The Berkeley Science Review: Articles

Faculty Portrait: Stephen Smale
Professor Emeritus of Mathematics (view PDF)
by Meredith Carpenter

Stephen Smale was born in Flint, Michigan, in 1930. He received his PhD from the University of Michigan in 1957, and soon after began teaching at the University of Chicago. In 1961, Smale published his now famous proof of the generalized Poincaré conjecture for seven dimensions and above, later extended to five dimensions and above. This mathematical problem lies in the branch of pure mathematics known as topology, the study of the properties of a space that do not change under continuous deformations (such as stretching and bending, but not tearing). For example, from the point of view of a topologist, a coffee cup, but not a ball, is the same as a donut—one can be turned into another because they both have exactly one hole. In 1904, Poincaré conjectured that any three-dimensional space (“closed 3-manifold”) in which every loop can be continuously contracted is just a three-dimensional sphere. Only in 2002 was the original conjecture completely solved by the reclusive Russian mathematician Grigori Perelman.

Smale’s proof, along with his earlier proof of sphere eversion (which involves turning a sphere inside out without breaking or creasing it), were the main contributions for which he won the 1966 Fields Medal, widely viewed as the Nobel Prize of mathematics. The Poincaré proof also nearly cost Smale his National Science Foundation travel funds when he joked that his best work was done “on the beaches of Rio.”

Smale was on the faculty of UC Berkeley’s mathematics department from 1964 until his retirement in 1995. During this time, he made advances in the fields of dynamical systems, mathematical economics, and the mathematical theory of computer science. He also became well-known for his political leanings: He participated in anti-war demonstrations at Cal in the 1960s, including co-organizing the 1965 “Vietnam Day” teach-in with activist Jerry Rubin, and the House Un-American Activities Committee almost subpoenaed him due to his involvement with the Communist Party.

Smale was recently named the 2007 winner of the Wolf Foundation Prize in Mathematics for “his groundbreaking contributions that have played a fundamental role in shaping differential topology, dynamical systems, mathematical economics and other subjects in mathematics.

Smale is a member of the National Academy of Sciences, a fellow of the American Academy of Arts and Sciences, and a winner of the 1996 National Medal of Science. He currently holds a half-time appointment with the Toyota Technological Institute at the University of Chicago. The Berkeley Science Review sat down with Smale in February 2007 to discuss mathematics, life as an Emeritus professor, and the Smale collection of minerals.

BSR: What drew you to the study of mathematics?

SS: I liked mathematics, but I was a major in physics in college. At some point I was doing badly in physics—I actually failed a course in nuclear physics—and I had enough math courses that it was reasonable to switch. I switched my senior year to mathematics.

BSR: How has the field of mathematics changed since you began your career?

What I've seen over all these years is that the motivation and inspiration for problems in mathematics are coming in different ways than they have in the past. It used to be physics that was the main source of inspiration for mathematical problems, and now it's much more engineering, computers, biology, and those kinds of fields.

BSR: Your most well-known work is your proof of the Poincaré conjecture in 5 dimensions and higher. Why was this such a long-standing and difficult problem in mathematics?

Probably because things weren't very well-developed when Poincaré made these conjectures. The foundational aspects of topology were not very strong for decades. It was not even clear what the definition of a so-called "manifold," or a surface, was. Those things were developed over the first half of the last century when some of the algebraic techniques for the study of topology were also developed.

BSR: Have there been applications for the Poincaré proof in other areas?

I guess so, but I don't see theoretical mathematics as simply a matter of applying it. Theoretical physics, too, as far as that goes, is not exactly something you want to apply. For example, Newton's laws—one doesn't think of applying them. Though they paved the way for engineering, it wasn't exactly the application of Newton's laws, it was the understanding that Newton's laws gave to engineering.... In the same way, for my kind of mathematics, I don't think of it as "applied," but it helps us to understand the world better, and thus it helps make a contribution to the other sciences.

BSR: Was there a sudden flash of insight when you figured out the proof?

I don't think so, no. There were a lot of ups and downs—sometimes I thought I had it, but it wasn't quite right, so I did some more work and had to fill in some more areas.... At some point you begin to get more and more confident that you have a proof.

BSR: What was your reaction to Perelman's solution to the problem?

It was really good to see that happen. And by now the evidence is pretty high that he's correct.

BSR: In 1998 you assembled a list of 18 problems in mathematics to be solved in the 21st century. Why do you think it takes so long, upwards of 100 years in the case of the Poincaré conjecture, to find answers to some math problems?

Well, a really hard problem, in some sense, is never solved—it just keeps reinventing itself in different terms. Some of the problems I gave probably won't be solved for a while, because one has to understand much more of the basic material, I call it the foundational material, which paves the way to...an environment where the solution will come. But the problems I gave provide focus to the development of mathematics—that's some of the main purpose to a hard problem. But also I'm a person who does not believe necessarily that mathematics comes from solving old problems. It comes even more by the understanding of some new subject, mathematics or outside of mathematics. So understanding does not necessarily tie into solving an old problem.

BSR: What has your daily routine been like since your retirement from Cal in 1995? Do you still actively work on mathematical problems?

I probably spend more time on research since I retired since I don't have teaching and I don't have so many committees. My main job is at a small institution, TTI [The Toyota Technological Institute] in Chicago, where I go during the week. Most of my time there is in research or in developing a research atmosphere: inviting people to Chicago, giving talks, working to help give an atmosphere of research.

BSR: What topics do you study in your current research?

I work with a neuroscientist, Tommy Poggio, on trying to find some kind of mathematical model for human vision in the visual cortex. Again, it'll be very mathematical, so it's not something one would apply, but it will have a more simple mathematical structure and give some kind of understanding of why babies can recognize an elephant pretty fast after just being told once or twice that it's an elephant....

I've also worked a lot on flocking with my colleague [Felipe Cucker] from Hong Kong...on understanding how when you surprise a bunch of birds they go up into the sky and they all pick the same direction and follow the same route. We did get some kind of mathematical understanding of how that can be possible, even though the birds can't see a leader or don't follow a leader. But just by looking at nearby birds they can form a flock all going in the same direction. It's called emergence—I also work on emergence in the origins of language, again this is a very mathematical, abstract, but universal way of thinking about these things. In flocks, the birds all have a common velocity, and in language, you start from different perspectives on the emotions and meanings of words, and converge to a common simple language.

BSR: You've been collecting minerals for over 30 years, and you recently published a book of your collection. Why did you start collecting minerals? Did it have anything to do with the mathematics of crystals?

No, I just enjoyed doing it. I also took about 30 percent of the pictures in the book and wrote all of the text. [The book is available from www.lithographie.org]

Meredith Carpenter is a graduate student in molecular and cell biology.

Want to know more?
Professor Smale's TTI website contains links to his recent papers. www.tti-c.org/smale.html

Sphere eversion movie: tinyurl.com/2k84dl

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