Transcripted by Zilin Jiang, Zhiyu Wang and Lizao Ye.

Edited by Zilin Jiang and Zhiyu Wang.

We were honored to have an informal meeting with Steve Smale, a Fields Medalist and Wolf Prize recipient, on April 16th, 2010, at Peking University. Here we present the notes from the talk.

Opening

Prof. Lan Wen: We will begin our informal meeting. Here is Steve Smale. Steve Smale has had a legendary life, as everybody knows. You can ask whatever you want to know about him, his ideas, or how to do mathematics. Or he will ask you what you want to know.

Prof. Smale: OK, what do you want to know?

On the Poincaré Conjecture

Yan Zou: The Poincaré conjecture in dimension three was not proven by topology, but by geometry. What do you think about it?

Prof. Smale: It is a proof by geometry, as you say. I don’t know whether it is an interesting question to look for another proof which is more topological. Grigori Perelman has produced a good proof. It is a new method, and it is fine to use differential geometry and PDEs.

Yan Zou: Do you think it might be possible to prove the Poincaré conjecture in dimension four using similar methods?

Prof. Smale: I thought about dimensions five and higher. I thought about three sometimes, but I did not really think about four very much. I just have limited resources. There are various things in the world for me to think about. It’s not good for me to focus on the Poincaré conjecture in dimension four. I’m not saying what other people should do. It’s fine that other people want to think about the Poincaré conjecture in dimension four. Great. For me, there are just many things I like to think about - other subjects, not just topology. The world of mathematics is very big. The dimension four Poincaré conjecture is an important part, but just one part.

On Complex Dynamics

Yan Zou: What do you think about complex dynamics?

Prof. Smale: I don’t know so much about complex dynamics. Like phase problems, the thesis in complex dynamics - in the 1970s, forty years ago, complex dynamics was at its very beginning. There are some very hard problems and many beautiful connections, especially in polynomial dynamics. It’s a very beautiful subject. It can be mathematically a different new land. I’m very enthusiastic about complex dynamics, but it is not immediately applicable to real-world dynamics.

On Mathematics in Economics

Yue Yu: There are a lot of applications of differential topology in economics and other related areas. Are they really profound applications of mathematics or just some definitions borrowed from mathematics?

Prof. Smale: I would say some are in between. I worked in economics in the 1970s. My feeling about my contribution was to give a foundation to economic equilibrium theory. I did not use deep results from differential topology, but some results, and gave different perspectives on equilibrium theory.

Yue Yu: Did those applications also contribute new things to mathematics itself?

Prof. Smale: Well, certainly economic theory has contributed to mathematics, especially game theory, and mathematical programming has been partly influenced by economics. There is some feedback, particularly in optimization. Yes, there is quite a lot.

On Personal Interests and Balance

Prof. Lan Wen: Do you know that Smale has a very big collection of crystals? He is also a professional photographer. I’m wondering why he would spend his valuable time on collections. Does it waste much time?

Prof. Smale: I’ll answer a reverse question: Why do I spend my valuable time on mathematics? I don’t need to because I have a good income. Crystals are worth more in terms of money, so maybe I am losing money when people appreciate the minerals more than my mathematics. Maybe I should not be distracted by mathematics or other sciences. I try to keep some balance.

On Education and Independence

Prof. Lan Wen: What is the most important thing for a student - hard work or talent? These are two extremes. Traditional Chinese seem to appreciate more a hard-working type; what is your opinion?

Prof. Smale: I am not a hard-working type, but I know it is good to work hard sometimes. One has to think about what one is doing. What am I doing in my whole life? What am I doing in mathematics? I do this all the time now. Every morning when I wake up, I make a kind of list of things I should pay attention to that day or the next day. Then I try to prioritize which are more important, which I should do first.

I spend time on mathematics, minerals, and most importantly, health. After minerals and mathematics, my health comes first. So I do a lot of research on medicines and health as it relates to my own well-being. I take many supplements daily, like vitamins B and C. Then the research comes in. I analyze what I should be eating to be healthy, what other things I should use to be healthy. I have blood tests frequently just to see how my body is doing.

I consider myself not just a mathematician but a mathematical scientist. I want to see mathematics in a broad sense, especially in what is happening in the world - engineering, statistics, and psychology. Mathematics often provides foundations for these areas. So that requires looking into not just mathematics, but mathematical sciences. Mathematics has changed enormously in the last 30 or 40 years. Biological science, computer science, and computational science have changed the face of mathematics.

On His Path to Mathematics

Prof. Lan Wen: I guess that when you were young, you were not only interested in mathematics. I’m wondering how, in your undergraduate years, you decided to devote yourself to mathematics. Did you envision mathematics as your career?

Prof. Smale: When I was younger, I played blindfold chess. I was doing many different things. In high school, I was passionate about chemistry. I read a lot about organic chemistry. I was a physics major in college. Finally, in my last year in college, I took nuclear physics. I wasn’t doing well anyway. I failed the nuclear physics course. So I thought maybe I didn’t want to be a physicist. I didn’t see myself continuing in physics. Then I changed my major to mathematics in the last year and I got through. I didn’t do great, and I didn’t do terribly either. I had a B average. So I went to graduate school learning mathematics. My teacher was Raoul Bott. He was a wonderful person. He paid special attention to me and two others as we were a little slower to grasp concepts. His courses were all advanced courses. Then I became more and more involved in mathematics. I was moving forward slowly.

On Blindfold Mathematics

Prof. Smale: I used to play blindfold chess with another player from New York, who was better than I was. We met in a national tournament when he was a high school student. We hadn’t seen each other for fifty years. I didn’t know how he was doing, so I looked him up on the web. He eventually emailed me, and he had just finished a book called “Blindfold Chess.” Last year we played blindfold chess together again. He had become a professor in psychology.

I used to compete in chess tournaments. In the national tournament for players under twenty-one, I was never in the top ten out of fifty people. In 1954, I entered the US Open Chess Tournament. There were one hundred and twenty-five people in the US Open, and without much practice, I placed fiftieth. I think that’s not great, but pretty good. Later, I more or less stopped playing chess because I found mathematics more interesting, and I could make a living from it.

Zhiyu Wang: As you mentioned, you can just sit back and think about math problems. I’m wondering whether you are playing blindfold math now?

Prof. Smale: Oh, yes.

Zhiyu Wang: What kind of math can we think about when we are playing blindfold math, and how can we focus on it without paper to write on? Will it be much more difficult for us?

Prof. Smale: It’s a good question. There are different methods of doing mathematics. I think the most important things are structures. What are the right structures one needs? In traditional mathematical problems, there are right structures. And there is another thing that is computing examples. These two aid each other. They are both important.

I tend to emphasize structures and concepts. What are the right concepts that are important in a proof? What are the stepping stones? What are the intermediate steps? That’s where my blindfold math comes in. If I work on something, I ask: What is the important thing we know about this? What is the important thing to see? What are the important intermediate concepts? Sometimes I work on a blackboard, sometimes I work with other people who scribble on the blackboard in my office. My blindfold math is sometimes aided a little because I look at the blackboard. What is the right map? What is the right correspondence? I scribble a lot. Some people can do blindfold math purely, and there are even some blind mathematicians.

On Learning and Teaching

Prof. Xiang Ma: You mentioned you had lots of hobbies in high school. What was the stimulus that made you concentrate on mathematics? Who influenced you most - your supervisor, your fellows, or your students?

Prof. Smale: I mentioned maturity. I became more experienced and more mature in life. The psychological aspect of doing mathematics is important. Graduate students can learn more from each other than they can from teachers. I was trying to think about doing things myself.

To some extent, I went to a one-room schoolhouse. It was a very poor school, one teacher for nine grades, and the teacher didn’t know very much. So I would go to the library and get the encyclopedia. In my math courses during my last year of graduate school, I began solving some problems and making contributions.

I think even today, one doesn’t learn so much directly from the teacher but can get inspiration from them. One learns a lot more by teaching. I think a teacher learns more about the subject while teaching than the students do. So if you put yourself in the position of teaching or tutoring something, especially what you’re working on, telling somebody else - that’s advantageous. You can talk to students about what you learn. I think it’s the best way to learn. Learning by teaching.

There is some feedback, too. If you teach in one situation, you get feedback from all around. Teaching forces one to think about things pretty deeply. I don’t teach nowadays, but when I taught, I had to prepare to defend myself in the classroom. To do that, you have to know the things pretty well. Today I don’t teach but I give lectures. So I have to learn something very well to be able to give a lecture, be able to defend it, and be able to be happy with it.

On Mathematical Understanding vs. Application

Prof. Xiang Ma: Now you’ve turned to applied mathematics. Where did the inspiration come from?

Prof. Smale: I am not an applied mathematician. I don’t think any applied mathematician would say I was. I theorize mathematics. I try to understand the world through mathematics, and that’s not exactly pure math either.

Prof. Xiang Ma: So you try to introduce mathematical frameworks or concepts to theorize something in the real world, to introduce good, elegant structures to them?

Prof. Smale: Here is the inspiration. Well, Newton, he wasn’t a mathematician. I don’t think he was a pure mathematician. But he gave the laws that many other subjects are based on. He gave the mathematical foundations of mechanics and gravity. In doing so, he developed calculus.

More recently, we have von Neumann. He wrote a book which I’ve looked at, called “Mathematical Foundations of Quantum Mechanics.” He did something that people don’t fully appreciate. He developed the concept of Hilbert space. Hilbert didn’t actually know what it was. There’s a joke that people used to tell about Hilbert being in the audience while somebody talked about Hilbert spaces. It’s very true, but it was a great concept, not due to Hilbert. It was von Neumann’s concept. This is von Neumann’s foundation for quantum mechanics.

So these are foundational things which eventually contribute to mathematics itself. Today, the most important things are not primarily from physics, but from biology, engineering, and computational science, especially computational science. Computation has become almost the paradise of science. Mathematics can play maybe the biggest role in computation. A very important subject today is understanding the foundation of scientific computation through mathematics.

On Recommended Books and Learning Resources

Student: Which books do you like best?

Prof. Smale: I don’t read books extensively, but I browse some books. The books I go back to review are fundamental works. In topology, one book which is extremely influential and still useful is by Eilenberg and Steenrod. Do any of you know that book? It’s on algebraic topology. It’s a fundamental advanced study that puts everything together in a clean way.

Another very important book is Jean Dieudonné’s “Foundations of Modern Analysis.” I should mention these are serial books. The first one covers calculus in Banach spaces. I think that is one of the most universal ideas that mathematicians are still slow to embrace - that calculus can be formulated in Banach spaces and made simpler. We can do all of calculus in Banach spaces.

I used to use Dieudonné as a textbook because it covers Banach spaces, Hilbert spaces, and spectral theory. There are some more accessible versions of these concepts that I think are good. Some books, not all, by Serge Lang are excellent. Lang has a book which might be considered a more accessible version of Dieudonné, called “Real Analysis” (third edition). I think that’s a very good book because it deals with things directly and gets to the core of mathematics.

I was on a committee at Berkeley to choose undergraduate textbooks in analysis. We had many discussions about books. I finally chose Lang’s for analysis. There are many other books, such as Rudin’s, but I like Lang’s because it includes Banach spaces. Today, I think only a few mathematicians really understand calculus deeply, because they deal with things in very traditional ways. Calculus in Banach spaces cannot be treated traditionally.

On Social Science and Mathematics

Dong Zhang: Professor, since last year, I’ve been trying to build a theory about social mobility based on social specification. Have you heard anything about social mobility and social specification?

Prof. Smale: Not exactly, but I’ve encountered related concepts through conferences and colleagues. I know some social scientists working on these questions.

I can say that among sociologists, there are both theoretical and practical approaches to these kinds of questions. They study how people form social groups, how they accept outsiders, and how social structures evolve. There have been some very good works on this by American mathematical sociologists using mathematical models.

On the Consistency of Mathematics and Reality

Zilin Jiang: I’m interested in your attitude toward the consistency of the world. In mathematics, we always deal with consistent systems. But in the real world, as in physics, sometimes physicists have to add new theories to old ones to fix problems that contradict experimental results. As you said, you are theorizing about the real world. But what if the world is not consistent? Are you searching for a theory of everything or something which is applicable enough for today?

Prof. Smale: I am against the theory of everything. Some physicists say everything can be reduced to small particles. I think that’s misguided. You cannot reduce biological phenomena to particles. You have to look at specific aspects of reality, some parts of it. Mathematics in some sense makes things consistent by putting them in a mathematical framework. Sometimes, inconsistencies can lead to the creation of new ideas.

I think one should focus on particular aspects of reality, whether in biology or physics. If you look at physics, classical mechanics works at a certain scale, but at other scales, you need quantum mechanics, and even then there are limitations. Everyone trying to create one big theory is missing the point. Different scales require different mathematical foundations.

On Pure vs. Applied Mathematics

Student: I know you are not an applied mathematician. But I still have a question about applied math. I believe there are always some things in mathematics that cannot be applied to other fields. What do you think about the meaning of studying this kind of mathematics?

Prof. Smale: Certainly there are some areas of mathematics that seem remote from reality, and I won’t force connections where there aren’t any. Number theory is an example. But other people have found applications, such as in cryptography. Number theory has influenced algebra and many other areas. In the history of mathematics, number theory has contributed significantly to pure mathematics.

People become excited about certain areas due to their mathematical importance and historical reasons, for example, the Poincaré conjecture. Different aspects of mathematics sometimes separate from each other, and some areas suffer from that isolation. However, there are certain aspects of mathematics that, while unrelated to the real world, are beautiful, concise, and important. I work on some of those myself sometimes.

[End of Interview]

Note: This transcript has been edited for clarity while maintaining the original content and meaning of the conversation.