Prof. Stephen Smale was born at Flint, Michigan on July 15, 1930. He received his PhD in 1957 from the University of Michigan. He was a faculty at Columbia University before joining the University of California, Berkeley in 1964, where he held a teaching position for 30 years. In 1995, he was invited to City University of Hong Kong as Distinguished University Professor. He was the director of Toyota Tech. Inst. at Chicago. His broad research interests include dynamic systems, geometry and topology, mathematical economics and theory of computations. He was awarded the Veblen Prize in 1965, the Fields Medal in 1966, the Chauvenet Prize in 1988 and the von Neumann Award in 1989. He is a member of National Academy of Sciences and of Econometric Society.

TPL:Steve, first thank you for coming all the way, I normally don’t say that, but we are honored.

SS:Thanks for inviting me.

TPL:We start with very boring question, which is how did you start your mathematical career?

SS:How do I start mathematics?

TPL:Yes, what makes you interest in mathematics, or you are interested in mathematics since very young?

SS:Sciences, when I was in high school. I was physics major in the college. My third year physics was very bad. So I was failing in physics, and I had some math courses for changing my major to mathematics at senior year, you know I really wasn’t doing too good. I was taking academic very lightly when I was in college, I had B average. Mathematics may be B+.

TPL:Would you have a good time then?

SS:Yes, It’s interesting, for me college was good. I was active in politics, other things, lots thing, chess. Mathematics I had some good courses and some bad courses.

TPL:I graduated from Michigan.

SS:You do, Ann Arbor?

TPL:Yes.

SS:Me too.

TPL:We were very fun to recall your story as a Ph.D. student, and took you a number of years and solve a big problem. You have supervised many Ph.D. students, right?

SS:Yes.

TPL:Do you always give hard problems like that?

SS:No. easy problem. Sometimes sometimes by accident I do, but I tried not to give hard problems.

TPL:The first time I came upon your name was when I was actually before Michigan; I spent one year in Oregon state University. A topologist called Wolfgang Smith. He said that, you know, we can see three dimensions, but there are people like Steve Smale who can visualize more than three dimension. Is that true?

SS:Oh, no. Well, in some way yes, sure you can project high dimensions to two or three dimensions, you can extend some notions you see higher dimensions. You see a knot in three space, you can somehow extend to see a 2 sphere in four space, right?No magic.

TPL:Clearly, that’s magic to a lot of other people. Recently we talk about Poincare conjecture, you are one who made the first major breakthrough, it must be very satisfying for you to see further development.

SS:Well, interesting. Yes, I am not so involved in topology for how many years, more than forty years I am not being working on topology since 1961. So you know it is not very big thing for me. I follow more academic interest to see what happens. It looks like this guy from St. Petersburg,列寧格勒（聖彼得堡）Perelman, it looks like it most likely has solved, interesting, sounds good, sounds good.

TPL:But otherwise you are more interested in applied mathematics these days.

SS:No, I am not. I don’t do it. I am not interested in so much applied math nor pure math, not applied , not pure, neither one. Mathematics, OK. But I am more theorizing something. Applied mathematicians don’t think I do applied math, because I theorizing very much theoretical scientific applied things. It is not exactly pure, it is not theorizing not about very old fields in pure mathematics, theorizing about things people are using today. So somehow doesn’t fit exactly either pure mathematics, doesn’t fit applied math. Applied mathematicians have certain traditional criteria for measuring success, consists more immediate kind of application usually. Like von Neumann doing the foundations of mechanics. Is he doing applied or pure? He invented Hilbert space, he was working out the foundations of quantum mechanics, that’s my goal. My idea was somehow looking foundations for some area. In his time quantum mechanics was the big thing. To give mathematical foundations, he did more than anybody I think. Now they answer things helping outside mathematics. Now I am talking about learning theory, I spend a lot time try to see mathematical foundations. That’s the title of this paper I wrote. It explains a little bit.

TPL:Applied Mathematics has certain goal and certain sets of problems to solve?

SS:Yes, philosophy which not exactly mine.

TPL:You had been Hong Kong, so you know this part of world, and of course most of time, you have been in the West, in US. It’s different society, right?

SS:Yes, in some way Hong Kong of course is a pretty advanced society. You see a lot differences I guess students maybe the biggest that I want to notice. Students in Hong Kong I didn’t find the king of independence I found in US students. In general rule they are hard working often time but not so pretty independent. That’s my impression that Hong Kong students lack the independence of the US students.

TPL:That’s the most clear difference.

SS:Yes, as far as being in the university.

TPL:But otherwise faculty were mostly trained in US.

SS:I am talking about students in my classes.

TPL:Sometimes ago I read an article about Russian mathematician Arnold. He mentioned you as very few heroes he had, you might even be aware of that.

SS:Yes I read that.

TPL:During all these many years, long scientific life so far, there are many more years in the future. For you, what are the exciting moments you feel?Either yourself or other, let’s maybe start with yourself.

SS:Exciting moment?When you make some kind of attempt many times, suddenly succeeded, something in the fact hold up which did sometimes that was very satisfying, you know the usual success I had like turning experience inside out, when I pull back, seems good, that seems still not being wrong after few weeks, few months, and then people like the results a lot, you know that, developing satisfaction, say my work on Poincare’s conjecture, higher dimension, whenever you do something seems to be hold up, both as far as correctness and as far as eventually getting some kind people be interested in. Sometimes it takes long while, sometime it takes many many years but satisfying, you know something like, I thought was very good, was the horseshoe that took ten, twenty years before being having very much recognition. So it is gradual but still gradual, more and more satisfaction as I see people like our results. So you know vulnerable mathematical things I did eventually catch attention. True, you know things I do, you look out a week later, a few weeks later, wrong, it’s maybe more common experience.

TPL:This scientific research is human activities.

SS:Yes, occasionally you are lucky, something hold up. Staying holding up, catch attention, that’s great.

TPL:You have been interested in various things, econometrics,（計量經濟學）learning theory, I mean after your topology theory, right?

SS:Right.

TPL:Do you talked to scientists or you just learned basic and then you want to find mathematical foundation?Do you talk extensively with people in economy?

SS:I was lucky to have Gerard Debreu to talk economics. Learning theory I sort partly worked with a friend of mine, Felipe Cucker. We spent a lot time reading it, try to understand many things that people have done, not mathematician. So it’s hard sometime to see what’s going on, you know there was very deep ideas we didn’t understand, took long time. That was about two years to write one paper. I think long time ago, because we correct our own misconception so much. We didn’t talk to people about that. During that day, we grabbed out literature, merely browse, we didn’t read but eye browse, look for a while things people done, eventually converge this area. Once we got basic things, we knew what we were doing, then we started building up some environment, we had conference in Hong Kong. I started got invited to conferences then there was a lot of discussion, other, generally not mathematicians, people working statistics, engineering, computer science.

TPL:These are the crowd you face?

SS:Yes, a number of mathematicians are also picking up subject. People mainly from approximation theory. So there is quite a lot people, mathematician who worked approximation theory are working now learning theory. I think it is little contribution we did. Things in approximation theoretician to learning theory, people like Ronald Devore, Albert Cohen, quite few people.

TPL:You are not going back to topology or geometry now?

SS:Well, actually my second talk here I am talking about topology and data, that’s had me going back looking at lots things I used to know forty, fifty years ago. Differential geometry and relearn second fundamental form, manifold, things like covering, algebraic topology, because we do take submanifold in Euclidean space. Suppose we take points random to that, that’s data, that’s general problem, data is some geometric curves… What’s the homology of the manifold, some questions we ask, we have some answers for that. I work with topologist, computer scientist Shmuel Weinberger, do you know him?

TPL:Yes, I know him. That’s a while ago. Was he in Courant or somewhere?

SS:Yes, he was a student there. Now he is a Professor of University of Chicago. He was one of three coauthors of us. I talked a lot about topology with him because he is interested in some of the computational aspects of topology. So I am coming back a little bit some of that, this paper we finished become second lecture.

TPL:You don’t distinguish things, whatever come naturally you see, but I think it’s nice for the student to know that Steve Smale needs to relearn second fundamental form.

SS:Yes.（Laugh）

TPL:Don’t you feel that it’s always important to admit that some basic things is somehow either has to remind yourself or needs to learn it?

SS:Right, but now I think, to take your point of view, I never relearn satisfactorily. I look at together, feel maybe simpler than I used to think, a little better, because of more experience, use submanifold in Euclidean space, higher dimension, so I don’t know, it is not easy either. I talk a lot with my son and learn a lot from him actually. He works on minimal surfaces.

TPL:He is now in Utah?

SS:Right. I called him a lot, and in this paper we need to know some basic questions about geometry, some manifolds, on the phone a lot to him, try to get him stress out.

TPL:It is convenient.

SS:So I’m lucky to have differential geometer son. I talk a little second fundamental form.

TPL:What did your father do?

SS:My father?He was a white collar worker, they called him. He has two years of college. He worked ceramics, an auto motor manufactory in Michigan. He is not a union member, white collar worker they called; but he eventually quitted and became a night watchman. His last job was night watchman. So he was not very successful. He wanted to be a writer. He wrote a novel, but he could never publish. Maybe he had sold it. I would say he was not so successful.

TPL:It is well-known to mathematical community in general that you have certain liberal thinking or maybe that’s wrong label. But in any case, you are against war, anti-war and so on, has some influence from your father?

SS:Perhaps, perhaps. He was a split from communist party in 1919, he split off the left, from another party, very small party called proletarian party. He was an armchair radical, very Marxist. For me, I was in communist party for a little while as a student. After that I became anti-communist. 50’I became an anti-communist and 60’I became very anti-war, anti-Vietnam war. At the same time I became more conservative in economics, so I was not liberal left when I came to economics. I was more market, marketing economics. I worked a little bit various issues so speak marketing economics. Economics point of view I was more on conservative side, not extreme, but more market. But on foreign policy the opposite, more though left wing side, when I came to foreign policy, anti-war, military, such things. So it is hard to say if I were liberal or left. But just on economics, a lot domestic things I am more conservative.

TPL:You are very much public conscientious, you think about society?

SS:Yes, I think a lot about these issues, strong opinion about a lot of domestic issues especially foreign policy. I mean things like social issues I am more liberal, but come to litigation I am more conservative, anti-litigation.

TPL:So, big royalty for lawyers.

SS:Some of those issues I am more on the side of Bush.

TPL:Be it liberalism or conservatism, mathematic you prefer to do, you enjoy do it, this is somewhat separate issue, right?

SS:Somewhat, I was interested in economics, maybe a little more relevant. Mathematically I tried to spend my time, my theorizing about mathematics could be important for society now. I tolerate type of questions for mathematics. Sometimes I like pure mathematical problem, sure, all right. My main decision has to do things could be exciting from point of view go beyond mathematics, you know, immediate feature like learning intelligence, how is brain work, things say something about that more interesting to me maybe than mathematics, very remove for me human than number. It is not clear cut, but you know I do.

TPL:But it has influence.

SS:Oh, yes.

TPL:Do you have suggestion for the young people going to the subject, or what would you say to bright young people should do these days?

SS:A lot of mathematician, you know, say they should learn mathematics. I wouldn’t say that, I wouldn’t give advice, but you know I was talking in lounge this morning people in National Taiwan University, they were saying how a lot very best people are not going to mathematics, but the mathematical courses are now advance courses are populated by many people out the mathematics, much much more so than twenty years ago. Twenty years ago, they have a class, the room is filled nine people all math majors, now they have a hundred people mostly outside the math. Mathematics, I don’t see mathematics in terms traditional mathematics department, because I see myself many many people who are great in mathematics and are not in math department. So I wouldn’t suggest somebody become mathematician traditional sense, maybe I won’t say it shouldn’t be, but there is a big place for mathematics, computer science, engineering, a big place, electronic engineering, a lot fields for advance mathematics.

TPL:You haven’t say that, but you like bright student to learn mathematics.

SS:I say so much modern life, more mathematics is big asset for students. I think so. Much more so than ever. So many fields with more advanced mathematics one has more powerful position that most person gets. But you know it is not just math. In medical or biological domain, if one also acquires a lot molecular biology, then one has a better position to do better research.

TPL:Just fundamental things?

SS:Yes, right.

TPL:More basic things.

SS:Right, right.

TPL:Mathematics is one of the basic things?

TPL:Yes, exactly.

SHY:Steve, What’s motivation you try to get into something you don’t know?

SS:Oh, it’s fun to learn something new. It’s fun, especially if you can see yourself make some contribution. So I am not little talk just reading Scientific American, but some places I see if I put some good effort in it, it could be rewarding by my making some contribution. So the motivation is for me to make contribution. Some area which are interesting for me, something I feel I could learn enough to relate to mathematics and making contribution, that’s the big part of it.

SHY:How about during your formation time? When you were young, how do you make decision that I want to engage this, maybe there is no reward?

SS:It is always a risk. I think the first thing because I worked in topology, but even after I have been working in topology, it’s way back 1958, I met some people who were working some kind global ordinary differential equations, and I talked to them. Then I say maybe we are using topology access, easy with my background, I could say something exiting about, what’s happening on ordinary differential equations. You know I were optimistic, which took much longer than I thought but I did start working that title called qualitative theory in ordinary differential equations. I did even back 58 or sort of 59 I started doing things very overoptimistic but I did write some papers. The feeling like I could say something, it is motivation for me to start learning more about ordinary differential equations. I took this book by Lefschetz “Geometric theory for ordinary differential equations” throughout quickly, you know I have some background on manifold, transversality, you know lots of differential topology. I believe maybe if I just spend a little more time, another few months, a year, I can really say something in this field. Then I did, that’s what happen, that’s an example though.

TPL:Is that one factor you are able to do it?

SS:I think it’s partly psychological. Some people approach something in new field like coming in as a student, not me. Oh, no, I wouldn’t have patience. I don’t know ordinary differential equations as a student, Oh, no I just started looked at advance research literature and subject. It takes a while, I learn only by successful approximation, try doing things almost immediately, try to do things usually failing, learning about things moving in parallel, not from bottom up, that makes big difference if you start bottom up, it’s too big. I say for me. I forget. So when I were working some new field, never learning as a student. Learning from my advantage point, my advantage point being successful mathematician in area which can help me learn.

TPL:Do you talk to people?

SS:Often time, I use library a lot, now it is more internet. So I scan, I go through a lot library, I do things very quickly and superficially first. Go through library, looking at dozens dozens books, try to find something I can learn. When I read a book, I just find certain chapters, pages, something that I can learn. So that’s the way to motivate. I don’t have a feel I can spend so much time as a student learning something, coming in using all the references I have. That is why I don’t like to read some subjects like learning theory, biology, I don’t like to read books by biologist, because I cannot related to them. It’s too slow. I try to read things by mathematician who already thought part of biology. I can use what they learned, use mathematical language they used, catch onto quickly, then makes a lot easier tell something in different field, using mathematics I know.

TPL:You must have learned and read a lot and choose only a few among the things you read and you actually do.

SS:Yes, so lots things I am working I don’t really know too well, a lot aspects. I have to learn some probability, statistics because learning theory. But a lots amount of subjects I know nothing about. I talked to statistician they have to be kind, because some areas I know pretty well I have some good questions, some general parts of statistics, probability. Something I know pretty well but lots things close by I don’t know. Yes, the way I am is sometime could be save a lot time about learning something to do, that’s the picture, but often time I talk to people too. Sure, You know, I talk to people and some people learning much more from my talking to them than I do. People like Dennis Sullivan, read hardly, always talking people to get information. I am sort part way in between.

SHY:How those information you gathered by talking to people turns out as motivation and intuition of the field?How do you reorganize them to become intuition?

SS:I first try to see some important problems I might be able to solve, make conjecture, make advance. Most time doesn’t work but then I adjust things. From the very beginning I try to do something new in the same as students. When a student came to me wants problem to work on, I try to give problem that he can make contribution in a week. I get say something special case or something could be interesting for him/ her to get new results in the matter of weeks. That’s my philosophy, keep surprising, usually does not be that simple. I try to get students working immediately thing they can have some satisfaction soon. It’s the same way I work. I try to get return pretty soon.

TPL:In other words you don’t suggest problems which need heavy machinery?

SS:Right, that’s exactly right. Sometime it turns out the students has to learn some kind of machine to do with it, that’s more a course doing it.

TPL:But this is not what you intended?

SS:Right, that’s exactly the point. So in fact I don’t work fields which lead half year study before you can do anything. I tend not do this kind things to students.

TPL:Mathematics has become popular, for example real analysis now the class is bigger advance calculus class is bigger. Over US and other places also, because perhaps related to what you said if you know more mathematics, you are in more powerful position. Have you paid attention to undergraduate education?

SS:I thought about those things, but it’s more what you called pie in the sky. One time I was sort very critical of math education, little completely different point of view in the math educators and mathematicians as far as I know. I think first year, second year courses are terrible in mathematics. Calculus course, I am very poor calculus myself. Most my work was always wrong calculus course, I do very poorly in calculus myself. Most my work was always wrong calculus. I think courses are terrible. It is too narrow. I think it is contrast to some texts in other fields. Economics, you have a great book covers the whole, Paul Samulson’s economics. It is a great book, covers economics. Physics. Physics, you have Feynman’s lectures, covers physics. Mathematics, calculus book is unmotivated and just technical, My idea would be having something course comparable mathematics like Feynman’s lecture, say, you know a kind course covering mathematics, it wouldn’t be just calculus. A year course mathematics where you cover some basic topics through all mathematics in one course, related to each other, to me that could be a great course, I talked about, I said pie in the sky. It would be almost a team of people to be writing a lot, because it will be a big job discussing a lot, so on. Just do all the basic things; linear algebra, number theory Galois theory, may be not so advance, but you know the basic, maybe Hilbert space but in the easy way, not like Hilbert space as a course for very technical things, just inner product spaces.

TPL:No measurable function, for example.

SS:Right, not spend time on measurable functions. Maybe also talking about probability. That’s will be one course covers all these basic things, like Feynman’s lecture in physics, Samuelson’s textbook in economics. That suppose good things to me in mathematics. I think it would be a great thing to have such articles. When you talk about calculus it becomes so unmotivated, so technical, I get angry mind. Why?Because calculus plays a big role in geometry. My vision such courses talk about calculus for all these purposes, optimization problem, geometry and so on. So they do optimization problem for their own writer. Ordinary differential equations, geometric Van del pol equation those kind of things would be basic one year course.

TPL:But it would have to be more than three credits.

SS:Four credits is OK. Let it be done very efficiently, put a lot thought, not a lot of unessential foundations talk about.

TPL:No ，too much ?

SS:Yes, not so much, whatever necessary. See, I think graduate analysis is the same problem. Graduate analysis starts off all these separate topics, general topology all these things. Students never get to real basic things, One book I like a lot, that’s Serge Lang’s analysis. It covers Fourier analysis, spectral theory, Hilbert spaces, calculus on Hilbert spaces, all these things he covers that of course doesn’t go into all the details, general topology, because more directly to the subject. Serge Lang’s analysis course is kind of spirit I like. Then they do some spectral theory, so often it takes three or four graduate courses before you get to basic spectral world. That’s ridiculous, should do it in one semester, first semester. I thought about writing book like that. I do it, I wrote ordinary differential equations with Morris Hirsch. Sometime I thought about doing second course partial differential equations on Hilbert spaces. That would be the spirit, try to get easy things, you know using Hilbert spaces efficiently and one semester mostly linear differential equations. Anyway that’s some of my thoughts in curriculum.

SS:Sometime I get turning away a little bit from those questions.

TPL:No, no this is good, but I was wondering how come it hasn’t been done? Is there intrinsic difficulty or it is just simply that we are not able to put up with the effort?

SS:Well, some effort and resistance, I think it’s partly political. One needs more than just write a book. You have to promote it you have to do a lot work to get sale of it. It takes a lot organization, politics, scientific politics to accomplish. You just cannot sit down write this and expect it to happen. Something like this will change curriculum a lot. You get a lot of resistance, and you know these things take this kind I use word political in broad sense, political not in strict sense, scientific political were how do you get people to do. You know ε −δ ε −δ Serge Lang’s book, analysis. I think it’s great, but I was in curriculum committee in Berkeley, we eventually convinced the committee that we should use the text. Once the text, eventually some other people taught the course never use the text again. So because people are used to it, I understand that, you know you get used to doing things one way, it is not moving ever to teach something completely different. Even teaching it takes a lot effort to accomplish this, that’s why pie in the sky.

TPL:We live in the real society.

SS:Yes, I am in the same way. When I teach a course, somebody else says here is a great idea, for me redo every item about teaching, teach complete course, I probably wouldn’t do it. Some other people would, that’s why we have to be patient. Make an effort, some people would do it. But a lot of people just have other issues working on, other agenda. I don’t blame them. It is not a question blaming people. It’s a question just wait things out. You just don’t change the language teaching. The language thinking probably is usually long.

TPL:Could we change topics a little bit?You have several decades of continuing outstanding research and you have met many people, right? You have met many mathematicians, scientists, do you mind to tell us some story, or some of the people you have met, whatever, some striking things, some interesting things or some of people you can describe.

SS:Story or what? I did this recently. I wrote a paper on Shannon theory, published in American Mathematical Society Bulletin to dedicate Rene Thom. Rene Thom died, I express a lot emulation. I said kind of nice way, how I enjoyed this complications. For example, his rejection 19th century mathematics, complex variables, that’s true. He was provoked. He did say19th century was step backwards for mathematics, he said things like that. We have big conflict over catastrophe theory. He was my good friend support and so on, that’s causes lots of friction. Chris Zeeman too. Strong capacity.

TPL:I read about that. That’s long time ago around 70’s.

SS:Usually I don’t have such conflicts, but occasionally I have some conflicts with other scientists.

TPL:Are there something that was done you feel really very beautiful?

SS:Yes, not exactly I have hero so much, I don’t think I have heroes. I see a lot exciting things some people have done. Sure, I appreciate a lot of Mathematics I see. I just recently this group in Italy cross a preprint on learning theory and there are something very interesting for me for my own development three years ago. Several people developing the paper I have done with Felipe Cucker, they did a new, better way. My other colleague read and rejected the paper. He is the referee. I get upset him. I said, look, this helps subscribe we think what we were doing in a good, complemental way.

TPL:This is your general upbeat feeling, mathematics is doing well?

SS:Not happen every year I forget most time in the past, but this was last year. You know those small level, what happens. Quite a day you learn something new from somebody, it’s still a lot, it happens frequently or not. When I was a graduate student, one graduate student, he was the very best. Chris and I ever wrote a paper when graduate student my life. It’s more common now in computer science to do joint paper, so I think it’s recent. It’s a joint paper. Doing that I started work out some myself, he came alone, started working again, pretty soon gets inside, it’s exciting for me to see.

TPL:Your mathematical life has been very fruitful. You are in Toyota center now in Chicago, just a brief description what you plan to do there, what the institute is about?

SS:It is a computer science institute. It’s more research than teaching I would say, even though they have three students. So I am doing a lot sort helping build up the center for not broadly computer science but more learning theory. Try to make a place for people want come to whole the world. That’s my goal. A world center for some aspects in computer science, mathematics. A lot effort to that.

TPL:Excellent. Thanks again.

• Tai-Ping Liu is a faculty member at the Institute of Mathematics, Academia Sinica.
• Shih-Hsien Yu was a faculty member at the City University of Hong Kong and is a faculty member at Academia Sinica starting July 2021.