傳播數學知識．促進數學教育

Interview with Prof. Richard Schoen

**Interview Editorial Consultant:** Tai-Ping Liu

**Interviewer:** Tai-Ping Liu (TPL)

**Interviewee:** Richard Schoen (RS)

**Date:** December 17th, 2013

**Venue:** Institute of Mathematics, Academia Sinica

Prof. Richard Melvin Schoen was born in Celina, Ohio on October 23, 1950. He received his B.S. from the University of Dayton in mathematics in 1972 and then received his PhD in 1977 from Stanford University. He has been a faculty at New York University, University of California, Berkeley and the University of California, San Diego, and, since 1987, he has been a faculty member at Stanford University. He is currently a faculty at UC, Irvine. He has been a world leader in differential geometry and partial differential equations. He was awarded a MacArthur Fellowship in 1983, elected to the National Academy of Sciences in 1991 and received the Wolfe Prize in 2017.

TPL: First, thank you for coming. People told me that they really enjoyed your talk for the Hsu Lectures. I am sorry that I was away. Could I just get down to one of the main things that people would like to know, which is when you started at Stanford with Yau, I have heard that you two worked together very hard and there was a major success – the positive mass conjecture. So, could you describe that period?

RS: Well, actually the positive mass conjecture was the 3rd or 4th problem that we worked on together. It was done a year after I graduated. I left Stanford in 1976. We actually did that in Berkeley in 1977/78. Yau was visiting Berkeley that year. Before that, when I was a graduate student we had also done some joint work, also with Leon Simon. Leon Simon was also my advisor. I had two official Ph. D. advisors. So, we got involved. We wrote a three-way joint paper. That was my first paper which I think was a very good paper. It was called “Curvature estimates for minimal hypersurfaces”. I think it had a lot of influence. So it was more of a analytical paper about how to bound the curvatures of minimal hypersurfaces under an area minimizing condition. We used stability which involves the second variation of volume.

TPL: How was it like to work with Leon and Yau first and later on very intensively with Yau?

RS: Well, it started gradually. I was a second year graduate student and taking Leon’s PDE course. I liked it very much in the first quarter. So, after the first quarter, I started doing readings with him. He proposed some readings on minimal submanifolds. Yau’s office was right across the hall. So, I started making some progress on generalizing these things. Yau got excited and got involved too. So that was how the collaboration started. I would say the two of them, Leon and Yau, had very different styles. At that time, Yau had only just graduated. I think he is only two years older than I am, maybe less than two years. He was pretty much living in his office. He had an apartment, but he was in his office from early morning until late at night, attending lots of classes and extremely ambitious. So, I don’t think I had ever met anyone quite like that before. So, it was very interesting. I also had a hard time understanding his language. So that was another challenge. So, I would say it was very challenging and very exciting, even though at that time he was not famous at all. He just had this ambition, dynamic property that was captivating him. I found that quite attractive. Leon on the other hand is much more systematic. He is also a very smart person, but approaches things more systematically. He is more focused, more narrowly focused than Yau was.

TPL: I was told this positive mass thing, you have actually read a lot of literatures from both the physics and mathematics sides. It was not even clear how to formulate the problem.

RS: There were some formulations available in the mathematical literature. Robert Geroch from University of Chicago gave lectures to mathematicians about outstanding problems in relativity and the positive mass conjecture was one of those. So, there were certain test cases which were very clearly formulated and one could get started. It’s true the precise formulation of the asymptotic behavior required, the condition was not completely clearly formulated. I think we did that. Initially, we had a slightly weaker theorem and we gradually improved it a lot. Eventually, we got the most general theorem. In terms of learning physics, I would say after starting to work on the positive mass theorem, I learnt a lot more physics in those following years than I knew before, even though I came from differential geometry. I knew just a little bit about Einstein metrics. It was really working on that problem that got me into relativity. Now I talk to a lot of people who are physics Ph. D., who work in relativity. I have a much better idea about the physical side now than I first started.

TPL: Before you and Yau worked together, Yau was not in general relativity?

RS: Neither of us was really in general relativity. We were both learning the background at that point. We had found a way to prove a special case of the theorem, and this led us to prove the full theorem.

TPL: That sounds really exciting, two young guys going into a new direction.

RS: It was exciting. Everybody was really excited about the theorem. Immediately it made a big splash in the relativity community. I think relativity is a subject with a good physical grounding, but ultimately the model is very mathematical. So the ideas connected with the gravitational mass and some of the other ideas from relativity were actually very close to ideas from differential geometry. So we were able to apply ideas about mean curvature variational theory to the problem. They fit very well to the subject. I would say because of that work, the subject has changed a lot. Now people working in relativity know about those things too. Just this morning I heard a talk where the speaker was talking about the same kind of ideas in a new context. Sergio Dain is a physics Ph. D., but he is very mathematical. So I think we did have a very big impact.

TPL: In retrospect, what would you say to make this success possible? Before that, you had solid ground in analysis. Then you two were so young that you were not afraid of anything and you learnt tones of new topics. You had to go through a lot of papers to figure out exactly what they were about and transformed into your own language. You also developed the theory. That’s a tremendous amount of things.

RS: I think it’s really important to broaden your perspectives as you do mathematics. Sometimes things that look like they are pretty far from what you are doing, are actually quite close. So I think in this case, there were people who were translating between physics and mathematics, like I mentioned Geroch. There were a few other people; very mathematical physicists who could speak both languages to some extent. There were a few in Berkeley actually, Jerry Marsden was a guy who was a physics Ph. D. but became very mathematical. He could explain the positive mass theorem and things like that to us directly. So we did have people right there in Berkeley to speak with about it. There was a relativity group. There was Abe Taub, Ray Sachs who was a really excellent relativist. They were in the math department. So that was definitely a help. I think talking with people is really important. Sometimes you can learn more just from understanding how people express themselves, than by trying to read a book.

TPL: Why you two could do it but those other people at Berkeley could not?

RS: Well, they did not have the mathematical foundation in the minimal surface side. That really was the difference. We had a deeper understanding of what you could do with minimization, mean curvature and so on and so forth. I think that is the difference. Physicists are very well trained in that subject, but they do not learn the mathematical tools. In the case something involving nonlinear PDE as you know, even speculations about how solutions behave are not quite reliable unless you can develop a theory and do something rigorous.

TPL: Then you solved the Yamabe problem, it seems to be somewhat related.

RS: I solved the Yamabe problem by realizing it is very closely related to the positive mass theorem, a special case of the positive mass theorem. The problem was around for a long time. People had a fairly detailed idea about how solutions would blow up. So there are natural ways of regularizing the problem. Then you can study; how the solutions blow up in problems that are very closely related. So the problem is to show the blow up does not occur. I realized when blow up does occur; you can scale things appropriately to produce a asymptotically flat solution of the Einstein equations. Then I realized the positive mass theorem gives exactly the correction term which, at least in the low dimensional cases, keeps the solutions from blowing up. So that was really the main idea. Definitely my experience with asymptotically flat manifolds, and positive mass theorem was really crucial in understanding the Yamabe problem.

TPL: Could I say that, yes, positive mass paper has been published and made a big splash, people are all aware of it, but you are the only one who had a deep feeling about it?

RS: At that time, yes. I don’t think people realized how important the positive mass theorem would be. Since that time, it has been used to do many things, particularly in relativity. Things like the uniqueness of the Schwarzchild solution amongst static black holes; there are many applications. The rotational symmetry of static stars was proven also using the positive mass theorem. People knew it as an important physical problem, but they did not realize it also could be used as a tool to solve other problems. So, I think I deserve some credit for recognizing the flexibility of the theory.

TPL: That’s because you are the one with deep understanding.

RS: Right, so I think it is true in the middle 80s when I did the Yamabe problem, I was one of the few people who understood the positive mass theorem.

TPL: Can I digress a little bit from this, we will come back to geometry. So, you are the one has done deep analysis and also important problems in geometry. How do you view these two fields, PDE and geometry? Perhaps you could even express your view of the kind of people in PDE and in geometry.

RS: Well, I don’t want to offend anybody. Well, at the time I started, most people working in differential geometry knew nothing at all about PDE, even though all of the problems in geometry were expressible as PDEs. They did not know anything about how to actually solve a PDE or get an estimate or anything like that; whereas on the other side, the PDE people do not understand any geometry at all. When you take a PDE that’s formulated without explicit coordinates written explicitly, they would feel very uncomfortable. So there is a gap between these two areas that I think is being slowly filling over the years, but I think it was really quite extreme around 1980s when I was a young researcher in the area. Frankly, I think one of the reasons I was able to do something that other people could not do was because I am able to combine these two different directions. I think that lots of areas in PDE are interfacing with geometry pretty well. I would definitely say that PDE people should learn more geometry and the other way around as well. Nowadays there is a subject called geometric analysis. Some people in that field are really very PDE oriented. Somewhat in the US but in Europe, there is a big group of people working on geometric PDE who are coming from the PDE side. So I think people fit in different places on a scale between PDE and geometry. I actually think that I have shifted a little more towards geometry over the years, but I am still a PDE guy. I have not done anything in pure PDE for a long time. I work mostly on geometric problems, PDEs that come from somewhere. I am actually interesting in other PDEs too, like fluid equations, Navier-Stokes. I am not an expert, but I know something.

TPL: One thing I know of you is that you have a broad perspective and you are open minded.

RS: I once taught a Navier-Stokes course.

TPL: The field of PDE of course has produced some very good mathematicians; so does the field of geometry, and these two kinds of people, are they very different kind of people? Do they think quite differently? Of course the subject is different.

RS: Well, I think all mathematicians are worried about being judged by people who are sort of philosophically in their spirit, right? I think part of the problem is that PDE people are worried that geometers are going to make judgments about their work which they don’t feel really fair and the other way around. So I think that is part of the problem. So people want to identify with a group, because it’s kind of a group support that they have. Of course, getting outside that group is a hard thing to do. I think for young people, of course they do have to be a little bit careful. It would be nice if people would make judgments about mathematician just based on their work not based on what field they are perceived to belong to. But I think there is always within a department some sort of territorial feature when it comes to appointments and things of that type. Some of it is natural. People vary a lot about as to how insistent they are on identifying with a particular group.

TPL: Your answer is very interesting, completely different from what I thought. You said that it is important for PDE people to learn geometry, but you seem to say that they do not partly because of this psychology of belonging to certain group. Isn’t it also due to the fact that geometry demands different thinking? If you are very much into PDE for quite a while, just seem difficult to learn this different thinking. Which shall we say, demand different kind of talents?

RS: There could be something to that. It is interesting to see how people work. When posed with a question, do they first draw a picture and try to get an idea of what is going on? Do they compute something? There definitely are people who are more geometric in their way of thinking; and people who are more analytic in their way of thinking. I think you are right. There is a difference in the thinking. But I think people can expand the way they think about problems as well, particularly senior people. I mean if you have a permanent job, in principle you have a lot of time to do mathematics. I think really top mathematicians tend to try to broaden their way of thinking. I think that is important, because sometimes something really good comes out of it.

TPL: Not to mention that this is important to educate the next generation.

RS: Yes, there is also a matter of what sort of role model you are for the students, both the contents of your course and the style of your interaction with young people I think it does make some differences.

TPL: You come from a farm somewhere in Ohio. I always have certain good feeling about people coming from a farm.

RS: So are you, I think.

TPL: Yeh, we seem to do different kinds of chores. I never go around with animals, except for my water buffalo. You are more versatile.

RS: I have done a lot of farm work when I was younger, that is true.

TPL: Still, during your growing up period, in addition to a lot of farm chores, you actually were a very good student, right?

RS: Actually, almost my whole family was excellent students. My family is very large. I think there were at least five or six who were the top student in their high school graduating class.

TPL: Good gene?

RS: Maybe. Also, I was one of the younger in the family. So when I was in middle school, my elder brother was in college. He was a math major in college. So he gave me math books. So that was one of the reasons I got interested in mathematics at such a young age. So I think that was a big influence early on. One of the advantages of growing up on farms, at least for me, is that I had a lot of freedom. I work somewhat but we have such a big family that I had plenty of free time too. So I could develop my own interest. I think mathematicians have to be independent. You have to be stubbornly pursuing something that somebody else might not be doing. So I think a farm background is consistent with that and helps with that.

TPL: There are enough periods of time in the day when nobody is paying attention to you.

RS: Yeh, almost the whole day.

TPL: How many children do your mother have?

RS: 13 and I am number 10.

TPL: Was your mother very healthy?

RS: She passed away at age 100 about two years ago.

TPL: I don’t think I know anyone else with 13 brothers or sisters.

RS: It is very rare. It is true. I have a former student who is one of eight.

TPL: I am one of seven. I guess in our generation it is more.

RS: Yeh, so I am really in a way from an older generation.

TPL: Through your career so far, you have in touched with many mathematicians. Some of them are leading mathematicians. Who are the ones give you strong impressions and in what?

RS: There are a lot of great mathematicians. First of all, we shall maybe separate very senior mathematicians from junior ones. Every time I am in a talk by Jean-Pierre Serre, there I am very impressed. The man is over eighty years old I think. He gives this spectacular blackboard talk. He is clearly a master of exposition and his subject. So there are people like that who I look at as being these very high role models. There are a lot of great people in my own field too. Yau is always being a role model of mine. Certainly in the early part of my career, he was definitely a role model. He tends to spread himself very thin these days so he is involved in so many things. I personally cannot follow that model at this stage, but he has been an important influence on me for sure. Misha Gromov in my field is certainly a very interesting mathematician. Sometimes he is a little vague, hard to follow, but he clearly has developed a field or subfield of geometry which has a lot of impact.

TPL: You have written some books. Do you have plans to write more introductory books?

RS: I have some lecture notes sitting around, of various types. I have a set of general relativity lectures. I also have a set of more general geometric PDE lectures which I might, when I run out of problems to work on. I do not really like to take the time to write the final version of a book. I think you have to choose what things where you can make most impact. I think if I decided that the materials of this book is important enough that getting it out there would have a big impact on the field, then I think I would probably do it. But at this point, I am not quite there. I would much rather work on new problems.

TPL: What are the directions that you have in mind these days?

RS: Well, I am still working in relativity on various kinds of things. I talked last week or few days ago about sort of localizing solutions of the Einstein equations. There are still several outstanding problems there. I got interested in some problems in Riemannian geometry a few years ago. It still is a subject which is very interesting to me, like the study of manifolds on positive curvature. So I proved a very good theorem on that. It is called the differential sphere theorem, four or five years ago. I was hoping that it would open the subject up more, but in a way it led to more difficulties. So I am still very interested in questions related to positive curvature for sure. So those are something I am thinking about at this moment. I have some other long term problems that I am interested in and will come back to I think. Often I come back to problems or get involved in new problems through my students. When I look for a problem for a student, I always try to choose something that is of an interest to me, something that I may not have worked on or may not have time to work on. Often those problems would develop into new directions for me too. So that is one of the reasons that I think that having students for me is very important. I am not a person who can just sit at my desk alone for ten hours a day and find new things myself. It’s a big commitment to work on a new problem. Having a student forces you to get started, to get somewhere,

TPL: That would be a good advice to many thesis advisors. How many students you have had so far?

RS: I think I have 37 graduates by this year. I am now over 100 descendants on the mathematics genealogy project.

TPL: You have some students beside Yng-ing Lee in Asia. You have some in Korea right?

RS: I have one student Jaigyoung Choe in KIAS in Korea, I have Sumio Yamada who is in Tokyo in Japan. Tom Wan is in Hong Kong. I think those are the only ones who are actually located in Asia. I have another recent student Martin Li who is going back to Hong Kong next year as well. I think in the future I will probably have more students in Asia. I think more Asian mathematicians are coming back. I have quite a few Asian students, many of them stay in the U.S.. I think of the 37, maybe more than a third, almost half are Asians. So the Asians like my field. They have good training in both analysis and geometry.

TPL: The students from Asia and the American students, are they different in the large?

RS: Well, first of all, right now, there are almost no American students who are working in analysis. That is my impression. I think it is a cyclic thing. Right now, number theory and algebra are the big area for American students. So there is a big difference in terms of background and what they want to do at the moment. I think in general, of course the university system is very different in the United States and in Asia, so American students are often less advanced but maybe a little broader in their thinking. Asian students often have some very specialized courses so they are better prepared in a certain way, from a technical point of view. But it does not necessarily mean that they will end up being better. Doing research is quite different from taking classes. So it is a question of how that transition goes.

TPL: So would you say that a broader interest is important in the initial stage like high school and undergraduate?

RS: Yes, I do think so. I think all math students shall learn some physics. They shall at least have some ideas about physics, especially anyone who is thinking of going into geometric areas. I think it is important for undergraduates to have a broad mathematical training, in the sense that they shall know algebra and analysis at a good level, at a high level. I do not believe in specializing too early, which I think some students are having that tendency these days in the United States. They feel like that if they have not done research by the time they graduate, they get their bachelor’s degree, they are somehow behind. I actually do not think that is true. Often the undergraduate research does not put them in a good direction. That is my concern. So I think the REUs and the heavy emphasis on undergraduate research is kind of mixed. There is some good but there is maybe also some not so good.

TPL: This Olympiad in high school, a lot of them are in discrete mathematics, for example. Does this explain the fact that some of the good students go to algebra and number theory?

RS: That is one of my theories. I think there is an over emphasis on this competition in mathematics. In the United States, it is very hard to get into top universities, so one of the criteria they use is whether the student has done well in those things. So the result of that is a lot of students who come out of Harvard or Princeton are aimed in those directions. So I do think that in certain ways it has a negative impact. On the other hand, of course anything that engages students is a good thing, so again, I wish that those exams were a little more broad, use a little broader set of techniques. The problem is to work in PDE or geometry requires a lot more background. So it is hard to cover things like that on exams of that type. So I think that might be part of the problem. When students ask me about it I always emphasize it is just for fun, but I can tell that they do not believe me. They really do think how they do in that exam is going to make a difference in their career.

TPL: At that tender age. I always compare number theory or even part of geometry or algebra with analysis; it is like in Chinese literature comparing the poetry with fiction. Fiction is very different from poetry. Chinese has a very glorious past in poetry in Tang and Sung dynasties. At that time, the Chinese society always praises the poets. Therefore, in fiction, China is far behind Japan, not to mention Europe. Fiction is kind of messy and you have to go through life and so on. You need to prepare yourself in life to write a fiction. Poetry is somewhat different.

RS: I see the point you are making. I also think that mathematics shall be connected to the real world. I think a lot of richness of mathematics is connected to the real world. I think part of the problem is those exams are almost totally disconnected to the real world. It treats mathematics as a game, a puzzle that you solve, which of course it can be and is very challenging. But I do think that of course there are lots of combinatorial problems that are connected to the real world. So it is possible to do something that is more concrete. I think the real world side of mathematics shall always be emphasized. When I was a kid or when I was a student, everybody regretted the Bourbaki era in mathematics. Bourbaki, the French took mathematics and they made it very abstract, so nobody could understand it except a few people who were in those subjects. Somehow when I was a student, mathematics went back to the concrete. People rejected the Bourbaki approach. So I am not sure we are going a little back towards it recently. It is certainly nowhere near what it was in the 50s or 60s, but I think that is something mathematics has to watch out for, is to stay connected with other disciplines.

TPL: Your growing up was in a farm, does this make you in some sense connected to the real world.

RS: I was connected to the real world in a way I didn’t want to be. Actually when I went to college, I really liked the abstract things. I liked functional analysis, logic, the sort of the abstract machinery. It was very attractive to me, as I think it is to a lot of undergraduates. So that is why I think it is important not to choose specialty so early. It was not until I went to graduate school that I realized other areas of mathematics that I had not seen before. So I would say this emphasis that mathematics shall be connected with other disciplines I think it is something that has of increasingly evolved in my career. I think the connection with relativity probably plays a role there. I think differential geometry is a subject pretty connected with physical problem generally. So I think it is important to be able to explain your work to some people who are not experts. Anyway, the connections with other areas I think are valuable. But I do think that mathematicians need to mainly look out for the quality of the mathematics, and not just be a service discipline also. It is kind of a balance there.

TPL: I have heard you giving talks. People also told me that you are a very good teacher. What are the things you can say about teaching?

RS: I think I am a good teacher in classes where content is the main thing. I would make a careful effort to explain the content in a way that the students can understand. So it is important to understand the background of your students of course. There are classes in the United States that we teach that are very low level and we are teaching it more as an entertainment. It is more about getting students engaged. So if I can make the assumptions that the students are interested in the material; they want to learn it. Then I think I am a good teacher at classes of that type. Mostly they are the classes that I teach.

TPL: So you prepare well.

RS: I prepare well. I try to explain things in different ways and on a level the students will be able to understand it. I have a teaching challenge coming up in the next quarter. I am teaching undergraduate Riemannian geometry, it is the first time we have ever taught it. So it is a follow up course to the curves and the surfaces in the undergraduate differential geometry. So I have ten weeks to teach them about Riemannian geometry. I think that will be a challenge.

TPL: Maybe that would be a good subject to write a book.

RS: Right, it is true I have a very hard time finding a book and I am not sure my choice is going to work well.

TPL: It is always nice talking to you. We shall get back on such an occasion in few years.

RS: Thanks! It is my pleasure. It is always great to be in Taiwan.

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