IMPRINTS: Issue 1 - May 2003

The Editor (Y.K. Leong) of Imprints interviewed Roe Goodman of Rutgers University on 11 February 2003 at the Department of Mathematics, National University of Singapore while he was visiting IMS and the Department of Mathematics. He was a guest participant in the IMS program on "Representation Theory of Lie Groups". The hour-long interview covered topics that range from teaching and research in mathematics to the influence of masters in mathematics and music. The following is an edited version of the interview.

Goodman's extensive research activities are centered around Lie groups. Together with Nolan Wallach, he has written a 685-page encyclopedic book Representations and invariants of the classical groups that is both an introduction to as well as an authoritative reference on the structure and finite-dimensional representation theory of the complex classical groups.

Imprints: Can you share with us some of the excitements of your latest research?

Roe Goodman: My own research started in the 1960s when I did my PhD thesis with Irving Segal in MIT. Segal himself was primarily interested in the mathematical problems of quantum field theory, viewed in a very broad sense: difficult questions in non-linear partial differential equations and their symmetry groups. My own interests and activities over the years have moved in the direction of representation theory and the symmetry groups although I maintain an interest in application to physics. The thing that attracted me to representation theory is that it lies at the crossroads of all of mathematics. You have the analysis side in connection with partial differential equations and you have algebra and geometry in the Lie groups. One of the things that is exciting about this field, as I have watched it developing over the last 40 years, is to see so many areas of mathematics come into this field - more and more of combinatorics, geometry and algebra - even though the subject started out with a lot of emphasis on analysis. Of course, one of the central things that make mathematics research so exciting is that, over the course of time, you see that problems that first seemed intractable examined by lots of people who find new ways to approach these problems. For example, problems that were originally posed as questions of functional analysis can now be approached using algebraic techniques, which simply avoid some of the difficult, maybe impossible, analytical problems. I have spent quite a lot of time over the last decade telling the story for the next generation, so to speak, in my collaboration with Nolan Wallach. We wrote quite a big book on representation and invariant theory, trying to make the basic results and philosophy of representation theory accessible to the current generation of mathematicians (and we hope to another generation).

I: Are there any unifying trends in the development of your field of research? Do you think that particular problems have to be solved first before some unifying theory can arise, or do you think that essentially new theories and concepts rather techniques need to be proposed before outstanding problems can be resolved?

G: In my field, it seems to be that there is this cycle of solving particular cases and pushing the methods that suffice for those cases as far as possible. At a certain point those methods often turn out to be insufficient or the computational difficulties simply become insurmountable and then there are new approaches that come in. One of the striking things about mathematics is the insistence to understand the subject from the conceptual point of view. For beginners of the subject, it is hard to understand the concepts without actually doing some calculations. But at a certain point, you discover that even if you have a very powerful computer doing symbolic calculations for you, the calculations alone are not going to tell you what the pattern is. You have to discern the pattern, and I think finding the pattern is one of the main purposes of mathematics. Of course being able to come out with an answer that can be translated into some of the applied domains is also very nice when you can get it.

I: Do you think that maybe sometimes the problems could be not so much notational or computational but conceptual difficulties?

G: Yes, certainly. There are striking examples where leaders in the field bring in new concepts and new approaches that are picked up by many people working on the subject and there is a kind of collective effort on the concepts which initially are very attractive but need a lot of work. I am thinking of, for example, Langlands program that links representation theory and number theory or ideas of Gromov in geometry or people like Kostant and Kirillov in representation theory. The individuals working in the field try to understand these leaders on their own terms, and that is where you get a range of results.

I: You mentioned leaders in the field who propose new concepts or prove new results. Very often they do so with a tremendous amount of intuition. Sometimes there are no real grounds for them to do so and yet their intuition works.

G: Yes, that is certainly true. The intuition is spectacular when you get into the area of mathematical physics. When someone like Witten proposes formulas one can often only interpret them as mathematical poetry. When one tries to give them literal analytical meaning, they become very difficult to justify. What has happened in recent years is that more and more algebraic and topological tools have been brought into the field.

I: Einstein is supposed to have said something to the effect that there are too many tantalizing problems in mathematics but few problems that are of central importance. In physics, how its central problems are resolved will determine and shape the future of physics whereas mathematics will still go on in very much the same way whether its numerous problems are resolved (one way or another). How much of this is true for contemporary mathematics?

G: I think that Einstein's point is well taken. In physics the relevancy of a model is paramount. If someone proposes a model for some area of physics, there will be an enormous number of papers typically written on that model. The model may be abandoned if it turns out not to be productive, and there is no point in pursuing every detail. I think mathematical ideas have a much longer half-life. The lectures I have given here over the last six weeks are based on more than 100 years of mathematical work. I mean, not my work, of course, but the citations in my lectures start with the work of Schur, almost 100 years ago. The real difference between physics and mathematics is that although physicists and mathematicians use the same symbols, their languages are different from the point of view of concepts. If I may be permitted to use the analogy of the Chinese language (I don't know any Chinese dialect), one has here a universal set of symbols for conveying ideas, but the spoken languages are very different. When a mathematician writes down an equation, he usually works hard to make it look easy. He wants to give it an intrinsic meaning, not some kind of semi-precise or poetic meaning, although mathematicians also like to have a poetic element in their work. In mathematics you have to keep expressing the language so that each new generation of mathematicians can talk among themselves. If you pick up a journal article from 100 years ago, an article by someone like Frobenius can be read very easily. Other articles can be hard to understand; even though the symbols are the same, the way the ideas are expressed has changed a lot.

I: In physics there is some kind of blueprint for the development of the subject whereas in mathematics there is no specific blueprint as to how mathematics should develop.

G: That's right. The remarkable thing in mathematics is that you have these extraordinary imaginative people (like Gromov, Langlands) who propose concepts that, to ordinary mortals like us, seem to just come out of the blue. Of course, they have a basis for those ideas but it can take the work of a lot of people to develop the consequences.

I: Do you think we have now reached the golden age of mathematics? Do you think it is largely due to the easy and swift access to and dissemination of information provided by information technology?

G: I certainly think that the possibility of going online and trying to get access to a large number of journal articles without having to leave your office or home or wherever in the world you are is quite wonderful. I know how different it used to be when I went to New Hampshire for the summer and carried a lot of journal reprints with me. On the other hand, I think that information technology or even a powerful computer is not enough to create mathematics, although it can give clues as to what is possible. Mathematics is basically created by mathematicians and not by the technology. The technology certainly plays an important role in communicating mathematics because now it is so incredibly much easier to turn mathematical ideas into printed form. This enables us to communicate ideas and to put mathematical papers in online archives so that people know right away when something new appears.

I: Do you see a continuing trend in which mathematical talent is being siphoned off into other areas like biology and computer science?

G: Yes, that is certainly the case especially at the undergraduate level in the US. I think that the beauty and attraction of mathematics are pretty obscure to most undergraduates. There will be a few who will have the natural inclination to like the rigor and purity of mathematics but others, if they have more practical instincts or are looking forward to monetary rewards, will not be attracted to a field that seems so difficult of entry as mathematics.

I: Pure mathematics is very demanding in terms of training and research standards. Even good students may find it too demanding and difficult to seek out careers in pure mathematics. Is your own university department making any efforts to attract mathematical talent to remain in pure research?

G: In US, we have this division between the undergraduate and graduate studies, especially in a university like mine which is a state university of quite good quality but not an elite institution like Harvard or MIT or Princeton. It is hard to get undergraduates into pure math because most of the talented ones tend to be in engineering or computer science or perhaps physics. We do try to attract students by seminars and honours sections of the basic courses. At the graduate level, I think one of the most positive new things in my department is the "pizza seminar" run by the graduate students. These seminars are held once or twice a week at lunchtime, and are only open to graduate students; faculty is not allowed. Students lecture on something they are working on for their theses or some other areas of mathematics that they are interested in. This seminar initiates graduate students into the process of talking about mathematics to others in an informal setting. Since they are talking to other graduate students, they are not intimidated by faculty. I certainly wish I had that when I was a graduate student.

I: This is quite unusual because mathematical activity is not often looked upon as a social activity.

G: Exactly. It is unknown to the public at large that mathematics is not just written down in books, but has an oral tradition, and that mathematics lives because people talk about it and they communicate it to each other. I think that this tradition in mathematics has many things in common with the tradition of folk music passed on by oral tradition. To do mathematics is a hard thing, and one needs the encouragement of a supporting culture. You can never get all those details right on paper and it is only the paraphrasing of things and summarizing them in speech that gives one the courage to continue writing things down in print.

I: I remember there are a lot of things in mathematics that are known to students but which somehow cannot be found in books. It is some sort of folklore.

G: Exactly. These are folk theorems. The point of mathematics it is not simply to set things down like writing laws. Mathematics is a body of knowledge that is passed on partly in print and partly by oral transmission.

I: There is some perception that pure mathematicians look upon practical applications with disdain. How much of this is true? Some of the best mathematical minds like Hibert and Poincare have worked in both pure and applied areas. Is it possible to achieve their status in the present age of specialization?

G: Judging by my own experience, it seems hard to establish links with applied science departments like chemistry, computer science and physics partly because the faculties in those departments themselves have quite a high mathematical level and they generally view the kind of mathematics they are using as something that they are reasonably competent with and they don't seem to have an enormous need for mathematicians. Of course, one can try to create the need, and there is also the tendency on the mathematician's side to think, as the phrase goes, "We would rather build fire houses than to put out fires". But there are remarkable counter-examples. My own personal hero is Hermann Weyl, who is not very well known to the general public. He was a student of Hilbert. He gave the first set of lectures on Einstein's general relativity theory in 1917/1918 and published basically the first book on general relativity theory based on those lectures. He also worked as a forerunner in understanding and explaining what was going on in the new quantum theory in the 1920s. Certainly there are many examples of mathematicians who have done this. In recent years someone like Irving Segal is an inspiring example. Another person who comes to mind is Michael Atiyah. As physics becomes more mathematical and uses a wider range of mathematical tools like algebraic geometry, physicists have to turn into mathematicians.

From the point of view of people completely outside of science, and in particular people interested in what is the worth of mathematics to society at large, they would like to see how mathematics can be turned towards more practical things. It is interesting to observe that in my own field involving harmonic analysis as well as representation theory very recent work in wavelet analysis is essential for things like image compression and data analysis. A lot of that grew out of what used to be thought as quite abstract kind of harmonic analysis and abstract Fourier analysis. It is a question of having links with applied mathematics. There are people like Coifman at Yale, who, when I first knew him in the 1970s, was working in the kind of representation theory and harmonic analysis that I was. He moved into wavelets and has been very successful in promoting its commercial technology.

I: Did you try your hands in physical problems?

G: Not so much. I always thought of doing it. I have another life as a professional musician and so I like to think that in a certain sense my practical side is in connection with making my instrument work and playing music. But that is not the same thing as actually doing applied science. As a teenager I was very attracted to physics and music. I thought that if I don't go into music as a profession, maybe I will go into a field like acoustics, but in the end that didn't work out and I ended up in mathematics instead.

I: What instrument do you play?

G: The instrument that I play most seriously is the bassoon. Originally I started out as a child playing the cello but then when I was fourteen I switched to the bassoon, which I consider the most non-linear oscillating system that is of any practical use. So every morning when I practice my instrument I perform experiments on a little non-linear oscillator in the form of a bassoon reed.

I: What made you switch from music to mathematics?

G: My father was a professional musician, a pianist, and I knew from personal experience the difficulty of making a living as a musician. Most of my adult friends in the orchestra that I played in as a teenager advised me that it would be much better to go into science. But it was hard for me not to go into music and composition because that was what I was most passionately interested in at the time. In retrospect, however, mathematics has been a very rewarding career, and I have still managed to maintain an active musical life.

I: It seems that a lot of mathematicians are musically inclined.

G: Performing music at the professional level has given me some insight into mathematics as a way of communication and performance. In music, you don't just walk out on stage and play a composition by sight. You spend a long time practicing off stage before you even start rehearsing on stage. I feel it is the same thing when you are dealing with mathematics lectures or trying to teach mathematics. There is a long period of private confrontation and there is a tremendous excitement of actually performing. As in performing music, giving mathematics lectures opens up the mind to other possibilities. I always find that after giving a lecture, I have a lot more ideas about the thing I am lecturing on.

I: Are you more interested in jazz?

G: No, the bassoon is a classical instrument. But I also played the trombone when I was in high school and I played quite a bit of jazz at that time. I play in a professional symphony orchestra (the Princeton Symphony Orchestra).

I: In music you can improvise when performing. Do you improvise when you give a lecture?

G: Absolutely. When I prepare a lecture, one reason for writing down the lecture notes (even for elementary courses) is so that I can try out things I would not say. I try to see where it is I would like to be at the end of the lecture and then I would leave things out of the lecture if they are going to interfere with attaining that goal.

I: Do you prepare your lectures thoroughly?

G: I prepare them but in the same way like jazz musicians like Louis Armstrong would practice and practice so that in the actual performance you can improvise on the theme that you are going to be using.

I: Do you think that part of the problem of overspecialization is a lack of communication between mathematicians and people in other areas like engineering, physics, and computer science?

G: Yes, that is certainly a problem and I think that is a real challenge for mathematicians. One can almost feel it as a drawback of mathematics that we have such a perfect system of notation that for us the notation serves all the purpose that we want in the same way as the written language serves our purposes. But students and people outside of mathematics often tend to view mathematics simply as a collection of symbols to push around. When the symbols get too complicated, only the professional mathematicians can read them and then people outside the field just turn off. I don't know how to get around that. I teach engineering students a lot and try to explain the concepts in a way that is acceptable to them. I view that as one of the biggest challenge when I am teaching. Of course, it can be quite frustrating because you know as a professional mathematician that with the benefit of an appropriate concept certain ideas can be quite simple. But this is only true if the person dealing with the concept has mastered it, and for people outside mathematics the notation and concept can be so obscure that it is very hard to get the ideas across. I think that is one area in which mathematics, as a profession, sometimes tends to be too narrow. We don't realize that the mathematical ideas are just too dry when used by students outside of mathematics.

I: How do you select the problems you intend to work on?

G: The problems tend to come to me when there is something I would like to try to understand and I realize that there is a barrier to my understanding that I must work through. Most of the problems that I have worked on in representation theory have their origins in mathematical physics. There you would have a formulation of some problem where, as is usual in physics, there is not too much worry about whether things are literally true. A power series or some mathematical expression is written down and it is assumed that it can be manipulated as much as needed to get the answer. Often some of my earlier work was involved in exploring in more detail whether some of the things that physicists would do by manipulating some formula was really valid. And of course the interesting thing about mathematics to me is that when you look more carefully at a problem, usually the correct answer is a lot more interesting than anything written down blindly. There is a subtlety in the problem that emerges as you look at it in more detail.

I: Do you have any particular strategy for attacking difficult research problems? How much can perseverance replace inspiration?

G: First one tries to see where and why the problem was difficult. Trying some simple examples and keeping in mind a quotation of Einstein, "One should simplify but not over simplify". The first test is to see where the difficulty is and then you realize that the problem is actually difficult and may be more difficult than you can deal with. It is a remarkable phenomenon that after working on it for a while and walking away from it, your mind keeps working on it in the background and then somehow you find some way of getting out of it. Sometimes the reality is that everything you try runs up against a dead end. Then someone else suggests a completely different approach that is successful. So one goes between the two poles of despair and enthusiasm as to whether your given line of attack is going to work and you keep pushing it for a long as you can.

I: When you work on a problem, do you first find out the relevant literature on the problem, or do you get into the problem straightaway?

G: I try to work on it first straightaway. I would say that technology has made it a lot easier now to find papers by doing a key word search electronically. When I was working on problems decades ago, there was a lot of library work. Nowadays some key words search can churn out papers that you might have no idea existed. That happened to me several times recently and I was able to get some perspectives on the subject much more rapidly than I could have had. So I think that is certainly one very positive aspect of mathematics even though mathematics has become, in some sense, much more complicated.

I: Do you think that by doing that you might be influenced in your own approach towards the problem in the sense that reading up what has been done may affect your thinking about the subject?

G: When I was an editor for a journal, I discovered that most authors have their own views of the subject and that if a referee suggests that the author should have written the paper in a different way, most authors are not happy about accepting the advice. I think it is the same in your own research. Your have a certain outlook on the problem and it is great when you find that somebody else's paper gives you insights you have not thought of and then you pursue that for a while. But I think you have a tendency to come back to the methods with which you are comfortable.

I: There are some people who would attack a problem from first principles. They develop their own understanding of the problem and then develop essentially their own methods for the problem.

G: I think the most spectacular example in my own field is Harish-Chandra who, starting in the late 1940s, simply came into the subject of representations of semi-simple Lie groups on his own. There had been very important preliminary work by the Russian school under Gelfand, but Harish-Chandra started at the beginning and created an incredible edifice single-handedly. For a period of about 25 years, starting from the late 40s to the mid 70s, he was so clearly leading the field. that it was only in the early 70s that there was a significant number of other people working in the field. In his case, the methods were always his own. He took what were, in some way, very classical methods and extended them to serve his needs. It has taken several mathematical generations to go beyond Harish-Chandra's methods. His ideas had tremendous depth. Of course, now more recent approaches to the subject try to understand it by other methods, but he basically set the direction in the field. The results achieved were so precise and profound that everybody in the field has to take his methods into account. A parallel instance in mathematics of someone creating a monumental edifice is in algebraic geometry. Grothendieck created very general machinery that has now become the language of algebraic geometry. So I think the absolutely strongest people in the field simply create the field by using their own methods and then the rest of us have to learn those methods and see what other results can be obtained.

I: It is really remarkable. It is almost like from nowhere. In some sense, this can be a bit demoralizing to students of the subject.

G: Oh, yes. In the mid 1960s when I had done my PhD thesis, I was looking for an area to work in. I asked James Glimm, who was on my PhD committee at MIT, and he said to me to try and understand Harish-Chandra. Using the latest technology at that time I made a big stack of Xerox copies of Harish-Chandra's papers and carried them with me wherever I was working. I read the first few of them and it was such hard going that I never succeeded in climbing all the way up "Mount Harish-Chandra". In fact, I worked in a different area of representation theory at that time because I didn't see that I would ever master the Harish-Chandra's theory. So it has been very interesting for me to see how the subject has evolved in the succeeding decades. At present, the Harish-Chandra theory is more accessible because many people have worked on it, but it is still a major piece of mathematics.

I: So essentially one has to start very young to learn the basics?

G: I guess so. I certainly did not start very young in mathematics. Since my interest as a teenager was in music rather than in mathematics, I only began doing mathematics seriously when I went to graduate school at MIT around 20 years of age. On the other hand, just like doing sports or being a performing musician, starting young when learning the language means that your body and mind are more flexible and more able to take it on.

I: Is there a role for perseverance? How much inspiration does one need?

G: Oh, absolutely. I think without perseverance you certainly can't do mathematics. If there are never any ideas that come along, it is pretty discouraging. It is an elusive thing. Solving a mathematical problem is trying to judge at any moment whether the track that you are trying is going to pan out. Of course, perseverance alone may not work, but even if it does,, you try to know whether you are moving towards a dead end. That can be very discouraging in mathematics.

I: How do you know when to stop a certain line of approach and try a new one?

G: I don't know. I think from the positive point of view probably the best strategy is try to at least get some result and not to wait for the final perfect results. To get results, you have to work in private, of course. But at some point you have to talk about the results and to submit them for refereeing and publication. The example of working in isolation for years and then publishing the final results in the case of Wiles on the Fermat problem is very exceptional. With such a long history and enormous interest for that problem and with so many false proofs announced over the centuries, I suppose it was important for Wiles to work in private. But for most mathematicians, publishing results frequently, even if partial, is essential.

I: Do you get results when you are not working on the problem - when you least expect the ideas to come?

G: I have never considered myself a particularly creative mathematician. I have to be thinking about a problem a lot. Once I am really immersed in the problem there is a mental momentum that builds up. Mathematics is about patterns and the human brain seems to be determined to find patterns. When we look at a cloud we see a face or a mountain or something. We are all looking for patterns. I think that is part of human intelligence, but the patterns you look for in a mathematical problem might not be the patterns that you know. So in thinking hard about mathematics or mathematical problems you are thinking about all the patterns that you do know and trying to see if the problem you are working on fits, in some sense, into those patterns. In that sense, sometimes a solution, or at least a strategy for the solution, comes out when you are not thinking about the problem. Especially sometimes when you realize that maybe a certain strategy has a chance of working. Then you have to sit down and see whether the strategy is going to pan out.

I: It's a lot of hard work sitting down and working it out, isn't it?

G: For me, certainly. My ideal in music is Bach and Mozart. Both composers could work out fantastic combinations of utmost beauty in their heads before writing anything down on paper. On the other hand, Beethoven wrote down the initial drafts of his compositions (he kept notebooks all his life) in a way that was so crude that you couldn't imagine that he could make anything significant out of them. So as I scratch along in mathematics, I get great comfort thinking that I am trying to follow Beethoven's footsteps. I try not to be timid about writing down an initial draft but the point is you don't just leave it at that. You keep revising it, you keep thinking whether what you have written down is really what you intended to write down and, even more importantly, whether it is actually true.

I: Do you think that mathematics is like a marathon race that is long, arduous and lonely?

G: I think there is a partial truth in that comment. But there is such a large social element in mathematics, public perception notwithstanding, in the sense that if you only create mathematics in writing and never tell anyone about it, then it is like running a long race where no one is even looking. I like to think that at least there is this aspect of mathematics as a communal effort. As Einstein commented, there are innumerable problems in mathematics. But I think the ones that have a life of their own are the ones that have a significant number of people (which, of course, in mathematics could be a small number) with some real interest in those problems. And then the joint efforts of people working on these problems make it interesting - you get some results yourself and compare yours with what other people have. So maybe instead of thinking that it is a long marathon race, it is more like a situation I observed once, to my surprise, at a rehearsal of the orchestra. A grand piano was on the floor of the concert hall but needed to be on the stage. I certainly couldn't lift it by myself, but with eight people it was very easy to lift the piano onto the stage. So I think hard mathematics problems may have some of that element of joint effort. Of course, it is one thing to get the piano onto the stage and another thing to get a beautiful performance. We do need the gifted mathematician to give the beautiful performance but the joint effort can play an essential role.

I: If you were to live your life again, would you take up mathematics?

G: When I went into mathematics, one of the reasons that made it attractive from a very practical point of view was that it was just at the time (1958) when the Russians had launched Sputnik. The US was in a complete state of shock that another country somehow could have taken the scientific and technological lead. This was also at the height of the Cold War. There were many financial incentives and other incentives in terms of postdoctoral research and teaching positions. I think the opportunities in term of intellectual development that mathematics offers are very great, and the community of mathematicians is an interesting bunch of people. I think the independence of thought and the insistence on careful analysis is admirable. I enjoy working with students, and I think that the human aspect of working with students is something that is missing in some jobs in fields like finance and business. Of course, different professions have different aspects but for me the human side of being a professional mathematician and working in a university setting in many parts of the world has been very rewarding.

I: It is very encouraging for students - the different aspects of mathematics.

G: I try to convey it. I have been fortunate in having some wonderful teachers in my mathematical career. Judged by narrow standards, in some cases the teaching was terrible but they were all very inspiring in terms of their passion for mathematics and their interest in their students. So I try to communicate this to my own students. And in some measure I think I have succeeded.

Copyright © 2003 Institute for Mathematical Sciences, National University of Singapore.