傳播數學知識．促進數學教育

Interview with Prof. David Vogan

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

**Interviewers:** Shun-Jen Cheng(SJC), Weiqiang Wang(WQW), Meng-Kiat Chuah(MKC)

**Interviewee:** David Vogan(DV)

**Date:** December 20th, 2010

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

Prof. David Vogan was born on September 8, 1954 at Mercer, Pennsylvania. He received his B.S. and M.S. in 1974 from the University of Chicago, and his Ph.D. in 1976 from MIT under the supervision of Prof. Bertram Kostant. In 1979, he became a member of faculty at the Department of mathematics at MIT. He works on unitary representations of Lie group. He was invited to give a keynote speech at the International Congress of Mathematicians in Berkeley in 1986. He was elected to American Academy of Arts and Sciences in 1996, and then to the National Academy of Sciences in 2013. He served as the president of the American Mathematical Society from 2013 to 2014.

SJC: First of all, welcome to Taiwan, I suppose this is your first time here.

DV: That’s right. It’s been delightful visit. The computer said that it was going to rain all week, instead, it’s sunny and beautiful weather. Of course, there are very nice people, very nice food and very nice mathematics. So I am quite happy.

WQW: You know all three of us took your class before.

SJC: I am not sure whether …

WQW: You didn’t?

MKC: I did.

SJC: I did. Not officially, but I certain did when I went to Harvard.

MKC: I think I took 18.757 Representation Theory of Lie Group.

WQW: I told you that you have a good number of students. Certainly when you were students you heard things about professors. As nowadays, other students will see things behind us. One thing I want to ask is that how quickly you went through the education system to get your Ph.D. from undergraduate?

DV: I was a student at MIT as a graduate student for two years. Near the end of my second year, in the middle of my second year, I thought things were going nicely that I had a couple of years to finish thinking about things and write carefully and slowly. Kostant stopped me in the hall and asked me “Do you want a job next year?” Of course the only answer to this question is “Yes”, you want a job. So, then, I had to work hard to finish off by the end of that year.

WQW: So it’s within two years. Undergraduate was four years or shorter than that?

DV: Three years.

WQW: So three plus two.

DV: That’s right. My wife and I were going to colleges far apart. We were planning to get married. The idea was we would get married after we were done with college. The idea was that should be soon.

WQW: That motivated quite a bit.

DV: That’s right.

WQW: Where did you go for undergraduate?

DV: The University of Chicago.

MKC: You got the Master’s degree in three years?

DV: Yeah.

SJC: You got the master degree and the bachelor degree in three years?

DV: Yeah.

WQW: When did you decide to choose mathematics?

DV: Maybe during my first year at Chicago. I have been interested in Astronomy from the time I was seven or eight years old or something, because it’s very easy to be interested in Astronomy. There are lots of beautiful things to see. You can read nice books about planets and stars and so on. So I like this very much. But then when I got to college and had to take real physics courses, they were hard. Physics labs were impossible for me. I couldn’t make any of the equipment do the things it was supposed to do, whereas, mathematics was easy and I found out that in college that it was also interesting and fun. I had really good teachers again and again, even when I was in high school. Just wonderful teachers.

SJC: So I mean, what you have just said is that you didn’t enjoy the lab, so maybe that’s the reason that you did not get results which you were supposed to get.

DV: That’s true. That’s certainly right.

SJC: Maybe that’s the reason you didn’t not want to pursue.

MKC: So you get your Ph.D. in an earlier age than most people.

DV: There was a time when I always enjoy being the youngest person at conferences, and then, many years ago, I was for the first time the oldest person. Now this is more typical.

MKC: What age did you get your Ph.D.?

DV: 21 I guess, just before my 22$^{nd}$ birthday maybe.

SJC: So your first position was at MIT?

DV: That’s right. Yep.

WQW: So Kostant offered you a job, and you are still in the same place.

DV: Yes, it sounds boring.

WQW: No wonder you were saying it’s an offer you cannot refuse. When did you choose Lie Theory to be your future area?

DV: Well, when I was an undergraduate student at Chicago, there were many wonderful teachers there, but one of the most amazing one, who is very famous even now, is Paul Sally . I mean, he is a famous teacher at many levels. He has many wonderful graduate students and he has a huge influence on undergraduate students. In Chicago he runs a number of programs. He runs programs for high school teachers of mathematics and also for high school students. He teaches everybody. He does representation theory. So it just seemed that's what I should do too.

WQW: It’s a coincidence. We had this colloquium speaker just two weeks ago who mentioned one story about him. If I quote correctly, “there is this thing you shall not let students take late exams or make-up exams unless he is attending his own funeral.” You have probably heard about it.

DV: That’s right. There are many stories about his dealing with students trying to get exams changed or something. Some student I think told him that he had a flight reservation at the time of the final exam. Sally took his cell phone away from him and opened it up and called the airline to cancel the reservation.

WQW: Soon after you graduated, this Kazhdan-Lusztig Theory was a major thing. You also certainly played a quite significant role in this part. Can you tell us something about this?

DV: Well, it was an extremely exciting period. When I was learning this subject in early 1970s, for infinite dimensional representations, even the classification of the representation was not well known. I mean, Langlands has done this work in the late 1960s, but he didn’t publish it. Not many people understood it. In fact, because I didn’t know about it and people at MIT didn’t know about it, more or less, I repeated some of it for my thesis, but it’s okay, it’s good to copy good mathematicians. So this huge advance there, Wilfried Schmid was doing work about the discrete series representations and the proofs of Blattner’s conjecture. Gregg Zuckerman had all these wonderful new constructions of representations, so that all of a sudden, you could build enormous new classes of representations that were completely mysterious before. So there was just enormous progress in how much you could understand this subject. In some sense, it went from understanding things not at all to understanding the way you understand Verma modules from an elementary point of view, which was a huge advance. But then, of course there are sophisticated questions about Verma modules like the character formulas for irreducible things which were just completely inaccessible. By the end of 1970s, Kazhdan and Lusztig came along and said, yeah, we could do those problems too.

WQW: Conjecturally?

SJC: Very quickly solved.

DV: I mean, I always think this formulation that Kazhdan and Lusztig made a conjecture that was then proved by Beilinson and Bernstein , and Brylinski and Kashiwara . I think this is unfair to Kazhdan and Lusztig. There are many difficult steps in the proofs and most of them were done by Kazhdan and Lusztig. So, well, you know, I don’t have to rehearse it all now, what they did was they identified some really sophisticated geometric objects which they conjectured were related to representation theory. So that’s the conjecture part. Then what they did, was they showed how to calculate this really sophisticated geometric things explicitly and completely. This, nobody could do anything like that. These sophisticated geometric objects, what Brylinski and Kashiwara, Beilinson and Bernstein did, was to prove that these geometric things were really the same as some representation theory things. The world is full of mathematical theorems that say two difficult problems are equivalent to each other, but you can’t still solve either problem. You know, if it weren’t for Kazhdan and Lusztig, I think this equivalence could still have been proved because it’s not very hard to prove this equivalence. But then to calculate the geometric things, that’s much harder. This uses the Weil conjecture and all this really deep work of Deligne , and that was done entirely by Kazhdan and Lusztig.

WQW: You are referring to the Schubert varieties side?

DV: Yeah, that’s right.

WQW: But was in the work of Kazhdan and Lusztig D-modules available? They serve as the bridge in some way?

DV: No, this machinery that makes the connection between representation theory and the D-modules is extremely clever, but it isn’t very deep. You know, this whole equivalence there is a beautiful and powerful idea but it isn’t so hard. The Weil conjectures are hard.

WQW: Also, the paper of Beilinson and Bernstein, at the beginning, of course was just announcement in a way. How did you think about that? It took them another 10-15 years to write a much fuller and stronger version.

DV: Well, I try to understand what they were doing in terms of the ways that I already understood representation theory. I mean, nobody understood very well. So Beilinson and Bernstein gave a classification of irreducible representations in geometric terms, and one of the things, in order to get a good connection with classical representation theory was necessary to understand the connection between this geometric classification that Beilinson and Bernstein made, and the classification that Langlands made ten years previously. Bernstein said to me that all these are obvious. I said, no, they aren’t. He said, well just think about the case of principal series. I said, I understand that it’s obvious for principal series. He said, well then think about the case of discrete series. I said, I understand that it’s obvious for discrete series. He said, oh. In fact, in the intermediate cases, he just hadn’t thought it through carefully. It isn’t obvious at all. You have to do some serious work. I mean, in extreme cases the Langlands classification and Beilinson and Bernstein classification are just different ways of saying the same thing. So as Bernstein said the equivalence is sort of obvious, but in between, they are not different ways of saying the same thing. They are quite different statements. You have to do something to prove them. So there are a lot of things like that to figure out.

WQW: So eventually, Bernstein and Beilinson's work is on the level of real reductive groups, not just category O?

DV: Sure.

SJC: So would you say that around that time of Kazhdan and Lusztig, that was the most exciting time for you? Were there many things going on?

DV: Certainly in my experience, that’s right. There was a change in the field. For the 25 years before I started, you know, 1950 to 1975, there was an enormous amount of progress. I can annoy many of my older friends. It was all made by Harish-Chandra. To a first approximation, he did much more than anybody else. So that was exciting to watch, I guess, or frustrating.

WQW: Did you have firsthand experience with Harish-Chandra?

DV: I have met him a little bit, but really his last, big, serious work was the Plancherel formula for real groups which he finished in the late 1960s. So it was before I started, but in this period after that, in the 1970s, there were half a dozen people or more making enormous contributions. That’s much more interesting. It was lots more fun to have a lot of people doing something to move the field forward.

WQW: But was the Kazhdan-Lusztig conjecture a surprise?

DV: Absolutely yes, entirely. These problems seemed impossibly complicated before. People more or less knew how to calculate these things by hand using no machinery, so more or less it’s a question of writing down some enormous and very complicated matrices and computing their ranks, things like that. I mean, maybe some matrix of rational numbers and you have to figure out what its rank is. To do this thing for $SL(2,R)$, you can manage with $1 \times 1$ matrices, so it’s no problem. If you want to work $SL(3,R)$, then, maybe you have to do some $5 \times 5$ matrices or something. So you can manage but it’s painful. By the time you get to $SL(4)$, the matrices have size 10 or 20, who knows what. It just looked hopelessly complicated. Kazhdan and Lusztig found that there were these really simple calculations that gave the ranks of these matrices.

WQW: Even in terms of Hecke algebra, purely algebraic in a way, combinatorial.

DV: That’s right. It was just shocking these questions had such easy answers. I mean not easy to establish but easy to understand, to state, as easy as they could possibly be.

SJC: So judging from what you are saying, I have figured that you think this Kazhdan and Lusztig is probably the most exciting thing in representation theory in the last 40 or 50 years? Does it go that far back?

DV: I would say that’s probably correct. It really was an amazing thing. It’s so widely applicable. Their ideas turn out to be usable in a lot of versions of representation theory.

SJC: I agree, even in Lie superalgebra.

DV: That’s right.

WQW: We [Shun-Jen and I] are very slow learners. After graduate school we thought that we would be a little more mature, and we go back to our older teachers' notes and to learn something from your work and Lusztig's. We realize that Kac ’s old problem can be solved by using other notes from other professors.

SJC: Even when we were students all the solutions were there, we just did not understand.

WQW: We did not understand. Lusztig was beyond me when I was at graduate school.

DV: Of course, with undergraduate students at MIT, one of the problems is, students who come to MIT, when they were at high school, they were the best students, and when they come to MIT, in almost all cases, they are not the best student anymore. It’s a complicated thing to go from being the smartest person to not being the smartest person.

SJC: I found this quite interesting, you are saying this. I mean, how was it in your case?

DV: See, you are talking about Lusztig. Every day I go to work and I go passing Lusztig’s office thinking “What am I doing here”. But you know, I can tell this to the undergraduates when they feel discouraged, because they feel like they are not as smart as other kids in the class. Well, maybe that’s true, though you probably don’t know, because people who act smarter in class are not necessarily the best students. But even if you aren’t as smart as these other people, still, should be "you can do mathematics or whatever and you can do something interesting.

MKC: Even students in graduate school.

SJC: Absolutely.

DV: That’s right.

WQW: MIT has such a long tradition in representation theory. Are the relations complicated, with so many big shots under the same roof?

DV: Oh, we are all good friends. You know, every mathematics department has some political structure, the way hiring decisions are made and so on. At MIT the structure is determined by representation theory, because for many years Irving Segal was a professor there. If you have faculty meetings to have a discussion or a vote on something, then, every faculty member has the opportunity to speak, which means, in particular Professor Segal would have the opportunity to speak. So it was necessary to have a governing system in which there were no faculty meetings so we could avoid this problem.

WQW: So when he was around, you would be busy listening to one of the speakers particularly.

DV: Yeah, something like that. I mean to be a student there, well, I have always sort felt like a student there. Lusztig is still there. But it’s wonderful, because there is always somebody who has some interesting ideas about any questions you could possibly ask. Someone can tell you where to look or why it’s a silly idea or how you should just change the question a little bit to make it more interesting. This is completely wonderful. It’s an important thing to learn to use libraries, but I never learnt this at MIT. Instead of going to the libraries, it’s so much more fun to go and ask somebody. You have to work hard to understand the definition which is in some textbook that’s badly written or maybe it’s in Italian or who knows what. But if you ask you could get a wonderful answer.

WQW: So you might need a correction to certain texts. I have just noticed that from your web. That’s unusual putting corrections for different person’s book.

DV: Yeah, that was not so nice of me to do.

WQW: No, no. I think I understand that. That’s undergraduate or graduate text books. Some of the misprints are a little too obvious or less obvious.

MKC: I remember once when I was visiting MIT and Professor Vogan was sitting in some other people’s course and Professor Vogan was asking questions which were good questions but hard to answer. At one point the question was asked and then someone said that it’s in a book therefore it must be correct. You immediately said that the book is wrong.

SJC: Yeah, I remember I was sitting in one of your classes, and there was this about the famous Demazure character formula. Then I remembered the comments you made there which really struck me. I shall not repeat it here.

WQW: Why not? I haven’t heard about it.

SJC: I think he said that was just “BS”.

DV: It was a very reasonable looking argument.

WQW: The first proof was wrong in his Inventiones paper. There was a gap.

DV: That’s of course one of the thing you learn in mathematics is when you read something or listen to a talk, you are trying to see what’s wrong with everything that’s being said. Trying to see why it isn’t quite right. Well, it’s part of understanding things, because if you can’t find out what’s wrong then you begin to understand why it’s right. Of course, this is one of the skills in mathematics which is dangerous to carry into the rest of the world. It’s not so good to listen to ordinary conversation with the idea of finding out what little mis-statements there might be, at least not with your family.

SJC: At least not with your wife.

MKC: Do you have any advice for students who could be interested in doing research in representation theory of Lie groups, given that, this is a notoriously difficult field?

SJC: Notoriously difficult but with very good people in that field.

DV: I don’t know. You always just have to find something that’s so interesting for you that you cannot help doing it. It doesn’t matter what kind of mathematics that is. Just find out what’s the most interesting thing and your idea of what’s interesting will be different from anybody else’s idea, and that’s good.

WQW: I read about Harish-Chandra , but never had the chance to meet him unfortunately. Certainly when he switched from physics to mathematics, he started doing representations of reductive groups, and revolutionized it. He always thought that he would have that done in a few years and changed, felt a little guilty doing that for two decades and more on discrete series and all these, all these long series of papers building one on top of previous one. Now when you come to the unitary dual, would you feel fair to compare, say, when you started out, you thought that this business could be finished sooner?

DV: Certainly, well, in fact, there are two things. One is a story about teachers, so my teacher when I was an undergraduate student, the most important one was Paul Sally. He got me started doing representation theory. So I went off and work with Kostant at MIT. I did some algebraic representation theory and I was very pleased with what I proved and came back and I told Paul Sally about this. He said, “Yeah, but what does it tell you about unitary representations?” The answer was not very much. Well, he’s absolutely right. This is what the original question in representation was, to understand unitary representations and to do harmonic analysis problems. It’s fine to have fun with these algebraic things but it’s important to remember where you are trying to go or what this subject is for. It’s okay if what the subject is for changes, but it should be for something. I spent too long. A lot of people spend a long time. But it’s okay if you can find interesting answers. You can find interesting mathematics. There are always new things to understand. The representation theory of the symmetric groups was worked out more than a hundred years ago and we are still hearing interesting new things about representations of the symmetric groups and new ways to think about them, and this is wonderful.

WQW: So in a way, it’s … I mean, at different stages, one could be more optimistic how this could be taken care of or not?

DV: I don’t know. I am not sure whether I’ve ever been really optimistic. It’s a little bit like the Kazhdan and Lusztig conjectures. Before the Kazhdan and Lusztig conjectures there were examples that showed the answers to these questions were pretty complicated and any straightforward way to calculate the answers were really hard. This is discouraging somehow. You just don’t think that you can come to an interesting answer. That’s the way it was with unitary representations. There were always examples, partial results known, which showed the problem had a really complicated answer and that was really difficult to find the answer even in simple examples. There’re always more discouraging answers than encouraging.

SJC: So from the impression I felt from your talk today is that this business would not be finished for some time.

DV: No. I think that’s right. What I have talked about today was a way to calculate the answer but what you want is to understand interesting things about the answer. That’s, you know, you can learn when you are in elementary school how to tell a number is prime or not. You have got the algorithm. That’s the end. You just start with each number and you check whether it’s prime. That’s the whole story. Well, except, it isn’t the whole story. There are still interesting questions to ask. You can try to analyze the algorithm and find out more interesting things about prime numbers.

MKC: Could you tell us a little bit about the project Atlas that you have been involved in the last few years?

DV: So this is the creation of Jeff Adams , University of Maryland. He decided. I have written in a paper in the 1980s that it was possible to calculate the unitary representations of a reductive group by a finite calculation. It was sort of a joke of a statement. I mean it was true, but the size of the calculation was just out of control. As I said, this thing before the Kazhdan and Lusztig conjectures. You couldn’t possibly do such a thing. But Jeff said, about 10 or 15 years after I wrote that, well, computers are a lot faster now and we understand more mathematics now. What can we do? We shall be able to write a program that really works on this problem at least. He recruited Fokko du Cloux , who was an amazing mathematician and an amazing programmer, to start writing software to work on this problem. He made a large group of 15 or 20 mathematicians just to meet and talk about these problems. To explain to each other what the mathematics was that needed to go in the program and how the program dealt with the mathematics. There was a lot of really useful stuff that happened in converting mathematics into computer programs. There are a lot of things that we say in mathematics which are a little vague and imprecise. If you want to write a computer program about your mathematics, you are not allowed to be vague and imprecise. You have to really understand exactly what all the signs are and everything. So Fokko du Cloux got these absolutely precise. In the process, he had to read, for example there’s a famous paper of Kostant from 1950s about classification of Cartan subalgebra in real semisimple Lie algebras. So that was one of the things that the program had to do, was to work with classification. So he read Kostant’s paper and he found two or three little typographical errors and he understood the paper just really well. I would say better than Kostant in some sense. Certainly this happened with me again and again that du Cloux understood my mathematics way better than I had understood it. He fixed a lot of things, just made it much clearer to me. So this business of writing the software made us understand the mathematics better. I hope almost all mathematicians have at least some experience with how much more fun it is to do mathematics when you are talking to somebody else all the time about it.

WQW: It’s better, I am sure. Collaboration is more fun than working alone. But I am not sure if we can convince Lusztig that’s the case.

MKC: Does the group still meet from time to time?

DV: Yeah, more or less. We have been doing two or three time a year. It’s been maybe once or twice a year now.

WQW: Do you think it’s moving at a constant or getting faster pace or slowing down?

DV: Well, I would say certainly slowing down when Fokko du Cloux who did all of the original programming, after he died, the programming was taken over by Marc van Leeuwen . To take over the writing of a complicated piece of mathematical software is like taking somebody else’s large paper when it’s half written. You know, it’s incredibly difficult. Van Leeuwen did enormous amount of really difficult work just too completely master the codes that du Cloux had already written so that he could then begin to extend it. He has done that. He has completely mastered all of the original parts and rewritten a lot of it, extended it substantially. In particular he has done many of the steps in the direction of this algorithm I was describing today. I hope in the next year or so, I don’t know, that he might have a program that can calculate unitary representations. That’s the plan.

WQW: A few years ago the E8 Kazhdan-Lusztig polynomials made headlines.

SJC: It made to some magazines. Was it even TIME magazine or?

DV: There were a lot of stories all over the place. My son was working in Africa at the time and he sent me an e-mail said “You were just on the BBC.”

MKC: A while ago, Professor Vogan sent me pictures of New York fashion models who displayed nice clothing with this Stembridge’s picture of E8.

DV: That’s right.

WQW: Was that a patent?

DV: I think the reason that the story was so widely circulated was John Stembridge ’s picture. The one that you mentioned of this geometric figure made by the E8 root system in 8 dimensions was known to Coxeter and a mathematician named Gosset (William Sealy Gosset) maybe in the 19th century. But Coxeter wrote about it in his book about polytopes. He got one of the persons who was working with him, another mathematician, Peter McMullen to make a picture of a 2-dimensional projection of this 8-dimensional polytope. It’s beautiful. John Stembridge made a color image of it in postscript, because John Stembridge is a very clever fellow. The picture is beautiful and a lot of newspaper thought we would like to have this picture on our webpage.

SJC: Here is the story to go by with it.

WQW: The beauty of mathematics is appreciated by those models as well.

SJC: Just couple days ago, I was happened to read that 900 page book of yours. The book is on Cohomological Induction and Unitary Representations. I cannot help to be impressed by people who are able to write such a big book. How long did it take you to write?

DV: One of my colleagues, not at MIT, another person in the field wrote to me after this book was published, he said “For such a long book, you didn’t get very far, did you.” Knapp did all of the writing. I did read almost every word, but he was the one who did all of the work.

WQW: How long did it take?

DV: He came to visit MIT for a year. That was the main thing he did during that year.

WQW: 1 year for 900 pages.

DV: He has started before that. He did a little work after. But that’s more or less.

SJC: 900 pages proofreading, that must be a tremendous job.

WQW: So far, your corrections are only 3 pages on your webpage, relative to the 900 pages is very small.

DV: We actually got an award , some sort of publishing industry award for typesetting in technical publishing. I don’t know what is was exactly. But this is again entirely for Tony, because Tony Knapp did all the TEX files that were used to produce the book. So everything about it is his work.

WQW: One thing we could have sensed, even as a student or later on, in this field of real reductive groups there are a lot of generous spirits around. In some other fields which we are more familiar with, this is not always the case. Occasionally, for that sake, really students wish to work in a different field, sometimes. But, of course, we are intimidated by the unitary dual.

DV: Once, long time ago, I was at a conference. I was walking back after dinner with Tony Knapp. I said, “This is a really nice life where you go to these professional conferences with your friends and do all these interesting works and you have these pleasant times with them in the evenings.” He looked at me and said, “That hasn’t always been my experience”. So I think, in some sense, real group representation theories, maybe before 1970s, was more like some of the other fields you were talking about. Actually, I went to a conference once where one of my non-representation theory colleagues came to one of the talks. He said, “There was blood on the floor. We don’t do that in algebraic geometry.”

WQW: Algebraic geometry I heard good things about: the leaders are very generous, and therefore, the whole field is very friendly. I guess, it certainly makes a difference how the leaders behave and how they write. That is “show by examples”. How many books did you write? Still counting maybe?

DV: Well, I would say three that I wrote, that I, myself, did the writing.

WQW: Three single authored?

DV: Not necessarily single author, but I was the one who did the writing. Well, the book with Tony is four. So I think maybe just four. You know, I am listed as an editor or something of some other books, but, they were everybody else did all the writing.

WQW: In contrast, your teacher did not seem to have written a book. He wrote many long articles.

DV: Not so much about books. He wrote an enormous amount of papers. I think Kostant’s papers are beautifully written. You can just pick them up and read them.

SJC: Better than some books.

DV: That’s right.

WQW: I have the same experience with that. Even 100 pages paper, if you are willing to spare a day or two, you can read from beginning to the end.

DV: It’s worthwhile because he has so many wonderful ideas.

WQW: Definitely. For the affine algebras there seems to be affinization of Kostant's ideas everywhere. Those people early enough [in the field] did all those “easy" things. Though that's not fair. Just a little jealous. Those were important things, easy but important.

MKC: So Tony Knapp came up with term Vogan diagram in 1996, but I noticed that you haven’t been willing to use that terminology.

DV: See, because I think they existed, they were certainly used by Sugiura . They were used by various people in the 1960s when I was in elementary school. I know I didn’t have anything to do with those. It’s one of the rules, maybe Arnold ’s principle that “nothing in mathematics is named for its real originator” including this.

WQW: I was sitting in a lecture by Arnold once, in Max-Planck Institute, he was probably repeating this principle around that time and talking about some various things. Hirzebruch was there. At some point, Hirzebruch was asking very innocently, “Was that the so-called Arnold conjecture?” Arnold was nodding. “Did you conjecture about it?” Suddenly, it took Arnold a minute or two to recover and tried to mumble something. Eventually he claimed every rule you set up you don’t apply to yourself. You have written so many books. There must be a reason. Do you feel expositions are important for a field?

DV: Absolutely. In each case, there was some collection of ideas that I wanted to write about which.

WQW: Sorry, let me correct my question, your earlier books were not expositions, more like monographs.

DV: I mean, in a sense, a lot of the things are structured sort of like research papers, but when they get to be 200 pages long or something, they stop to be reasonable papers. I don’t know. Actually, I have been writing the paper about this unitarity algorithm, the manuscript is more than 100 pages long now and I don’t think it’s half done. So I don’t know if this would become a different book or not.

WQW: I see. I guess different authors treat these things differently. I remember a story about Harish-Chandra. His papers are almost always around 55 pages, as was observed by some Russian mathematician. He was wondering. He eventually conjectured that Harish-Chandra probably had sequels ready or knew what was coming, and so he cut some parts off.

DV: One of the many stories about Harish-Chandra is that he wrote up almost everything he did, write by hand on papers. If it was part of his project to prove the Plancherel formula, then it was typed and sent to a journal. If it wasn’t part of his project to prove the Plancherel formula, it went into a drawer, which is okay. Except then if someone came to Harish-Chandra and said, “I’ve done this piece of mathematics, let me tell you about it.” Harish-Chandra would go into his drawer and take out a manuscript. I mean, this is not okay. If you decide you are not going to publish the paper, when somebody come to tell you that they have discovered it you can’t just bring out the manuscript.

WQW: Did similar things happen to you? Only publishing unitary dual?

DV: Well, I am sure that everything I have criticized another people, I am sure, I have done similar thing.

WQW: No, I was not saying about the second part [of Harish-Chandra's story]. I was referring to the drawer. One reason that you have so many students is maybe your drawer [of unpublished ideas].

DV: They do all the work, that’s right.

MKC: Do you have 40 students almost, right now, I mean accumulated?

DV: I think it’s less than 30. I think so. Victor Guillemin has more.

WQW: But he is older, to be fair.

DV: You see, when I was a student Victor Guillemin was older than me. As you get older, these differences shrink somehow. Now he is hardly older than me.

MKC: But the entrance is higher in representation theory of Lie groups.

DV: Than symplectic geometry? I don’t know. I don’t know.

WQW: Has Guillemin retired?

MKC: No, he still has students graduating this year, I heard.

DV: Yeah, I think that’s right. Certainly he hasn’t retired. He certainly still is an interesting person to talk to.

WQW: I think I remember one thing you said, quote “Basically, in physics when they have a student, they have one extra paper. In math when they have a student they have one less paper to publish.”

DV: I’ve heard this statement. That’s right. In fact, the reason I have many students is that I have some really good students. There have been various times when I have some really good students who were really good at explaining mathematics. Immediately, I have five or six students because they all go and learn from these people who are so good at explaining. More than 10 years ago, Peter Trapa was my student. All of a sudden, I have many students because they could all listen to his explanations, a nice person and a generous person.

WQW: How many students do you have now?

DV: Three, I believe. It’s terrible I can’t just instantly answer such a question, but, I think the correct answer is three.

WQW: As a student when I was watching, of course, you almost answer every question, always with a pause. Sometimes I even suspect that because you are thinking so fast you try not to offend the person who asks the question so much by slowing down a little bit.

DV: You must know this story about the mathematician and the balloon.

MKC: No.

DV: So, there were two people flying in a balloon and they were lost in the clouds. So they didn’t know where they were. Suddenly they caught a glimpse of the ground through the clouds. There was a person down there. So they shouted down “Where are you?” So there was a long long pause. Finally, “You are in a balloon.” So the person in the balloon said to his friend that “This is a mathematician down here.” “How can you tell?” First, it took him forever to answer the question. Second, the answer was absolutely correct. Third, it was completely useless. So I always try to keep this is as my standard for answering questions.

WQW: As a student, I think that I have found MIT a somewhat cold place. But certainly your door was always open.

MKC: I also thought that in your old office the closet with all the papers inside was connected to the cookie closet. Until one day I realized that they are not connected.

DV: I know. I am very bad at this kind of metric geometry. I never understood how that closet failed to be connected to the cookie closet. But you are right, it isn’t.

SJC: But I mean, did you think it was when you choose the office?

DV: No no, but somehow, I kept hoping. I would keep opening this closet and looking to see if there was a little door, but it’s just not connected.

MKC: Yesterday, you told me that a problem of being the chairman at MIT it’s not easy to improve but it’s quite easy to screw up. It’s ready right there. Any interesting story of achievement happened while you were the chair?

DV: As I have told you if you have chaired at MIT, you can’t possibly succeed in any way. You can only fail. One of my great failures was that Mike Hopkins went from MIT to Harvard while I was chair. Also, while I was chair, I move from a large office to a small office to help convince the dean that we really needed more space. So my old big office went to some new person we were hiring because we didn’t have any other space. But after I stop being chair, the new chair gave me a big office again. But in order to remind me every day of my failures the new office he gave me was Michael Hopkins’s old office.

WQW: I also remember that you ran for president, not of the United States, but of the AMS. This has to do with changes of your perception from research to education, how did that work?

DV: You know, I certainly never wanted to be head of the math department at MIT and I didn’t want to be president of the AMS . But I thought these are the jobs that need to be done and I would do my best at doing them. I am not even sure that I think I can be competent at either of these jobs. I am sure I could tell you the names of many people who would explain all the ways that I was not competent being head of the department. But as I say, I think the jobs have to be done. Well, it’s clear that the American Math Society does some really important things for mathematics, maybe the most important is the publication of math reviews. The whole publishing program of the AMS is very important, but if it disappeared, there’s still mathematical publishing in the world. That’s a very valuable thing but it’s not irreplaceable somehow. The conferences that the AMS organizes are again a very valuable contribution to the world of conferences. So I think it’s a very valuable organization. I wanted to succeed. It would be nice to sit and talk to people about mathematics all the time and never do anything else, but, this means somebody else has to do the organizing work. So if sometimes I can do some of it to help a little bit, it feels like taking your dirty dishes back to the right place in the cafeteria. You are just supposed to do that.

SJC: I think it was a very interesting talking to you today about all these things, we learnt quite a bit.

DV: Thank you so much for your hospitality. I think this is a small price to pay for all of the taking care of me.

WQW: For young students, it’s interesting for them to hear different mathematicians and different views. I am always interested in reading those things myself, even now.

SJC: I think so. I certainly have learnt quite a bit.

- Shun-Jen Cheng is a faculty member at the Institute of Mathematics, Academia Sinica.
- Meng-Kiat Chuah is a faculty member at the National Tsing Hua University.
- Weiqiang Wang is a faculty member at the University of Virginia.