Professor George Lusztig
Planning | Liu Taiping
Visit | Cheng Shunren, Wang Weiqiang, David Vogan
Time | January 6, 2016
Location | Institute of Mathematics, Academia Sinica (Taiwan)
Organize | Huang Xinpei
Source | Mathematics Communication December 2016 (160), Fun Math is authorized to reprint, thanks!
Professor George Lusztig was born in Romania in 1946, graduated from the University of Bucharest in 1968, and received his M.S. and Ph.D. degrees from Princeton University in 1971. He is currently the Abdun-Nur Chair Professor of Mathematics at the Massachusetts Institute of Technology, best known for his work on representation theory and algebraic groups. In 2005, he was conferred an honorary fellowship by the Institute of Mathematics of the Romanian Academy of Sciences and is a fellow of the Royal Society, the American College of Humanities and Sciences and the National Academy of Sciences. He was awarded the Shaw Prize for Mathematical Sciences in 2014. Winner of the 2022 Wolf Prize in Mathematics for "his pioneering contribution to representation theory and its related fields."
Cheng Shunren (hereinafter referred to as "Cheng"): First of all, George, thank you for your willingness to be interviewed. Let's start with the following question: When did you know you were interested in math? At what age? What point in time?
Lusztig (hereinafter referred to as "L"): Probably 14 years old, I was interested in mathematics until then, but at the age of 14 I decided to become a mathematician.
Cheng: So before you go to college, you know you want to be a mathematician.
L: Right.
Cheng: Understand. However, you may also know that in Chinese culture, parents usually...
L: It was because I was in the eighth grade in the Olympia competition, and I didn't have any idea of this kind of national Olympia, I didn't expect a good performance, and the result was unexpected, and I decided to spend a lot of time on it.
Wang Weiqiang (hereinafter referred to as "Wang"): When you were in junior high school? 14 years old?
L: In the first year of high school.
Cheng: When you were studying at the University of Bucharest, how was the math department at that time different from the Western schools?
L: Well, it's hard to compare, it's very different. First of all, college is going to last five years, but in the last two years you have to do some special topics, maybe more like a master's program.
Cheng: I would like to ask, do you only need to take math classes to get into college, or do you need to take other things?
L: You must enter the Mathematics Department, pass the entrance examination, and decide your favorite school department.
Cheng: So unlike in the United States, you have to take other classes other than math...
L: No, for example, students have to practice Marxism. Every year, certain Marxist courses are taken, and about two years out of five years are taken.
Cheng: What did you learn? I mean, what are these lessons taught?
Wang: We have a similar course in Chinese mainland, to learn the history of the Chinese Communist Party.
L: Yes, there's a discipline of dialectical materialism, which is a certain Marxist philosophy, and there's a theme like political economy, but it's all Marxism. In addition, we have classes on teaching methods. Because this department is basically to train teachers, we have to go to teaching, and the first year is to go to physics, and mechanics.
Cheng: So it's the Department of Mathematics and Mechanics, like the Russian system, which answers my second question: Was the Romanian education system influenced by the Soviet Union? Obviously the answer at this university is yes.
L: One of the implications is that there are a lot of Russian math books on the market that translate everything from the West, and every volume of Bourbaki has a Russian translation, and these books are extremely low priced, and I can easily buy them, which is more affordable than they are now.
Cheng: None of us can afford it right now, especially Elsevier's journal.
KING: Can you read Russian?
L: Yes.
Cheng: I think you are required to take Russian in high school or college?
L: We have to learn Russian from grades 4 to 11, and we have to do sophomore year.
Cheng: Full of.
King: Is Russian the second official language?
L: No. We have to practice two foreign languages, Russian is compulsory, and the other can be English, French or German.
Cheng: So what did you choose?
L: Although this is theoretically the case, it also depends on the actual status of the course, not every school has these options. I practice French.
Cheng: George, when you read the University of Bucharest, did any professor influence you the most?
L: Yes, there's a professor named Teleman. There are several people named Teleman, from the same family, who are related by blood. The professor had four brothers who were mathematicians, and one of them was my professor.
Cheng: Why did you go to the West? How did it go? Did you leave Romania and go directly to the United States?
L: No, I actually went to the West once before, and then I came back to Romania. In 1968 I went to a summer academic event in Italy and went to the UK after the conference, which was not allowed, but I went to the embassy to apply for retroactive permission. After returning to China for half a year, I went to Bonn, Germany, and then to the United States. Actually, I received an invitation from Princeton Because of Atiyah's relationship. I first met Atiyah in Oxford.
Cheng: I don't think everyone knew Thattyah was at Princeton at the time.
L: When I first met him in Oxford in 1968, he was on his way to Princeton. At the same time as applying for the job, I could recommend some young people to the Institute of Advanced Study, and he recommended that I work with him. So he was already at Princeton in 1969.
Cheng: No, you said you went to Italy first.
L: No, that was another time, my first time abroad. First to Italy, then to England, then to Romania. When I was in Oxford, I received an invitation from Princeton, and when I returned home, I applied to go to Princeton, but I was rejected.
Cheng: So what do you do?
L: But at the same time I also received an invitation from Arbeitstagung in Bonn. That invitation was approved because it was short, only one week, which was the competence of the local authorities, and the long-distance trip required central approval. So at the same time, one approves, one rejects. So I went to Bongan first and took a detour from there to go to the United States because I didn't have a U.S. visa.
Cheng: But do you have an invitation from Princeton?
L: Yeah, actually I went to Canada first and I went to a conference. Applied for a U.S. visa in Canada and stayed for two months before arriving in Princeton.
Cheng: What was it like to do research with Michael Atiyah?
L: Of course I wasn't an official student of him, he was a very brilliant mathematician, very nice, and he helped me a lot. Actually we discuss, we discuss, we talk about it almost every day, and that's really...
KING: You were in Princeton for a year before you went to Warwick?
L: Two years.
Vogan (hereinafter referred to as "V"): He must get a Doctorate.
KING: Yes, you can teach. But you didn't do all of your paper, didn't lie theory, did you?
KING: How did you gradually or quickly turn to the study of Lie theory, or the Lie type finite groups?
L: Atiyah certainly knows some Lie theory, and knows a lot about classical group representation. But he didn't think much of the exceptional groups, felt they were unimportant, and could be ignored. So all I know is index theory stuff, and Michael Atiyah is actually an iconographer. But then because of a lecture by Quillen, who was also in Princeton at the time, he solved some of Topology's problems with representation of finite groups; in fact, he used Brave lifting to elevate the modular representation of finite groups to normal with complex coefficients Representation, similar to what Green had done before. It's actually a very esoteric theorem, and no other topologist realizes it's doing that, and it's amazing. He applied that method to solve something rustic, and I was interested, asking Atiyah if he knew anything more clear about Brave lifting. Because Quilen starts with the natural representation of the finite body, somehow connected to some complex representation, its virtual representation. I asked him if he already knew the components of the virtual representation? Atiyah told me that characters are known, but the actual performance is still unknown, and that traits are known through Green's work.
KING: In type A?
L: That's just the standard performance. One of these components is discrete series representation, which is still unknown, and only its feature markers are known. I started to get interested, so learning basically what I needed, and then I became more and more interested in performance theory.
KING: Was there already Green's signature theory?
L: Yes, it came out in 1955. Interestingly, I got a letter of appointment from Warwick University before leaving Princeton, and Green was right there. I thought it would be a good place for the problem I wanted to solve. But I was wrong, he was actually not in good health, and he almost never came to the department.
W. What research did you continue to do in Warwick after that?
L: Yes, but Carter was there, and I talked to him, and he explained the various algebraic groups for me, and we co-wrote some papers. Finally I was able to solve this problem.
KING: At that stage did you just study type A?
L: No, the first thing I did was to find a way to construct this discrete series representation by using some definite model of the Witt vector. And then pretty quickly I started asking the same questions about classical groups, because they also have a natural performance. While doing this, through the scrutiny of Brain lifting, I began to understand some of the theory of performance.
V: With the technique of the time, it was impossible to make a specific representation of these classical groups; but on the other hand, with your work for the next ten years, these became surprisingly simple. But thinking about these problems used to be so complicated, simple is actually just seemingly simple. Is this true?
L: Basically what happens to classical groups under The Booster lifting? For example, a symplectic group, which has 2 components, has a dimensional representation in the modular setting, and increases this dimensional representation, one of which is actually a discrete series corresponding to the Coxeter torus, but the others are also very difficult, not discrete series, but must be decomposed with discrete series. In , all the others are directly induced, completely undecomposed. In other classical groups, decomposition is not a simple task, and I have to understand that Hecke algebra with unequal parameters with unequal parameters is something I have to use so it's more complicated than that. This method can only construct one series , but classical groups have multiple discrete series , several non-isomorphic torus. In fact, I put a lot of effort into it, and I ended up with some results, but I never published it, and I just wrote some very short abstracts. Actually, that's pretty good theory, and it's not published because it's outdated. But all of this actually helped me gain a lot of experience, and it helped me a lot in other ways.
V: So it sounds like you might have done something almost like writing down a list of character tables for these classical groups, and in any case, you've very explicitly calculated a lot of representational signatures, understanding all the conjugacy classes. Everyone who does math has to do some calculations like that, but some people like it more than others, do you like calculations? You're counting a lot.
L: Well, I do love computing.
KING: Well, those calculations are things I've never really read, but I've learned your braid group calculations, and after 20 years it's still a torture. I've taught braid groups a few times, always choosing the simpler type to teach, but even then it's very complicated. Even today, I have never tried to calculate.
V: You said you had to take a teaching course at Bucharest, did you learn anything from that course?
L: No.
V: But on the other hand you mentioned Teleman, how did you learn how to write a good math paper and make good math? Who are you following and emulating?
L: I did write some papers in college, but I don't think I learned anything from Teleman. I learned a lot from Atiyah, and I tried to use him as an example; I should say that I learned more from Deligne, and I learned his way of writing.
King: I've read some Atiyah articles, and I remember that your style is not very similar, let me say so!
L: Not really.
V: Reading papers, listening to lectures, so far you have read and know that many people do mathematics, is there anyone you think you should follow?
L: Milnor was the number one in my mind, and I've been told he's the best mathematician in this area. His books are absolutely the best and so are his speeches, which are my paragons.
V: Have you seriously considered going back to topology? In the 1970s and 1980s, after you started doing finite groups, reductive groups, etc., and you were solving problems like that, and at the same time you were opening up new problems so quickly that it was hard to imagine that you would go back and do something else, because there was so much more to be tapped into.
L: That's still the case. But there's another reason, I thought I was going to be an iconographer at Princeton, and I knew Dennis Sullivan, and I had some contact with him. In fact, I had the impression that he was solving all the major problems of Topology, and that was why I wasn't doing Topology. In fact, even now, I still think that performance theory has more interesting problems than topology. I find Topology as a tool to be very useful.
WANG: There's something more interesting about the classical group.
L: But I find topology as a goal that's not that interesting in itself.
V: I forget who said that mathematics is like a garden, and fertilizing the land is very important.
L: Hironaka said.
V: Yes, but some parts of the math are flowers, so maybe the groups and representations are some flowers. There are many new ideas in mathematics that you are personally involved in creating. My impression is that the vast majority of new ideas come up when you have a problem to solve and you can't solve it with existing mathematics, so you build something new to solve an old problem. Have you ever unearthed something just because it looks like a very interesting idea? I always feel like Kac-Moody Lie algebra is artificial, because after Serre relations are written down, we might say, "Well, we don't need it to be finite-dimensional algebra or something, all the formalisms are still done." My feeling is that there doesn't seem to be much reason to do this math that can be done. But you introduce something new on the subject, always seems to come to tell you something about the old problem, at least at first.
L: Actually, my impression is that when I was pursuing certain goals, I was always lucky, and something happened by chance. For example, at some point I was very keen to understand how to classify the representation of classical groups, and after co-writing the paper with Deligne, I was tempted to know that this was a really important question. Although the situation is well known, how classical groups are classified is still completely unknown. I didn't come up with any new methods, I already know the methods in the paper I co-authored with Deligne. But then I met Two Korean mathematicians, Ree and Chang, and you know these two people?
V: Ree group 的Ree ?
L: Yes, there's another guy who is his collaborator. They studied characteristic markers on finite fields. I met them in Vancouver, and Chang told me that he had a student who had just figured out the classical type of classical type with arbitrary unequal parameters, and he had their construction and degrees, and the kid actually stopped doing math. The news came at the right time, so I studied his stuff and found that it was almost what I was missing. There have been times like this, and I just happen to get exactly what's missing, and they just happen to be delivered to me.
King: But if you hadn't had that method back then, you'd probably have to solve it yourself.
L: Possibly. These opportunities came one after another, and as soon as I learned about these classical groups, I found that in a copy of a paper someone had found the generic degree of Hecke algebra.
Wang: Interested in the exception group, you are not a good student of Atiyah.
L: I really want to do classification Going back to the generic degree mentioned earlier, I got some ideas on how to construct a non-abelian Fourier transform matrix, because the degree table in the paper above, you can see that the first few of them have coefficients like this, 1/120, 1/24, and I see some patterns in it. However, the timing of this copy was just right.
KING: You do it yourself, right?
L: No, I didn't do those calculations, someone else did.
Cheng: But what I'm saying is, if there weren't those, there's a good chance you'd count yourself, wouldn't you?
V: Well, you have some friends you can send. I mean, eventually you're going to need all the branch tables of weyl group performance, you don't have to do it yourself, you can bother them with it, but you're responsible for the results.
L: Yes, even before that, the first time I used a computer was to calculate the fade degree, and in this study I did the fake degree of the classical group, and I wanted to know the fake degree of the exception group. So I went to someone in Warwick and told him the rules to calculate, and he helped me calculate. That was the first time I used a computer.
KING: I think I know his name, Beynon? He was not a mathematician.
L: No, he's in computer science.
Wang: So sometimes there is no computer for the calculation of the abnormal type, and even you feel hopeless.
L: Yes, computers have made a big difference. Around the 1960s, classical groups were thought to be simple, while exception groups were harder. Because of the computer, I think it's the other way around now.
V: I want to ask a very different thing, but maybe not so different. Joanne Jonsson is an administrator in the Department of Mathematics at MIT. Of course, there were a lot of people in the department who went to meetings all over the place, and they used to talk to her about places where they used to talk, and she said that she liked to hear you share it with her because she said, "He saw everything, he remembered everything." Do you know about it? I'm referring to the fact that obviously you know that you have more insight into certain mathematical things than other people, but are you aware that you're doing the same for other things?
L: I was amazed!
V: You mentioned that in the case of various karmic coincidences, there are ready-made results that are exactly what you need. However, in the mathematical world, ready-made results abound, and it may not be so easy to find suitable and usable.
L: Actually, I started to get interested in quantum groups because Borel wrote me a letter saying that Michio Jimbo has a job that I might be interested in because Jimbo does something that looks like Hecke algebra. He knew I was interested in Hecke algebra, and he said that he actually showed up directly in Jimbo's work. He pointed this out, so I studied quantum groups and gave a course.
Cheng: When was that? What year?
L: 1986。
King: That paper was published in '86. I think we probably know a little more about your work in quantum groups than you did in Lie-type finite groups earlier. Even if I've flipped through your early books once or twice, I'm still not sure how much I've absorbed.
L: But I think lie finite groups are in some ways more interesting than quantum groups, because they don't know the result, and it's mysterious.
Wang: Yes, these problems of the Lie finite group are 100 years old, which can be said to be a classical old problem. So can we say that the introduction of geometric methods is one of the keys to lie finite groups?
L: Indeed, nothing can be done without a geometric method.
V: Surprisingly, Green had a complete list of non-performing performances in the 1950s, so with enough effort, it seemed that anything could be achieved. But it's because of the geometry you and Deligne introduced that some of these results are clearer, that it makes it clearer what has to be right, not just the result of some terrible calculation.
L: Oh, all of Green's articles are amazing, he's the only one who's proficient in mold representation, so he's able to use Brauer's approach. He was a student of Hall, so he knew all about Hall polynomials, and he used that to define Green polynomials. He knew all this stuff and before him Steinberg was doing research on that but only doing it.
KING: I'm pretty sure of symmetric polynomials, Hall-Littlewood, and how to make all these things work, and I really need to specialize in them. But going back to the quantum community, for several years from '86 to '94, you were very focused on the quantum group, and it was your main job.
L: Yes, almost 10 years. I study two theories, which are actually two things: one is a quantum groups at a root of unity, trying to understand their contribution to the representation of the module; the other is a canonical basis.
Cheng: We were still students at that time.
KING: Yes, I didn't understand canonical basis at the time, even though I copied notes in your class.
L: But now you know a lot.
King: But only now do I really need it. I've been trained well by students for the last 5 to 10 years, and somehow I've been forced to be re-educated by different students for different purposes.
L: Actually, Borel was right, and he could see that knowing Hecke algebra helped me, and I was able to move something to a quantum group, which wasn't bad at all.
V: You talk about working with Atiyah and Deligne, who can be said to be your teachers and collaborators. Do you enjoy working with students? I mean you have two choices: you can write a beautiful book and send it to a publishing house, or you can take in a student, which is very different to satisfy. Have you had a wonderful experience?
L: Yes, I have some really good experiences. I have some very good students, and I'm very happy to have them. One of them is Spaltenstein, and the other is Xuhua He. I have several of these students, but not all of them are, and some of them are really good.
V: Have any students really changed the way you do math, changed the way you think about doing certain problems?
L: Yes, I learned a lot from Spaltenstein, more than I learned from other students. He was actually a very early student of mine, probably when he was in England.
Cheng: Was he the first student or the second student?
L: I started with three students and he came about a year later, probably the second, the first one was de Concini.
Wang: From your bibliography, it's not hard to see that the vast majority of your papers are single authors. When it comes to working like this and collaborators, is it basically your own choice, or is that naturally the case? What do you think?
L: Yes, I usually study alone, with some exceptions, but I think it's natural for me to do research alone.
KING: Even those lengthy calculations, would you rather do it yourself?
Cheng: George said he likes to calculate.
V: I don't know if working with Deligne spoiled your appetite for working with others.
W. Probably a lot of people tend to have multiple collaborators instead of doing independent research in a long academic career?
V: I've asked people who work at Math Reviews to easily count whether the number of co-authors is actually increasing. My impression is that the number of people has indeed increased.
L: Even four people, I found that papers with four authors have become commonplace, at least as I saw in this conference.
Cheng: Yes, that's right, three or four people. Very common, because communication becomes easy.
KING: Yes, it also became efficient. A single person has only one profession. We don't learn fast enough, even if we know what to learn. Collaborators often amaze me, they bring a lot of good and different methods, and I may take a longer, or a lifetime, to learn alone.
V: But there are all sorts of jokes about collaboration, and I probably don't remember them right. Camels are a horse designed by a committee, and cooperation is always problematic.
L: I've never been in a team of four, and three people is the ceiling.
V: You once told me when you were in middle school at Bucharest
L: Not in middle school, in Bucharest when I was in college.
V: Yes, you join the rowing team. How do you allocate time for rowing, studying, and doing math?
L: I don't draw much at Bucharest. It was actually the last two years of high school that I was rowing, and it didn't conflict with schoolwork. About twice a week, I'm very poor at sports, and I get involved in rowing by chance. I forget how it started but it improved my health so it was good.
V: But then you didn't go on.
L: Later in college, the first year still lasted, but it was not very convenient, it was a long way to go, and I didn't like it for a year. There is a river in our hometown which is convenient and I loved it.
V: Have you found any other activities? I mean, in order to do math, do you feel like you have to do something different than math?
L: Yes, for example, like yoga. I guess I can't function without yoga.
Cheng: Do you think about math when you do yoga?
L: No.
Cheng: Can you easily shut down your head? For example, when you are ready to go to bed, can you quickly fall asleep?
L: I can fall asleep when I go to sleep, and the problem is that I often wake up in the middle of the night, and there are always problems in my head that keep me awake for the next two hours. But there was no problem when I went to bed, because I was tired.
V: You said you enjoyed reading Milnor's books, are there any non-mathematical books that you think are worth reading?
L: Yes.
V: "Love When the Plague Spreads"?
L: That's a great book, I love it. Do you like it?
V: Well, I remember you mentioned it about twenty years ago.
L: Yes, I mentioned.
V: Look, be careful what you say, we'll write it all down.
L: Well, I love this writer, I read several of his books.
Cheng: One Hundred Years of Solitude?
Wang: I think Shun-jin may have recommended these books to me.
Cheng: Very good book, Marquez's book.
V: Do you still read non-English novels and the like?
L: Yes, in fact I just read some Balzac French novels, and sometimes I read Books in Romanian, occasionally in Italian, but relatively rarely. Books in Italian have been read a lot in the past.
Cheng: You said you only had a short time in Italy.
L: One year.
V: How far are Italian and Romanian?
L: Yes, but still have to learn, no
V: Well, I know Italian is not a dialect.
L: But you can still transition from one language to another, to a certain extent.
V: Did your previous Russian math books come with you to the United States? Are these books still there?
L: No, my sister brought it, and my sister came later, and she brought the books, so it was all there. But I can't read Russian literature, and I'm not that good enough. Actually, there's a book I recently read that I think is very good and recommendable, and it's Life and Fate by the Russian Vasily Grossman, which is about the Battle of Stalingrad. The book is really good, somewhat similar to War and Peace, and it talks about certain battles that involve the survival of the Russian state.
V: What about music? Do you have many opportunities to listen to music?
L: My wife Gongqing plays the piano and practices every day.
Cheng: It's nice to have live music at home! Do you read a book before going to bed? Does it make you feel relaxed?
L: No, not before bedtime.
KING: You've written several books yourself, have you enjoyed the experience of writing books?
L: Those are not real books, the first one is written about discrete series representations, which were not originally set to be a book, but a treatise. I submitted Ann.~Math., and they proposed to publish a book instead of as a paper, so I did.
V: The Characters of Reductive Groups over Finite Fields?
L: I also think of it as a thesis rather than a book, and the words of the book should make it easier to read, but I didn't do that. I write it the fastest possible way, so I write everything down and don't care much about the reader.
V: I was going to ask you how many books you've written, but I think it's easy to calculate, the length of the articles you write is the normal distribution, and any paper that is more than 200 pages is a book.
But" Introduction to Quantum Groups, Progr.in Math.110, Birkhauser Boston 1993, 341p. (Reprinted 1994, 2010.) It must be a book.
L: Yes somehow but
Cheng: Is an introduction.
V: If it is a paper, there should be no "introduction" in the title.
KING: But this book you wanted to write from the beginning as a book.
V: 那么Hecke Algebras with Unequal Parameters呢?
L: That was asked, and I gave a series of lectures at Montreal that they asked me to write a book. I've talked about that at MIT, and I already have some notes that I just need to expand on. So in fact, "Introduction to Quantum Groups" is probably the only book that was originally scheduled not to be a dissertation.
V: It's funny.
Wang: Yes, "Introduction to Quantum Groups" takes a lot of effort to get it through, and I mean, it takes another introduction to get into this introduction.
Cheng: I've been thinking about its title, ~ I mean, ~ did you deliberately take the title of the book as "Introduction~ to ~ Quantum Groups"?
L: Do you think it's misleading?
V: The title follows the tradition of André Weil's foundational number theory, and I assume you believe that good books of this kind are important enough to allow the next generation of mathematicians to delve into these questions. You said that the signatures of the limited Chevalley group are the most exciting or extremely exciting work, and Roger Carter has elaborated on this in his article, and I think a lot of people have benefited a lot. Have you thought about this? Wouldn't you feel like writing a book like this isn't your job? I mean, the quantum group book is kind of like that.
L: I know, but the theory of the representation of finite groups, I may not think I know enough. So there's something that's still to be studied, and I'm going to take it to heart and write some articles to elaborate on it when it comes to fruition, but I don't think I'm ready yet. I don't think the theory is complete, and I don't think I'm good at writing this kind of article.
V: You just have to learn LaTeX.
KING: I think George is using TeX, right?
V: AMS TeX, 对。
L: For example, I'm not very good at proofreading, I've ever made mistakes.
V: Of the 4 books we just mentioned, there are 7 errors
WANG: I think your equations are reliable, not just impressions, but I've used different things in very specific ways.
V: Around 1980, you asked me to photocopy a few pages of the nilpotent element that you counted, and so on, and the exception group. Having these calculations is very good, they're perfect. It's bad that you write in light blue ink, the copy is poor and difficult to read, but the calculations are reliable.
King: Now most of your work is still handwritten, or do you type very quickly and lose your handwritten notes?
L: No, I type, and I really don't write the whole article by hand, maybe I just do some calculations and type them out after accumulating enough.
Wang: When learning LaTeX, compared to AMS TeX, the advantage of automatic processing of numbering convinced us that we all chose to use LaTeX. But it seems like you never bother with cross reference? Because we sometimes need to change the ordering drastically, the numbering of the equations is of course messed up, and LaTeX automatically takes care of that. Is AMS TeX good for you?
L: Because I have different numbering methods, some sections are numbered with A, B, C, D, it doesn't matter if you change the chapter number, the number of the equation doesn't need to be changed, so the number of the chapters is not included in the number of the equation, just a letter, such as 1t section, equation A or something.
Quantum Group is a very good book, and the more I learn about the subject, the more I appreciate it. Other than that, I found almost no mistakes, and it's the book I've known best in the last decade.
Cheng: I think we both have two copies each.
Cheng and Wang: One at home, one in the office.
L: But I think the quantum group is relatively simple, and the representation theory of finite groups is more exciting and interesting.
Wang: That's why I can understand a little bit of quantum groups.
Cheng: When you mention Atiyah and Deligne, are there any other mathematicians you admire? Besides the above two, is there anyone else?
L: Gauss, for example.
Cheng: Oh, indeed! We all admired him. Anyone with Lie theory?
V: Have you read The Papers by Lie or Élie Cartan?
L: Lie's really not, but I've read some of Élie Cartan's papers.
V: Hermann Weyl呢?
L: I have his classical books, and I read most of them.
V: When it comes to mathematical style, I'm not sure if Roger Howe is saying that weyl has a million dollars' worth of mathematics all in pennies.
L: Maybe I should mention Chevalley. In fact, when I was at Warwick University, the library had The Chevalley Symposium literature, which was so hard to obtain, somehow it wasn't officially published.
V: Well, a lot of the literature for these French seminars can be found in school libraries and in some places.
L: So I'm going to study that.
WANG: Was it the lecture notes for the seminar?
L: No, not handouts, it's Chevalley's exposés, some of which are Borel's, some of which are of other people's. But the workshop was moderated by Chevalley on the classification of reduced groups.
KING: Do you have plans to write another book?
L: No, but if I knew about lie-type finite group performance, I would.
Cheng: There will be an introduction.
V: Do you know About Macdonald's paper? He rewrote Green's work on the Weil group performance.
L: I don't know.
V: He demonstrated that the performance corresponds one-to-one to the equivalence classes of a -adic Weil group. -adic bodies should have residue fields, you just have to make the Weil group behave in wild inertia is trivial, and the equivalence relation he defines on these representations is the equivalence limited to inertia sub-groups. So basically, leaving aside the fact that there must be an extension of the Weil group, it's nothing more than a manifestation of the inertial subgroups, the modero wild inertia. He also showed that these corresponded to performance one-to-one. Anyway, I asked this because I only learned a week or two ago that this article was published around 1980, and that the formulation was perfectly feasible for any limited Chevalley group.
L: No, it still requires special classes, so it won't get it right
V: Look, obviously that's what Bill Casselman used to say, and it can't be right, but I haven't been convinced yet. I mean, there's something -packet that has to be described as difficult or impossible, but in my opinion those ideas provide a way to explain some of your past work.
L: I don't think so, I think you still have to say the word "special representation", otherwise you can't expect to do categorization.
V: I don't think so, we can talk about it again. What makes you happiest about doing math? Thinking about your plans for the next five years, you might expect to have a very outstanding student, teach a wonderful class, solve some rigorous problems, or write a good paper, and so on. It's normal to have a vision, these are all things you've done before, so what are your goals?
L: Teaching, well, but I get the most pleasure out of writing a dissertation. I don't think I'm going to solve any important problems, I'm just doing smaller things to get pleasure out of it.
KING: But in general, are you a problem solver or a theorist? Of course, categorization is difficult, but some people still tend to solve problems, solve famous problems, and they set some goals.
L: No, actually I like to look for conjectures, to find patterns of conjectures, which are probably more interesting than solving problems.
KING: Especially to solve your own conjecture?
V: I think It's Gelfand who said that making accurate guesses is not only more interesting, it's more important. Anyone can write some proof, but only if the guess is accurate and interesting.
L: That's why I like to guess.
KING: Maybe they shouldn't be so easy to prove.
V: Well, sometimes the conjecture is properly stated to prove that "indeed" is simple. Perhaps the best conjecture is that it's okay to say something in a way that no one before used
L: But sometimes if a good conjecture seems obviously right, I'm less interested, and proof is less important, and I think so.
V: Yes, that's right. The Riemann hypothesis may be an extreme example, and it is obviously worthwhile to do number theory, which can be used to see the meaning behind the hypothesis, because it is probably right, and its connotations are rich and profound. I think I've asked pretty much the question I listed.
Cheng: OK, I think it's almost over, George thank you again!
L: Thank you.
exegesis:
[1] Michael Francis Atiyah (1929~), British mathematician and winner of the 1966 Fields Medal, has been hailed as one of the greatest mathematicians of our time, focusing on geometry.
[2] Roger W. Carter, Professor Emeritus, University of Warwick, UK. Defines Carter subgroups and is the author of the important book Simple Groups of Lie Type.
[3] Pierre Deligne (1944~), Belgian mathematician, one of his most important contributions was to work on the Weil conjecture in the 1970s, winning the Fields Medal in 1978 and the Abel Prize in 2013.
[4] Hirinaka Hiriko Hironaka (1931–), a Japanese mathematician, won the Fields Medal in 1970 for his work in algebraic geometry.
[5] Armand Borel (1923–2003), Swiss mathematician, studied algebraic topology and Lie group theory, and made fundamental contributions to Lie, algebraic and arithmetic groups.
[6] Philip Hall (1904–1982) was a British mathematician. His main research area is group theory, which is known for his work on finite groups and solvable groups
[7] George Lusztig, Introduction to Quantum Groups
[8] André Weil (1906–1998), French mathematician and one of Bourbaki's founding fathers, made fundamental contributions to number theory and algebraic geometry.
[9] Roger Howe (1945~), professor in the Department of Mathematics at Yale University, is best known for his work on performance theory.
[10] Claude Chevalley (1909–1984), French-American mathematician and one of bourbaki's founding fathers.
[11] Bill Casselman (1941–), American and Canadian mathematician, studied group theory.
[12] Israel Gelfand (1913–2009), Russian mathematician, is regarded as one of the greatest mathematicians of the twentieth century. He has made significant contributions in various areas of mathematics such as group theory, representation theory, and functional analysis.
Note: The visitors to this article are Cheng Shunren at the Institute of Mathematics at academia sinensis, Wang Weiqiang is teaching in the department of mathematics at the University of Virginia, David Vogan is teaching in the department of mathematics at the Massachusetts Institute of Technology, and The finisher Ishinpei Huang is an assistant to the Institute of Mathematics of academia Sinica
