傳播數學知識．促進數學教育

Interview with Prof. Robert Griess

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

**Interviewers: **Shun-Jen Cheng(SJC), Ching-Hung Lam(CHL), Wen-Fong Ke(WFK), Hsuan-Pei Lee(HPL)

**Interviewee:** Robert Griess(RG)

**Date: **August 18th, 2010

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

Prof. Robert Louis Griess, Jr. was born in Savannah, Georgia in 1945. He received his bachelor and PhD in 1971 from the University of Chicago under the supervision of John Thompson. He is a professor of mathematics at University of Michigan. For his seminal contribution on the existence of Monster group, he received AMS Leroy P. Steele Prize in 2010.

SJC: Okay, you said that you have some questions which you want to ask.

HPL: No, just the usual questions, we would start with how you become interest in mathematics and when?

RG: Okay, let me ask you first what is this interview for. Is this for publication?

HPL: Yes, for publication. We have a journal in Chinese with readers ranging from high school students to anyone who is interested in mathematics. So it's sort of a popular science magazine.

SJC: It's general, for people who are interested in mathematics.

HPL: So my first question is the usual one we have, how and when did you start interest in mathematics?

RG: When did I start? Well, I went to public schools and I found that I was good at mathematics. My father liked mathematics when he was a student and he showed me a lot of simple things in geometry, factorizations of integers, trigonometry formulas and so forth. These are elementary things I could play with without much specialized knowledge. So I played with them a lot and I just enjoyed the feeling. My teachers in all subjects emphasized careful answers and accuracy. These qualities are relevant when you study mathematics. There isn't always a feeling of excitement to being accurate in answering questions, but in science you have to get answers right and express them clearly. Later, one learns that one needs a vision of where to look for the truth. You don't get that when you are a child normally.

SJC: Speaking of your father, you mentioned that he was interested in mathematics. Did he study mathematics in college?

RG: No, not really. He enjoyed literature and I think eventually he got a degree in business administration. It's not very scientific. He had a lot of pleasure in studying and analyzing language and logic problems. He communicated those feelings to me.

SJC: I see.

HPL: I notice that you speak French and German fluently.

RG: Oh, fluent is an overstatement. I can make my way to the train station.

HPL: But you are interested in languages?

RG: That's right. I think that language is an opening to other cultures and the way other people think. I get more out of my contact with other people in my travels by learning other languages. I have picked up small amounts of many languages by getting to the train stations. Solving a problem while traveling is kind of fun and it's also fun to make comparisons of words and sounds in different languages.

HPL: I see. So you went to Chicago for college?

RG: Yes, I got all my degrees in University of Chicago. I entered college in fall 1963 and 8 years later I had a Ph.D. Then I took my job at the University of Michigan. They were unable to fire me so I am still there.

HPL: So you majored in mathematics from the beginning?

RG: I did. I was open to other things at the beginning but I kept coming back to mathematics. It was engaging.

HPL: Were there any teachers that influenced you to be major in mathematics?

RG: Well, at what level do you mean?

HPL: I mean when you entering into the university you decided to be math major?

RG: Yeh.

HPL: Oh, already?

RG: Well, I took a year or two to make up my mind finally. But I would say that by my second or third year I was pretty certain that I wanted to study mathematics. There were some good teachers I had an undergraduate. One was Arunas L. Liulevicius. He was an algebraic topologist. He was a very friendly and very clear teacher and his course in linear algebra was very popular. When I was a fourth year student, I took a Group Theory course from Jonathan L. Alperin. It had a lot of content and he made it so interesting. I got a global picture of the subject.

HPL: So that's why you get into Group Theory?

RG: Yes.

HPL: What about graduate school.

RG: The evolution of Finite Group Theory was in the air. University of Chicago had a tremendous number of visitors and instructors, post-docs in that subject. There was so much stimulation. Furthermore, near to the University of Chicago was the University of Illinois in Chicago. They had a lot of faculty interested Group Theory. The seminars, which were on Tuesdays, attracted visitors from those universities, Northwestern University and University of Notre Dame. There was a lot of excitement. People were talking about things all the time. There was a sense of mystery how the subject would develop. So it was very easy to get interested.

SJC: Natural in some sense.

CHL: Could you tell me more about some people who influenced you.

RG: Oh, okay. The regular faculty who were Group Theorists were Jon Alperin, George Glauberman, John Griggs Thompson, whom I asked to be my adviser eventually. Some visitors were rather influential. I mentioned post-docs, Martin Isaacs and Leonard Scott. Roger W. Carter visited one term from England. He gave a course on groups of Lie. Carter's exposition was really quite interesting.

SJC: Looks like it was a very exciting time. Seems to be one of the main directions in Chicago at that time?

RG: Chicago at that time had a reputation of perhaps the number one place in algebra. I mention the three Group Theorists. Basic Ring Theory was represented by Irving Kaplansky, Israel Nathan Herstein. Saunders Mac Lane. Mac Lane and Richard Swan taught courses in ring theory, K-theory and group cohomology.

HPL: Okay, so maybe I will ask about the monster. Everybody knows you and Bernd Fischer discovered the monster. Can you tell me some story about the discovery? How did you discover that group?

RG: How did that happen? Well, when I was in graduate school (1967 to 1971), there was growing excitement about possibly classifying finite simple groups. John Thompson’s 1959 thesis was a breakthrough. Abraham Adrian Albert arranged a special year in Group Theory at the University of Chicago (I think it was 1961-62). Jacques Tits visited that year. Jon Alperin was a graduate student from Princeton who came to spend the year there. Daniel Gorenstein and John Walter were there. Eventually they were able to classify finite groups which have dihedral 2-sylow subgroups. According to Alperin, that result looked so hard and so technical, it seemed at the time to be as far as people could go. That turned out not to be the case. Tits was developing the Theory of Buildings and many applications to Finite Group Theory. He did some amazing things. He told me how he heard about the series of Suzuki groups. After he learned their orders and that they were doubly transitive, he constructed them himself, before seeing details of what Suzuki did. This is an example of the high level energy at that time. The ball really got rolling after the Feit-Thompson odd order theorem. During the next few years, there were some theorems proved, classifying certain finite groups as the only ones satisfying particular hypotheses. Around 1965, Zvonimir Janko came up with a new finite simple group. It turned out to be a case that was missed in some analysis that Thompson and he had done jointly. The mistake, in a sense, it was not really theirs. They depended on some outside information, a character table, which turned out to be incorrect. So it shows you shouldn't always believe what other mathematicians claim. Anyway, Janko was off by himself when he found this group, which we call J-subscript-1. Moreover, within a year or two, he found two others; one of these was also found by Marshall Hall, independently. The Group Theory community was very startled by this eruption of new sporadic groups. This discovery of J-subscript-1 was the first appearance of a new sporadic group in about a hundred years, since the Mathieu groups in the late 19th century. People began to wonder if there were more. The year 1968 saw the discovery of about five or so additional ones. Their discoveries used techniques from Combinatorics and Number Theory, Theory of rational lattices, and Internal Group Theory. There were so many ideas involved.

You asked about the monster. It was discovered in 1973. Discovery means that evidence was found. At that time there was no existence proof.

People really wondered what was going on during the decade or so after J-subscript-1 came along. It was exciting when new candidates for finite simple groups were announced. But after an announcement, sometimes, a year or two went by without any more claims of new sporadic groups. This made people sad. But, just when they thought the flow of new sporadic groups was all over, another discovery would be made. This was the kind of atmosphere that we were working in.

Of course, people thought it would be fun to find new simple groups. They contain lots of beautiful mathematics. People were trying to find them, sometimes publicly, sometimes privately. A mathematician who finds evidence for a candidate would have to know enough group theory to deal with the issues. But even if you are very skilled, but it doesn't mean you are lucky enough to find a good candidate. I guess it's kind of like the search of sub-atomic particles. You work hard, you find evidence for existence of a new particle, but you really need experimental confirmation.

SJC: But in your case, did you feel that there were some other simple groups left? Did you have any evidence? Did you believe that there should be others?

RG: Others beside that one? No evidence that there would be more. However, it was unclear when the process of discovering more would close, or if it would ever close. I wrote a review, which I hope you would look at some time, of Mark Ronan's book called Symmetry and the Monster, One of the Greatest Quests of Mathematics, a very publicity-generating sort of title. I explained in my review that, at that time, it wasn't clear what to believe, and how a researcher should invest his or her time. Because of the size of the monster (about 10^54) and the high degrees of all its matrix representations (the minimum size would be 196883 x 196883), it would be technically challenging to attempt a description.

Even if one did so, it's unclear what the effort would be worth. Possibly, there were half a dozen more yet-undiscovered groups that were absolutely gigantic compared to the monster. If so, what would an effort to construct the monster be worth, if one had to give up other research programs? Working on an established research program instead would be safer.

Well, these points illustrate judgments that everyone makes in his or her career. Computers were involved in some of the sporadic group constructions.

I was rather young and I didn't quite know what to do. So I did what everybody did. I just waited a while to see what happens. By the late 70s, there was a stronger feeling that the classification of finite groups may actually be completed.

Janko, whom I have already mentioned, produced a fourth group which he announced in May, 1975. So after the monster and some of its close relatives were discovered in 1973, two years later, there was another discovery. Now, such news might shake one’s faith that a classification would close off. Well, it did. So the classification theorists gambled, in a way. It was risky, facing the prospect of years of work, only to be unable to resolve the program.

SJC: How long did you work on a construction of the monster?

RG: It depends on what you mean. Over a few years, I turned over ideas, tried small experimental calculations. I didn't seriously try to construct it until fall of 1979. The actual working period before I announced the results (on 14 January, 1980) was approximately three months.

SJC: I see.

CHL: What made you believe you can construct the group without using a computer? I think at that time nobody believes somebody can construct such a big group without using a computer, purely by hand.

RG: One of my motives for getting involved was craving for excitement. Also, I noticed that new ideas kept coming into play in finite group theory. I thought, well, if I get involved in this, there maybe I would find some other new ideas that would be very useful. So I decided to try and see what happened.

HPL: You were in Princeton at that time?

RG: I was on my first sabbatical from the University of Michigan. I got an appointment for 1979-80 at the Institute for Advanced Study. So I thought this would be a nice place to work, and it was. They were very supportive. You have lunch every day with all kinds of mathematicians. That academic year was a special year in Geometry at IAS. I met Shing-Tung Yau, Karen Uhlenbeck, Richard Schoen, and others. Members of this group were always talking geometry and geometry. They had potluck dinners, volleyball games and parties. They were very sociable. I listened to them and tried to learn something.

HPL: Did you really get ideas from talking to them?

RG: Not useful for my work. I attended lectures of Armand Borel on arithmetic group and cohomology. So I was exposed to a lot of geometry, analysis and the Lie theory that he did. This gave me some insights.

HPL: How did you start?

RG: In autumn, 1979, I was ready to work. The summer before I arrived, I went to a couple of meetings. I kept playing with these large dimensional representations. The dimensions were 196883 and there was supposed to be a product on it, which would be invariant under the action of the monster, if the monster is present here. There were all assumptions. So I kept just playing with simple situations like this just to see how things would work, just to clarify what the questions were. I started in the fall after arriving at the Institute. I just kept creating sample situations of groups acting on algebras that were more and more complicated and trying to explain them to myself. There was not much literature on finite groups acting on such algebras. I tried to understand such a situation by making a calculation in the algebra equivalent to some statement involving group multiplication, because that's where I was most comfortable. So I would try to deal with the questions there. Later when I converted the answers to formulas that define what I wanted, I threw the earlier experimental calculations away because I did not need them anymore.

CHL: So I think that Gorenstein announced the classification of the Simple Finite Groups in the early 1980s?

RG: 1980s … well, I am pretty sure that he announced that in a meeting which I attended. The title of his talk did not specify the final classification of the simple groups, but I think he made his announcement there. I am not aware that his words are recorded anywhere. I think it was the San Francisco American Mathematical Society meeting, January of, I think 1981, but I have to check the records with the American Math Society.

CHL: So, as far as I have heard of this, a lot of people feel very shocked, when they heard about the classification and many Group Theorists kind of got lost because they didn't know what the next project to deal with is. How do you deal with this problem at that time?

RG: Oh, another words, what to do, now that the central problem is supposedly finished? Well, you might call this faceing a career change, something many people deal with in their lives. A normal healthy process would be to resolve something, then move on. I tried different things. What I was most comfortable with was continuing to study the finite simple groups and how they arose in algebraic situations. So I studied non-associative algebras, I studied embedding of Finite Groups into Lie Groups, worked on non-associative systems. Then, when Vertex Operator Algebras came along in the mid-1980s, that was an opportunity to study finite simple groups in a new way. So, I chose to do that.

Other people had different responses to the career change issue. For instance, Representation Theory of Finite Groups has grown into quite a big subject in recent decades. Now it's strongly connected to Algebraic Geometry, Category Theory and even Algebraic Topology is coming into play. So that was one of the choice that people made. Finite Groups are relevant to Coding Theory, Galois Theory.

WFK: So what if some students, graduate students wants to study some Group Theory and got interested in it, what would be your advice on what direction to study? What direction of Group Theory you would advise such a student to study?

RG: Well, I mean partly that depends on the student. I would say what do you think you want to do. What kind of mathematics do you like. But if they don't have any idea, then I would just show them some of the things I have been working on recently and ask them to try to extend that. But further down the road, I mean, once they establish their work by writing a few papers in some areas, they should try to look for connections in other areas because that's a good source of inspiration.

SJC: How many Ph.D. students did you have?

RG: Let's see. I had five.

SJC: So I was wondering whether you think having Ph.D. students is inspiring in some ways for research?

RG: Yes. Discussing active research projects with a bright student can be very stimulating and enlightening. Teaching undergraduates helps me understand basic topics more clearly. I have had some excellent undergraduates doing summer research projects. I've been impressed by my graduate. Several of them remain active. My first one graduated 35 years ago and he is still writing lots of papers and so on.

SJC: What's the name? Who is it?

RG: His name is Arnold Feldman.

HPL: You have mentioned about a summer program, could you elaborate on it?

RG: Well, the main one is called the REU - Research Experiences for Undergraduates -- funded by the US National Science Foundation. There are some companion programs because I think a foreign student can't get money from the REU program so there are other sources of funds. But basically what it is they get an amount of money and they are supposed to do something for eight weeks. Before making a contract, they go around visiting professors to ask, if I work with you, what you would ask me to do. I tell them something of current interest and usually they go away and never come back. But some of them do return. I had a good one this past spring. I gave my REU student a model for constructing some lattices which include ones your colleague Ching Hung Lam and I needed in our research. He really did a great job of studying the literature, reading graduate texts in the subjects and generating lots of new results. I found it quite interesting to read what he did. There was a lot of pleasure for me to talk to him.

HPL: They were undergraduates?

RG: Yeh, I think he was a third year undergraduate. Some summers I get an REU student, and some I don't.

SJC: So I assume correctly that you enjoy teaching.

RG: I do.

SJC: I mean, certainly from the talks I heard from you, especially the one you gave in this annual meeting, I think that you must have spent a long time preparing. It was very clear and a very good talk.

RG: Well, thanks very much.

SJC: So do you spend a lot of times preparing for your lectures?

RG: Well, that depends. For an undergraduate course it's usually not necessary if it's something like elementary linear algebra, multi-variable calculus or elementary probability. Even for the graduate courses in Algebra, I don't usually have to prepare too much. If it's a more specialized course, then, people quarrel with me about proofs, so I have to defend myself.

SJC: You mean they quarrel during your lectures?

RG: Yeh,

CHL: They give you simple proofs?

RG: Well, they are very sensitive to nuance and logic and exactly what I am assuming. These points are openly discussed. People from different mathematical traditions bring up a viewpoint that I haven't thought of. So it's kind of fun to have debates like that with students.

HPL: Did you teach probability?

RG: Just elementary probability, I am certainly not an expert in Probability. The beginnings of elementary probability is lots of counting.

HPL: So it's not measure theoretical probability.

RG: Well, we have several courses that deal properly with Measure Theory and Probability, but the one I teach is introduction where you kind of wave your hands about set theory issues and real analysis. I don't know if the term “Measure Theory” is written in the text book, but I tell them about this background in class. They kind of accepted it. This course is for students who may not go on with formal mathematical training.

HPL: Do you feel that students right now and students 35 years ago are different?

RG: Yes, they are different. Well, mathematics is different. I think I see the difference mostly in the undergraduates in their behaviors. The graduate students tend to have more developed professional attitudes and these are not too much different from 35 years ago. What's interesting is mathematics has changed from 35 years ago. The undergraduates now come in with a different sense of mission. When I went to the college, people were interested in freedom and breaking down unreasonable barriers, promoting a new kind of civilization where things are fairer and more just. In college, we felt we should explore anything we want. Certain realities set in. People are more aware of environmental problems. When periods of economic stress set in, people got a little bit more sober about earning a living and being realistic. But you see, we need both viewpoints. There will always be an oscillation between these two tendencies. I wouldn't want to see either of them killed off. I think it's healthy to have a mixture like these. Society needs creative people. Freedom to explore subjects of interest, as well as rigorous training, are necessary for progress.

HPL: You were in Chicago for the 1968 disturbances?

RG: Yes, in the math department we have our own excitement about Finite Simple Groups, but, across town, there was the Democratic National Convention in August 1968. There were police actions. People were coming into Chicago. Some were angry and wanted to make public statements. This probably contributed to the Democrats’ loss at the elections. It was a really troublesome time for a lot of people. The Vietnam War was still going on.

HPL: The school was not influenced by the riot?

RG: This was late in the summer. The classes were not in session. I don't think.

WFK: Just coming back to this research for undergraduate students in summer program, I would like to know how it is granted. For example, here, I can apply to National Science Council for a project for undergraduate students, but I have to find the students. Was it the same in your case?

RG: Well, we faculty are encouraged to ask for an item in the grant applications to support students, and the department is aware of the money that's coming in. The money that I have won, and my grant, doesn't necessarily go to my students. If I don't have a student somewhere, then I would give it to some else's student. As I told you, there are other kinds of funds. The university administration gives money from their overhead sometimes.

WFK: So, how students got selected to these programs?

RG: University of Michigan undergraduates are asked to apply. One of my colleagues is the administrator. He finds out what the students want and tries to encourage them to meet potential faculty advisors. Undergraduates students don't know all the faculty members, so they are usually given some directions about which ones to talk to. The faculty member has to be willing and also the student has to be willing. So usually it's a bunch of happy matches. This year we had a problem because an unusually large number of students applied. I think there were about 60 applicants and only 30 faculties volunteered. So a lot of students had to be turned away. That was not expected.

HPL: So usually about 30 students?

RG: Well, I can't say usually. I don't talk to the administrators very much, but 30 sounds like a about right. Some student applicants were rather weak. If a student has done only Advanced Calculus and they get a C, most colleagues would not want him or her. Another category of applicant is a very good who is interested in engineering type problems, so they probably wouldn't want to come and talk to me here. Some applicants were interested in mathematical biology so they want to talk to one of the inter-discipline people. So usually it's pretty clear how to recommend potential matchups.

SJC: Maybe I should ask this question and maybe some of the readers probably would be interested in this question. If there is a high school student who said he is interested in mathematics and if he had asked you what are the most important pre-requisites to be successful in mathematics, what would you tell?

RG: Well, okay, one is easy. One is to take all possible math courses you can in high school. Do a good job with those basics.

In addition, there are summer programs for middle school and high school students. They are located around the United States. Our math department runs one. They have been running it for about 10 years. So students can come for one or two weeks and study coding theory or mathematical modeling or election theory or modeling fluid flow or number theory, lots of topics like these. The students are mostly very bright. The workshops tend to be hands on. The teacher comes in with the materials and gets the student to puzzle something out. I observed some of these classes. The kids tend to be very active and very talkative. They are very social. So that's good. When I was a student, I remember that the math students tended to work separately. They weren't social. Now the group work is emphasized. I think that's mentally healthy. What's wrong with group work? A weak student can hide behind the performance of strong students in the group. Well, if you test someone and you tell what they really know. There are other ways to find out who's good and who's not. But the interaction is good for mental development. I cannot teach kids everything they need to learn by the end of the course. In a small group discussions outside class, the additional dialogue is quite educational over the term.

SJC: The summer institutes are very important.

RG: Yew. Now, not every university runs them. I guess the math society probably keep track of the list of summer programs. So that's one thing to try. But I don't know about foreign students, whether they could get in or not. There may be special concerns about young foreign students coming to another country. I don't know. The University has many students and some of the dorms are available for students to live in if they come from a great distance. So that's another thing. There are a lot of websites that offer useful information, sometimes entertaining things about stuffs for kids. There must be math clubs in high schools around here.

SJC: Are there?

CHL: I am not sure. Not every school has it.

RG: Well, I know. If you make contacts with some of them and express your interest, people talk. Helpful information will get back to you. Networking like this was not so common when I was a kid

SJC: I see.

RG: Also, there is more publicity about mathematics these days. It's covered ore in the media. There are more books about the human side of doing mathematics.

SJC: The human side?

RG: Yeh, the stories of people who did this and that. For instance, there was this movie about John Nash, A Beautiful Mind. There was a movie in the early 1980s about a female mathematician, called It’s my Turn.

But I think, if you are at (49:11) your own university here, if you run these summer programs, you can start modestly. Just do one or two weeks. You've got to publicize it to the high schools or their kids won't know to apply. You might be teaching empty classrooms for a year or two, but once they find out that you are there and you are ready for them, they will come. Publicity about the social side of doing science, mathematics is a good idea, because it helps the students to engage and feel a sense of belonging. I don't know what it's like for females in mathematics and science here. I am guessing that they are still a minority. Is that right?

HPL: Yeh, that's right. But there is an organization for women in science.

RG: Yeh, there is Association for Women in Mathematics in U.S.. That started about 30 years ago and I think that's a good idea. People feel more comfortable talking and getting support. We have a large percentage of females and of minorities in our department, but it fluctuates. I don't think we should worry about whether it's 25 or 30 or 35 percent in a given year. I've wondered if the field does have some effect on percentage of gender involved. But as long as people think the mathematics community is healthy and welcoming to join in, then people should rise to their potential. I think that support is very important.

HPL: What do you do in your leisure time?

RG: My leisure time? What leisure time?

HPL: Beside mathematics.

RG: Well, I like to travel. I try to learn languages in my own slow manner. I am fond of art museums, paintings and drawing. I am very fond of dance, especially modern dance. I have learned that Taiwan has an excellent dance company, the Cloud Gate. I saw them in Tainan few years ago. I saw some more traditional dance companies there, but I forgot their names. They did the traditional Chinese tales. There was a monkey and others.

HPL: Was that dance or Peking Opera, I mean, Chinese Opera?

RG: I think there was singing, but, sometimes it was combined with dance. In fact, the Cloud Gate Company is really first rate. I have seen lots and lots of dance. I can give you a long list. I think they are really excellent. I think it's a pity that they don't travel more. I think they were in Vancouver recently. But they need a better agent. If they travel North America and Europe, South America. They would be a hit. I don't remember company names besides Cloud Gate.

HPL: I think there are a lot. There is UTheater (優人神鼓). Maybe the next time you are here.

RG: Well, I have few more days in Taipei. If there is anything interesting, I would go.

CHL: You would come very often.

RG: Well, I hope so. So there is dance. I also enjoy classical ballet. I have seen some excellent dancers from classical ballet. I have saw Baryshnikov few years ago. In 1984 I saw a very old lady, Galina Ulanova, the famous Russian ballerina. She was on stage for about one minute. She went up on her toe and went away. She introduced the piece and then the young dancers come it. I think it was called the troupe of the Bolshoi. It was just a subset of the Bolshoi so they had the younger people. I remember seeing Ulanova in a film from 1955 doing Romeo and Juliet, and you can get it on the YouTube, if you care. So that's it.

I think of creative work being rather similar despite being in a different fields. I don't like everything and not everyone would like dance. But if you find something in a creative field that you appreciate, and you wish to understand, you study the two components, the analysis and the synthesis. The analysis is about figuring out what the rules are, how they work. The synthesis is about putting together the ideas to express the artistic vision, whether it's going after a certain theorem in mathematics or some kind of study or creating a work of art, a performance piece or a piece of music. So you need both of these together. I think a good education would teach young people how to make both components work together. You don't just memorize formulas and you don't just do wild stuff until something clicks. Neither one of those is the best way. You need to learn how both of them together. You get to understand these by studying creative work, art, ballet, music, science, politics, business activities, . . . they are everywhere.

HPL: Well, another question. When you get stuck in your research what do you do?

RG: When I get stuck? I go for a walk or I switch to something that I like to do. I am personally very bad at dancing but I like to watch it. So I go and let them do the work. I just enjoy myself.

HPL: You change the topic or something?

RG: Yeh, if you just change the subject, even if just something, it just changes my viewpoint. For example, I was stuck on a research problem at the end of last week. So I took a day off and went to the National Palace Museum. I have been there many times. They don't change the exhibits much. So I saw a lot of old friends and some new friends there. At the end of the day, when I was waiting for the bus, I just thought of an idea that I could use in my math research. So you have to be patient. Sometimes I think about it in my dreams. I wake up and say okay now I have an idea for doing something.

HPL: You could remember about your dream?

RG: Well, I remember some dreams without special effort. If you make a point of remembering dreams, you could remember more dreams. It's a skill that you could teach yourself. If you get up immediately, take a shower and go to work, you would probably not remember. If you lie there, instead of getting up immediately, stay physically still, allow yourself to remember what your brain has been thinking about, and you can remember them. I find that I remember dreams in a reverse order. The most recent one first and then before that. I have remembered up to 5 dreams one night.

HPL: Really?

RG: Yeh, people who do sleep research say that's about the right number. You don't dream continuously. There are periods then you stop, then you start again. So this can be done. The dreams are usually not about mathematics. I won't tell you what they are usually about.

CHL: So you don't prove a lot of theorems in your dreams.

RG: Not usually.

SJC: But they have been helpful in your research also?

RG: Well, sometimes it can tell me what to do. That's nice. But it can also be about the process of doing research. In other words, why can't I figure something out? Well, it may turn out that I am thinking one piece of research may remind me something that has been annoying me a lot. My thinking is all mixed up. So if I kind of untie all the pieces I feel a little bit freer to think about it. Or I sometimes make up little dramas about places I am travelling, about doing mathematics, or about a recent meal that's very interesting, or meeting a friend and I didn't have a chance to talk and I thought of something that I wanted to say the next time I see them. This mental material can all get merged together. The reason is I am too busy when I am awake to resolve these thoughts. So my brain stores them and when I sleep, they all come out. They can come out in a very strange way. So if you just take a little time, you can remember.

I think some formulas proposed for interpreting dreams are useful. Your dreams are really about you. It's your personal material. If I see that part of a dream corresponds to something I was just thinking about, and another part corresponds to that annoying piece of research that I can't figure out, etc., then I can untie the threads in the dream and sometimes get clarity.

Successful research involves patience. You have to put in hard work, but you also have to be patient. In time, the secrets will be revealed.

CHL: I think we should end here. So thank you very much.

- Shun-Jen Cheng, Ching-Hung Lam and Hsuan-Pei Lee are faculty members at the Institute of Mathematics, Academia Sinica.
- Wen-Fong Ke is a faculty member at the National Cheng Kung University.