybe I can do something with this problem, but couldn’t quite get my hands on it. And then I remember one particular night, I realized that basically, when you strip away all the easy cases and really get down to the essential case, the essential case of that problem was when things were changing very slowly. The easy cases were when things were changing fast; you could handle that case. But when things were changing really slowly, that was the difficult case. But then I immediately realized, much to my initial dismay, that meant the problem was governed by a parabolic equation up to small error terms, which seemed to suggest that the asymptotic limit result I was looking for might be false—there examples like “goat tracks coming down the side of a mountain” in an infinite spiral, spiraling out to a circle, where you don’t get a unique limit, it just keeps going round and round. It was late at night, and I my wife said: “how’s it going?” and I said “I think it might be false.”

TPL: After all these months . . .

LS: After all this effort. But then that’s the way mathematics is. Just when my spirit was at its lowest ebb, the next day or a few days later, I realized, aha wait a minute though, there are some added ingredients here. All the counterexamples involve gradient flows of things that are only smooth. This on the other hand is a problem involving a real analytic functional, and maybe there’s some . . . then it took me a few more months or a few more weeks, I’m not sure how long is the time scale. And I realized it required an inequality involving real analytic functions which vanish, along with their gradient, at the origin—what was needed was that, for those functions, you need to look at the length of the gradient of small power of the absolute value of the function, and check that that length is bounded below in a neighborhood of the origin. And I remember thinking, that just might be true. And I asked around among various people, but no one could tell me whether it was true or not, but nobody had a counterexample either. So I had a couple of students looking at that problem, and eventually one day, after a few weeks, one of them came and showed me a paper, and there it was, the exact inequality I needed. It was a paper by Łojasiewicz, showing that for each real analytic function which vanishes together with its gradient at the origin, there is a small number, theta, between zero and 1, such that if you take the absolute value of the function, to the power of theta, and you take the length of the gradient of that, that’s bounded below in the neighborhood of the origin. Of course I needed an infinite dimensional version of that, but I was very confident that that was just sort of routine, and indeed it did work out. I was very happy about that.

TPL: When was that?

LS: That was back in about 1982, I think. And it was, you know, that tremendously euphoric feeling that you get which lasts for a couple of weeks after all your effort.

TPL: Only a couple of weeks?

LS: Only a couple of weeks. You’d bang your head against the wall for months and months, and you get a couple of weeks of euphoria. But I remember being very euphoric about that result. I was just really, just totally enthused by it, I just loved that result after all that struggle, realizing how that worked. It was just really something.

TPL: The feeling that it is tough, and because at any moment, people will think that, oh this field is now getting hard, so I suppose this is true before you get into it right? Maybe Jim Michael says oh this field is getting too hard.

LS: Yes right, that is true at least in my case. I had feelings like that when I was starting out, because I could tell I knew no geometric measure theory, and I had trouble reading Federer’s book because it was so difficult. And I was thinking, well ok, but at least I can do some PDE. I understand that. And it sort of kept me going. I was working on things that were rather easier than some of those problems later. So I think that’s important, that somehow you get something that you’re passionate about, which you feel you can do something with, you don’t have to feel like, oh this is just too hard, I can’t do any of it. Somehow, I think that’s the sign of a successful research mathematician in general. They, most of the time, they’re in that mode where they are doing something that they feel reasonably, emotionally, like they can probably do something with the problem. And sometimes it doesn’t work out and you have to admit: no, that’s not going to work out, I’ve got to do something else. But all in all, I think it’s important that you sort of have that sense of what you can do and what you can’t do, so that your enthusiasm keeps up.

TPL: Let me tell you a story. This great Swedish harmonic analyst, Lennart Carleson . . .

LS: Carleson, oh he is a fantastic mathematician. I mean there’s an example of someone that, oh I mean that’s the man who works on hard problems, not just for a few months, but in his case, he went on for years .

TPL: So he said something quite similar to what you just said. Namely, a really good mathematician has to have a certain hunch whether this is true or not. But there is also the story, he told me when he was in Taiwan in the 80s: For nine years he tried to find a counterexample for . . .

LS: Exactly, everyone believed it was false, all the top people, Kolomogrov also thought it was false. They all thought it was false, Carleson included also thought it was false. And the story I heard may be the same story that you heard, was that indeed, he thought it was false, but then he got to a certain point, he was working extremely hard for a long time on trying to construct counterexamples.

TPL: It’s nine years.

LS: Yes, right, there you are. Fortunately not the only thing he did in those nine years, but was the main thing he wanted to do. And fortunately, it got to the point where he could actually, at some point he had this incredible insight that he could basically disprove the possibility of the kind of counterexamples that everyone was thinking could exist. That was sort of the great turning point for him. And then he realized, my god, I should be trying to prove it, not disprove it.

TPL: So what he says is that it took one year for that.

LS: Yes, then it was a relatively short time, but still a year.

TPL: Still one year.

LS: Well yes, but I mean that’s an amazing story. But he is so incredibly deep, it’s really . . . but also a modest man, do you notice that? He’s just one of a kind, really.

TPL: Very interesting to me that I took him to Palace Museum, and I would think that what he has done is really very deep and difficult. But when he went to Palace Museum, the thing he liked was just a painting of a simple bamboo branch.

LS: Yes ok, no no, I think that, somehow that fits with my concept of Lennart Carleson actually. Yes, that he would see something in that, that really he finds absolutely enthralling and deeply impressive. That somehow doesn’t surprise me, that story.

TPL: I see, so that bamboo tree . . .

LS: I’d look at it and I’d see a stick of bamboo. Right, big deal, so what? But he is seeing more.

TPL: Yes, yes.

LS: I think that’s true of his mathematics. He was seeing more.

TPL: Yes, because there was a Chinese landscape and he preferred this bamboo tree.

LS: Yes, somehow that seems about right.

TPL: I see. That’s great. So now if you have other story about other people, it will be good. I’d like to hear from you.

LS: Yes, well, mathematicians are quite a spectrum of people. And their egos, for example, vary immensely. We all have a certain pride right? There’s no question about that. Don’t believe no one has got an ego, that’s simply not true. But if you compare Lennart Carleson with some of the more egoistical mathematicians, it’s just a total dramatic contrast.

TPL: So what you would like to do in the future?

LS: Well of course, my research has slowed down, but I still struggle to keep it going. I don’t want it to die out completely, so it’s important that I somehow keep it, keep working and keep trying. I’m working on something at the moment that does seem to be working out, a rare event which is pleasurable.

TPL: On what?

LS: It’s on some, constructing some examples of singularities . . . about thirty years or so ago, Bob Hardt, Luis Caffarelli, and I proved a result about constructing a whole family of minimal submanifolds, which were asymptotic to a given minimal submanifold at an isolated singular point of the given submanifold. And that result itself may not be so interesting, but the concept of being able to, so to speak, perturb the current to give other solutions which were asymptotic to it, I think, an important question, and you’d like to generalize that to higher dimensional singular sets where instead of an isolated singularity, but it’s much, much harder, and it’s not at all clear what’s true. So now I have some partial results, I think on that, in the case of a line of singularities. So I like that result. But it’s much more specialized, not nearly as general as the result back with Bob and Luis, so there is still much to understand there, and I’m trying to do that.

TPL: Sounds exciting. Just one more thing. Everybody said Leon Simon is a great teacher. Ok, so can you offer some secret?

LS: Well of course, preparation is obviously extremely important. I think yes, probably out of everything, I think that’s the most important. But somehow just preparation is not enough—there are probably some natural abilities here. I think the human brain is interesting in this regard. There are some people who, no matter how much they prepare, won’t be able to give a decent clear lecture. It’s just that their brain doesn’t work that way. They somehow fail to be able to explain things in a down-to-earth simple way that catches the attention of the audience, and keeps them involved. But nevertheless, I think there are some rules that you should follow during your preparation and during the lecture. In preparing, you should spend some time philosophically thinking about not just what you should say, but how you should say it, preparing for example some simple discussion of how a key idea might have come about, how it was motivated. Usually it’s some bone-headed, down-to-earth idea that people really appreciate if you can tell them what it is. But many people just tend to sort of present the material and that’s it. And of course when you’re teaching five lectures a week to undergraduates, then probably you have to do that some of the time. But you should, whenever possible, be trying to give that pithy insight without getting bogged down in convoluted explanations which won’t work anyway, because, you know, you’ve got about ten seconds to grab their attention at most stages during the lecture, right? And then they’ll say to themselves, what the hell is he talking about? What is he trying to say? But I think it is important to think about those things before the lecture, and try to intersperse them with more formal aspects of the lecture. And it does make a huge difference with your undergraduate class, and graduate classes too. One other point about what to say and how to say it: perhaps equally important is what not to say. Sometimes too much discussion can be counterproductive and increase the chance of causing confusion. So one should always strive for clarity and simplicity—don’t overdo the explanations.
And also don’t forget that you’re a human being, not a robot. And your audience are human beings, too, so you’ve got to interact with them. You can’t just be talking to the blackboard or something like that, which I’ve seen many a time actually. People forget—they are so internally focused they just end up talking to the blackboard. There could be no one behind them for all they know; the whole audience might’ve left, they wouldn’t notice. And so they’re the extreme cases that I always try to avoid, get as far as possible from those cases, and try to make as much reasonable contact with the audience as I can. And always keep in mind that overall you should be striving to organize the course and your lectures so that, at least for most of the time, you are enjoying it—if you are not enjoying it then it’s pretty certain your audience won’t be either.

TPL: Let’s finish with this upbeat note.

LS: Great, right.

TPL: It’s really good, you come back.

LS: Ok right.

  • Tai-Ping Liu is a faculty member at the Institute of Mathematics, Academia Sinica.