傳播數學知識．促進數學教育

Interview with Prof. Stefano Bianchini, Sijue Wu, and Shih-Hsien Yu

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

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

**Interviewees:** Stefano Bianchini(SB), Sijue Wu(SW), Shih-Hsien Yu(SHY)

**Date:** November 2nd, 2012

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

Prof. Stefano Bianchini was born in Pavia, Italy, on May 13, 1970. He graduated from
Politecnico di Milano (B.S., 1995) and earned his PhD from the International School
for Advanced Studies (SISSA) in 2000. He worked at Max Planck (Leipzig), Institute
of Mathematics, Academia Sinica (2000-2001) and CNR-Rome (2001-2004). Since
2004, he has served at SISSA as a professor. He won the 2004 European Mathematical
Society (EMS) Prize for his contributions to the theory of discontinuous solutions of one-
dimensional hyperbolic conservation laws

Prof. Sijue Wu was born on May 15, 1964 in Ningbo, China. She finished B.S. and
M.S. from Peking University respectively in 1983 and 1986. She received her PhD in
1990 from Yale University. After serving at NYU, IAS, Northwestern University,
University of Iowa, University of Maryland, she became a chair professor at the
University of Michigan in 2003. She was an invited speaker at International Congress
of Mathematicians in 2002. For her outstanding works on water waves, she won
the Ruth Lyttle Satter Prize in Mathematics in 2001 and the gold Morningside Medal in
2010 .

Prof. Shih-Hsien Yu was born on February 16, 1964 in Taichung,
Taiwan. He graduated from National Taiwan University (B.S., 1986;
M.S., 1989), and completed his PhD at Stanford University in 1994. He
was a faculty at IMA, UCLA, Osaka University and the City University
of Hong Kong. Since 2007, he has been a professor at National
University of Singapore. He has made pioneering works in bridging the
conservation laws and Boltzmann equation. He was an invited speaker at
the International Congress of Mathematicians(ICM) in 2014 and a
distinguished professor at NUS.
He joins Institute of Mathematics of Academia Sinica in July 2021.

TPL: The speakers for the meeting this week are ordered alphabetically. Stefano did not want to start alphabetically; he said it’s unfair, so let’s start alphabetically, but in reverse order. Shih-Hsien, you know about this interview.

SHY: Yeah, I have interviewed many people with you.

TPL: We usually start with this question: Why you ended up in mathematics; how you ended up in mathematics?

SHY: How I ended up in mathematics. I think that life is magic. I mean, I had no choice. At first I tried to do physics when I entered the college entrance exam, and I met someone from Taipei who made me feel that Taipei was so interesting, so I changed the order for my entrance exam. Fortunately or unfortunately I entered mathematics. I was not able to go into physics, and my scores were too high for mathematics, so I ended here. I tried to transfer to the physics department, but I was not able because all my scores were too bad, so I stayed in math. Eventually I realized this is my true love. This was determined by God. I just realized that if I had gone to other subjects, I would have been very miserable.

SW: So it’s not because you liked mathematics.

SHY: I really liked mathematics. But at that time, by accident, God gave more opportunity to find my real love.

SW: By elimination of choices.

SHY: Now I have more complaints about science. In the old days, physicists and mathematicians were the same people. So at that time I thought I would go to physics; I could entertain my math as well. So I would prefer to become a physicist. And nowadays it’s different.

TPL: You thought going to math and going to physics make no difference. They are both mathematical. That’s what you thought?

SHY: Right. In all the examples from the 18th century and the 19th century, they were essentially the same. But physicists have more freedom of imagination, that’s it.

SB: I think that’s a little weird. In set theory you can find something which is related to reality. And that’s really the key point in discovering the universal sight if you go deep. I really believe that the way we think, which is that we just use the set theory rules, it’s something we learn from the universe, maybe it is a true fact about the knowledge of the universe.

SW: I don’t quite understand what this set theory thing is talking about. Can you translate?

SB: The inference that we use to deduce theorems from hypothesis is something that is real. Why we learn this rule is because of the experience of thousands of years of history that leads the human brain to understand these simple inference rules and to deduce something about reality. The way we deduce something, not we created, but deduced because of natural selection.

SW: Because actually this is a universal truth.

SHY: We carry out a lot of information we can know. We don’t need to create from nothing. And on top of it, we can have more input. For example, we speak a language, but we do not create our own language.

SB: Even for languages there are universal rules, that point is true.

TPL: You always want to do math, is that so?

SW: No, not really. I always think mathematics is not what I want to do.

TPL: I asked a wrong question. How did you enter mathematics?

SW: When I was growing up, there was really not much study in China. Then after the Cultural Revolution, science became very much promoted by the government. There was a mathematical competition, which was the most highly regarded among all competitions. In high school, first, I skipped a grade, so it put me at a weak point because I knew very little about physics and chemistry. Actually I was always good in literature and humanity subjects. And that’s what I like, still. But I took this mathematical competition, and I won, first the city one, and then I was able to go to the province one, won that, and then the country one. But in the process, I had to skip classes to be trained, to go to the next level. I had already skipped a grade, which put me at a weak point, and then I skipped further classes, going to the competition. At the end, there were not many choices left. At the same time, we had the separation of humanities and sciences. Sciences were regarded as better/higher than humanities. Because I was a good student, I was in science without any choice. Because of the competition, mathematics became the only thing left. After the competition in the country, which I didn’t win, I had to take the college entrance exam. I did quite well. Well enough to enter the mathematics program at Beijing University, which was considered the best math program in the country. And it was my first choice. At that time, it was also one of the most competitive programs to get in. But at that time I didn’t really know if I study mathematics, what it is for, or what kind of career I would want to have, and what can I do with it. I had no idea. But I learned very little other subjects in science. I didn’t have confidence to choose any other subjects in science. So that’s about it.

TPL: Shih-Hsien said that he found his true love. How about you?

SW: Yeah. This is not my only love, but this is one of my true loves.

SHY: Actually, I think the environment in Taiwan is good. My undergraduate days at National Taiwan University were so good. They gave us freedom. I had many friends, and we lived in the dormitory. There were a lot of people in different subjects. When we came to the department, there were also many friends, and we got the opportunity to explore each other. We got some strong students, but we also got weak students. You can see different people. You got the chance to realize what your true love is.

SW: But you realize what your true love is whether or not you interact with people, right?

SHY: Well, sometimes you need to.

SW: I think I only find mathematics is really what I like very recently.

TPL: That reminds me of what S. S. Chern says. Some people talk about the interest in mathematics. But one is interested if one can do it. I guess you can do it.

SW: I always try to maybe subconsciously tell myself I like something else better.

TPL: That’s a healthy attitude.

SW: I probably will do better in other things, even though I have never done anything else, but I think I have other options. That probably also makes one feel good.

TPL: That’s called “hope.”

SW: Okay, I could do something else better.

TPL: Or “wish.”

SW: “Wish.” That’s true. But only very recently I find math is my true love. When you know why you do math, and that’s very important, then it becomes interesting.

TPL: Sijue, when you gave your talk this morning, I could see that you were very happy talking about the thing.

SW: Even though I didn’t know if I liked math or not, when I first started teaching, my students always said, “You obviously like your subject.”

SHY: So you love it, right?

SW: Yeah, without even knowing it. Without even trying to admit I like it.

SB: When I finished high school, I wanted to do physics, but my parents said, “No, no, no, you don’t get any job. Do engineering.” In fact I’m an engineer. But during the course, essentially I changed my direction because I discovered that what I like about mathematics is that you do not have to know a lot of notions to understand things, you just need to know the starting definitions, and then you construct by yourself a lot of steps needed to do the proof or do the theorems. Maybe you just need the statement. Then you can figure out if you’re in the wrong direction. It’s something like a game. And you read the proof, and you can see if you’re right or not. So I decided to apply for a PhD in mathematics and met many great mathematicians. That was very important. This is how I realized that all the probabilities would have different paths in front of you. And why I like mathematics. Math is not just a rule or a competition. Math is a way of thinking and to interpret life. It is the only knowledge of life that we can state for sure because we cannot know the universe. This is the assumption every philosopher knows. In this case it is a method, not the description you can have, for example, in classical mechanics, quantum mechanics, or even more complicated theories, but really the way, we did use theorems from hypothesis, and this is the key point of the real fact. This is true in the universe. If I should be young to choose again, well, I don’t know. Certainly I recognize the importance of this way of thinking. If you change the subject and lose this way of thinking, then of course I would say no. I need this understanding of life. So I would do mathematics again. Now in the known universe only men can solve mathematical problems. No computer or animal can do the same as we do. But maybe in the future this will not be the case. Like playing chess, now computers are much better than human. So I wonder if in the very far future, it would be important for human beings to develop algorithms and machines which think better than us.

SHY: How do you think of Galileo’s saying that “math is the language of God”?

SB: This is exactly my thinking. But I’m on a deeper level. So it’s not just the model of reality. You may never realize/prove if it is a correct model. But the language is the real knowledge we have about the universe.

SHY: That is the language of the creator. This is something common in all fields.

TPL: It’s great. You visited us for a few months and I remember you were doing mountain hiking and T'ai Chi Ch'uan, right? In the meantime, you do this center manifold coordinates. Could we talk a little bit about that?

SB: There are two parts. One is the idea of center manifold coordinate, which came to me a little bit before coming here. For some reason, there are some places where you feel at home as soon as you exit from the airplane or the car, and Taipei is one of the places for me. You let me stay quiet and do work on my own, which is something not real because I was in Germany, and there were these other people, and we had this seminar. For some people this was a good distraction, but for me this was a bad distraction. While here, I really worked hard for three months. Since I enjoyed the place, I could also hike or do T'ai Chi Ch'uan. All these things fit together into the right environment for me to work.

TPL: How did you think of this center manifold coordinates?

SB: This is just the fact if you have a nonlinear equation, a quite standard tool, as everybody here knows, is just to find essentially a good part of what you can, have estimates, and then the remaining term should be something that does not break this estimate in some sense, so that this estimate can be continued. In that case, if what turns out to be traveling waves, since they are constant solution in time, constant in the right coordinates, cannot be considered the source term. So you have to find a way to move this source term into the good part. So the center manifold exactly does this work, because whenever you are in one of these invariant solutions, it is on the invariant manifold by the way, the source term is zero. And this is the key point.

SW: So on the invariant manifold, the source term is actually zero. This is like, when you do it right, there shouldn’t be any source terms.

TPL: But nobody thought about that before.

SW: Maybe people thought about it, but they just didn’t know how to carry it out.

SHY: I don’t think so. Stefano is the first one.

SB: I did a master thesis on dynamical systems, so I had all the background to understand the ODE stuff. Otherwise I need to read the idea of the invariant manifold.

SHY: So it’s a mix of gene.

SW: I think my sentence should be understood in a broad sense. Invariant manifold is a choice of Stefano. In general if the situation is that of no sources, then when you do things right, there shouldn’t be any source terms.

SB: This is the philosophical point of view. The dynamical system, I could see immediately what was there. I had the background to understand what the right part was.

TPL: You work on water waves. By the way, your talk this morning was great. You were in harmonic analysis first, right?

SW: I agree with Stefano. That’s what I’m thinking now. Philosophically, I also think that my equation shouldn’t have any source terms. It’s just how to put it in the right way so that things just cleanly come out. It’s still not clear, but he has invariant manifold, I don’t know what I have for the moment. You mean why I work on the water waves?

TPL: I have heard people saying that, harmonic analysis, is a core subject, and then one goes to partial differential equation, and then people like Bourgain , for example, they really use harmonic analysis in a direct way to apply to PDE. But in your case, it was not a direct and obvious way to continue from harmonic analysis to PDE, it seems to me. Is that so?

SW: So you mean why I start to work on water waves. I started working on water waves after hearing the lecture by Thomas Beale in ICM, and I thought the subject was very interesting. And somehow he was talking about boundary integrals, and I thought I have the tool to understand the boundary integral. This is something I really like about fundamental subjects like harmonic analysis. Like Calderón , Coifman , McIntosh , Meyer’s work, which determines the class of curves on which the Cauchy integrals or Hilbert transforms are bounded from $L^2$ to $L^2$. It’s a very fundamental question. You don’t have to think, oh, what would be the applications. The interesting thing is if you can prove something so fundamental, there would be a lot of applications for it. When I heard of water waves, I realized I have the tool to understand the thing, so I decided to get into the subject and to look at it in detail. But then, of course, in the process, you always focus on your problem, on what you want to understand, and think about it, not that because you have this tool, then you must apply. So you look at what you want to solve, and you think what should be the essential things there. You don’t have to apply this tool. But I just felt at that time that I may have some advantage in certain ways because I’m a student of Coifman. I know the subject of harmonic analysis. I know the theorems a lot of PDE people may not know. But then, to solve precisely which problem, and to solve in what ways, that’s another matter.

TPL: You said you have a problem, and you should not always think about how to apply the tool you have; rather, you try to understand that problem, right?

SW: Yeah. You understand the problem. You have a starting point. You don’t work on a problem you have no feelings at all. You have to have some feelings.

TPL: That’s true. That’s a truism. This can be written on our front door.

SB: But sometimes I work on problems I don’t know from the beginning.

SW: I didn’t know at the very beginning also. I started working on water waves because I had been thinking about the vortex sheets for a while. After my PhD I somehow realized harmonic analysis was kind of mature. One of the reasons harmonic analysis has developed is its applications in PDE. I thought I should to go to the source to look for problems. So I decided to study PDE after my Ph.D. and was wandering around by myself and trying to find problems. And for a while I couldn’t find any problems to solve. Then I encountered the Euler equation, which looks simple enough, and the vortex sheet problem. But at that time I didn’t have any idea on what to solve, what do people care about. I didn’t know what I should do. That’s where I got stuck. And then there was the water wave problem that looked very similar, so I thought, why don’t I look at this one? In the process, I became aware that the Taylor sign condition is very important. It’s assumed in the paper of Thomas Beale, Tom Hou and Lowengrub . I was in Northwestern University, and I had easy access to the waves; it’s just next door; the university is on Lake Michigan. I went there looking at the waves and thought, even if it’s overturned, it doesn’t break. It doesn’t have to break. This condition should be true no matter what. Then I proved it.

TPL: Many people wish they could say that “it should be true, and then I prove it.”

SW: It looks true to me. But actually when it was first proved, some people didn’t believe it. They say that once it’s overturned, it should break. It should be instability. But for me, when it’s overturned, it’s still fine. Because you have to separate the effect from the wind and other things when you look at the picture. I gave a lecture in Hong Kong just before coming here and somebody there asked me: “do you do experiments?” I said, “No. But I watch the waves.” Then he said, “It would be difficult to separate the effect of the wind.” It’s true, but I think if you look hard and often enough, you would sense what’s going on. I would say, since you brought up harmonic analysis, harmonic analysis gives you the language. Not really, it’s a tool…

SHY: It’s a vehicle.

SW: Yeah, a vehicle. It’s not the only thing. The key point is, if you know harmonic analysis, when you see a singular integral, you will not be afraid. You have the basic feeling on when it’s bounded, why it’s bounded. But to really solve the problem, you need to understand the nature of the equation. Always, in every step, you have to understand the nature of the equation. This is not only harmonic analysis.

TPL: Because it varies from situation to situation.

SW: Yeah, this is the most important part. Once you reduce the water wave problem to the boundary, you encounter singular integrals, you have to deal with it. For example, you know when $u \times v$ is bounded. You know if $u$ is bounded, $v$ is bounded, then $u \times v$ is bounded. But now you have an integral form. You also want to know when it is bounded. Harmonic analysis gives you some basic feeling on what is bounded or when it should be bounded. Normally, if you have a straightforward equation such as $u_t-\Delta u = u^2$, you just solve it as is, you have nothing to worry about. But if I give you an equation containing some integral forms, then you start to worry, what does that integral mean? But if you are familiar with harmonic analysis, you say, okay, that’s a term I can handle. I don’t need to worry about it.

TPL: So you would not direct your attention to the place where there’s no need for attention.

SW: That’s right. So you just focus on what you need to understand, that’s the equation itself, instead of whether this integral is bounded or in what sense it is bounded.

SHY: That gives you the confidence.

SW: Exactly. You are not distracted by some basic stuff.

TPL: That’s a little bit different from the way that dynamical system helps you, right?

SB: Yeah, a little bit different.

SW: Maybe the same.

SB: I think you want to underline the fact that in your case, it allows you to concentrate on the part that really needs to be studied. They are not standard harmonic analysis. In my case, I changed subjects many times. I also did some paper in measure theory and linear transport. When I started as a mathematician, I did dynamical system as my master thesis. And then I did some measure theory. Then I decided to switch and did some hyperbolic equation. So I started from the scratch as I never thought I would. There was no direct, or even close application of the dynamical system to have hyperbolic equation before. In fact, this center manifold is just one piece because dynamical system is a huge subject with many parts, and invariant manifold is one small part of this subject. I was just lucky, in some sense.

SW: I think there’s no such thing as lucky. Perhaps intuitively you have the feeling that this may work.

TPL: You can be lucky on a more superficial level. With such depth, you cannot be lucky. I think that’s true. You come upon something, and that’s luck.

SW: You have to go that way to come upon it, right? That’s not lucky. You can go so many different ways.

TPL: Shih-Hsien, it’s not clear to me how you end up studying this boundary relation. Because in PDE you have initial value problem, boundary value problem, but you are doing something different.

SHY: And people think that I will try to go the opposite way. I believe if some problems can last for that long, there must be something wrong; people don’t do it right. And they do not end up with something, so you go back to the very fundamental step. So I decided to redo the problems again from the very scratch. For example, when people work problems in one way, I always try to see what happens if I do it in the other way. When I checked this thing and that thing, I found there were so many things mixed together, so I got the opportunity to think what would be the right thing to look into. So I gave up many things people already established. Then I started from scratch. If you’re starting from scratch, you don’t have the burden of knowledge, you get the freedom to see things freely, and eventually everything becomes so simple.

TPL: I’m sure every young person wants to have freedom, but they often end up as homeless people. So this urge for freedom is definitely not sufficient.

SHY: It’s emotion.

SW: Establishment may have some sense. You cannot destroy everything established.

SHY: I think one should. Establishment doesn’t mean it has to be that way. For example, Aristotle lasted for two thousand years, but after chemistry came out, we really need to have deep understanding.

SB: It’s the interpretation they give of Aristotle. I think Aristotle is a guy open to new ideas and so on. People misunderstand.

SHY: Establishment is fine, but don’t make it into a religion.

SB: People like to have somebody to tell them what to do.

SW: Yeah, that’s unfortunate. Actually when I was very young, I thought I should follow. Or at least I didn’t know what to do. That was a miserable period. But then, I worked on the water waves. I did it my way, and I felt very different since then. I really started to enjoy my work.

SHY: Going back to the question Tai-Ping asked, so I started from scratch and checked everything, and I realized that every step had a difficulty. Something really destroyed all the procedures. I also tried the way the people doing inverse transformation, etc., and you have initial data there. And I realized that you simply go nowhere; you are stuck for a long time. You should think, why not kick out the initial data. After you kick out the initial data, you realize that everything is changed. Our education does this. When we go to the PDE class, separation variable, and kick out boundary data to initial data, all in homogeneous term. Our knowledge started this way. And we go backwards and realize that the initial data is really troubling you.

TPL: You come to this because of the difficulty you encountered with those initial, boundary data, and you do without any of them.

SHY: Yes. In high school and junior high, we learned about control of parameters. When we did the physical experiment, we didn’t put in complicated things together. We took out this and that. I think math can perform similar to the way we do experiment. You just try to realize what the main cause of the difficulty is.

TPL: I’ve heard all three of you say about it, but I think fundamentally it is still a mystery how you do what you did, how you get in, how you get out that way.

SW: I think like Shih-Hsien said, you have to really give yourself freedom, to start with.

SB: You take a problem not only because you like the problem, but also because you know you are going to learn new things solving the problem.

TPL: You have that desire.

SB: Otherwise you’re not going to learn anything even if the problem is solved.

SW: You are able to unearth some secret.

SHY: You don’t create a new thing, you just make something.

SW: It’s really that you want to understand something that you have not understood.

TPL: I know you’re doing some measure theory, that the Pisa School, or the Italian School thing. You have been saying that measure is the real thing.

SB: I suspect the influence of measure theory into all, or many open problems of multi-dimensional hyperbolic equation, will be fundamental. That’s why I started with problems which are completely unrelated. I also had a little bit of background because the first two years of my PhD I did measure theory, then I switched. What I like is that, on the one hand, it’s something completely theoretical. You can re-formulate mathematical logic using measure, so you have all these problems of contradictions. As soon as the space becomes very weak, and as soon as you have very low assumptions, then you have completely wild situations. On the other hand, you can interpret many equations as a transport of measures, in this case the solution is not any more a classical solution, not even a weak solution, not even you can compute anything, and the nonlinearity of measure is meaningless because they are not functions, so you have nothing. On the other hand, this obstruction lets you focus on the important thing, for example, dissipation, because dissipation is something defined on the measure. In some cases, for example, compressible Euler, it has now been done, but this is my interpretation. It should be possible to rewrite the equation as the movement of some measures, some space, which describe this local status, but it really is not local any more, being as measures, a more complicated object, which should be interpreted. And this is something, in my opinion, very nice. So you give meaning to things which are meaningless a priori. This is a very important part of mathematics. So the inner core of mathematics is the fact that you are among many facts. What I want to wonder right now is that even if there are just formulas hence something not existing in reality. In any case you are able to interpret it in a correct way, if the theory is correct. If you interpret in a correct way, you get new insight into reality. There are many problems which are correlated, I think now no one is able to give meaning to these problems. However, many problems are dodged, which are open from ten to fifteen years, which are interesting on their own. They just lead you to acquire the knowledge, or the tools which may suggest what to do next.

TPL: I think your point of view for compressible flows is very welcome because right now we have a crisis, what a function space should the solution live. Now you’re opening up a different interpretation.

SB: It can be that even if the equation is not correct. So what you need is a time path of an object which has the right properties. So why should it be the equation?

SW: I didn’t quite understand. So you just want to study measure theory for the sake of measure theory, or it’s actually something else?

SB: Now I don’t know measure theory as a very wide subject, but I have the tool to take out the book, open the middle, and to read it. But there are some PDE problems related to geometric measure theory, which I really like a lot because measure theory is very clean. Since it is very close to mathematic logic, you can make it very clean, on the one hand. On the other hand, what is surprising is this simple fact: a function has the following fundamental property, it can be localized. So if you have a point, they give you a value. It’s not for a measure. For a measure, you need to have a set, or some particular set. However, they can be localized. It’s fantastic, so you have the notion of the derivative, with respect to another measure which has a lot to look at. Essentially this is my vision of this theory. If a function is not an object sufficiently rich to describe your system, for example, if you have a function multiply a Lebesgue measure, is a measure, so this is an extension in some sense. So maybe in this new space, you can solve problems which cannot be solved. For example, in the talk of F. Golse, if you approach without using measure theory, it’s very complicated. But once you interpret the flow as a flow applied to a measure, then it becomes very clean.

SW: It just gives you the language you need to describe. So you still have a purpose. Everything has a purpose.

TPL: So, during your PhD training, you need to have a certain core subject which you are trained well and have a good feeling about, not just to solve a good problem.

SB: As you become older, the energy decreases. But the most important part is that you have less time. This is really the biggest difficulty. So you have less time to study for free. You can definitely do it, but you have less time.

SW: I think as you get older, you become aware of so many problems you can solve, it’s so much easier to go into the direction that you already have the tool. You are less motivated to learn something completely unrelated.

SB: But this is also related to less time.

SW: You have to really conscientiously learn new things if you feel you have to. You have to be very motivated.

SB: As a student, doing courses is fundamental. Because then you choose a subject you should know absolutely. And since you have to give a course on it, i.e. you have to teach, it is a very important way to force you to study something.

SW: But then I have a feeling that if you want to teach well, this has to be a subject that you can somehow relate to. Say, right now, if I teach algebra, I don’t know what it is for. My lecture will be very dry. It just will not be very interesting.

TPL: By the way, talking about algebra, these days Shih-Hsien is thinking about algebra.

SHY: It is Galileo’s belief that mathematics is the language of God . I think algebra is the essence of math.

SW: No, that’s true. I agree. After I gave my public lecture, which was shortly after I showed that my equation actually contains no quadratic nonlinearity, the chairman of my department, Mel Hochster , came up to me. He’s an algebraist, a very good one. And he said, “This is algebra.” I said, “You are right.” You have to do algebra to reformulate your equation into that form. It’s an equality. Not an inequality.

SHY: For example, in algebra, you can do multiplication, subtraction, and division, everything. It is not only equality. It’s an object. It can do operations. But if you break this into several parts, then the hand, the head and the leg are in different places. You move one of them, and the leg goes away. But you keep algebra. One equation is an object. Then when you move another one, everything balances together perfectly. I will say it is polynomial, in particular.

SW: I don’t know if we are on the same wavelength in terms of algebra.

SB: Algebra is a wide subject.

SW: At least for me, I always think: I’m so stingy, I don’t want to give up anything. So I preferred to derive equality instead of inequality as much as I can.

TPL: Maybe one can rephrase another sentence and say that “behind every success there is an equality.”

SW: Maybe not. But when I’m talking about teaching algebra, I’m talking about teaching, say, category theory, or field theory, or group theory. For these topics I will not be able to tell students what are these for, and my lecture will be very dry. When you know why you are doing something, what you’re doing acquires life.

SHY: Let me give some examples. In Roman times, if one wanted to do multiplication, it was very complicated. But after one knew how to do multiplication and division, everything changed. So the language given by God has some kind of simplicity. Most of the time my brain is lacking new ideas and the understanding of a new problem, but at some moments new ideas are generated in the sub consciousness. There must be someone that puts it in my brain. It’s a mystery. In the future there are still many unknown phenomena of physics to be solved. It’s time to explore new ideas. We can explore new ideas in terms of mathematics and physics together.

TPL: You mean in the future there are many natural phenomena to be explored for which math plays the central role.

SHY: Right. Mathematics can still become the mother of science.

SW: Not can. It is. It should be, right? Because mathematicians or physicists, if we think about it, we really want to understand the laws of nature. So if this is the goal, you know what to do.

SHY: Yeah, but look at the reality. Nowadays, if you go out and talk to physicists or engineers, none of them will talk to you seriously.

SW: Sone Sensei and I were talking this morning. We agreed on many things.

TPL: I like the way Shih-Hsien put it. You have to exaggerate to make a point. You said “none of them,” but of course you didn’t really mean none of them.

SW: I think if some of them do, that’s good enough.

TPL: That’s all you can hope for. But you said to discover the law of nature.

SW: To understand the laws of nature. To understand is really the word because if you look at the phenomena, it’s all there, but you want to understand what’s really happening.

SB: Newton’s equations are very easy. However, you can arrive from Newton to Boltzmann to Euler . It’s very complicated. Reality is very complicated even if the model itself of reality is simple. Even inside classical mechanics, it is extremely rich. On the other hand, you cannot describe/deduce every proposition, it’s complicated, and it’s interesting. As soon as you don’t have a finite set, which is our case. You never assume that the universe is finite.

TPL: Shih-Hsien, do you have another final conclusion?

SHY: Yeah, mathematicians still have hope. The hope of occupying the core position in science.

SB: There’s a distortion in the media. In the high energy physics, there are many mathematicians working on models. Many models do not have inner coherence. To prove the coherence of the model from the assumption, the law. This is a mathematical world. No model will do it. Now we have this CERN new machine, you don’t recognize how many mathematicians are working there. They allow the machine to work, and they analyze the results. People think there are just physicists working there, but this is not the case.

TPL: Thank you all. And I hope to see all three of you individually or together very frequently in the future here.

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