An Interview with Barry Mazur

After high school I went to MIT with the idea of studying electronic engineering. But the first day at MIT I went to the library -- and was surprised that there were so many books on mathematics: I was surprised to find that math was actually a subject. I had thought of math as a tool, and I had intended to learn it to prepare myself to study electromagnetism. That first day while browsing through the shelves I realized that mathematics and not engineering was really what I wanted to do.

Instead I tried to learn calculus on my own, from this wonderful book that was either called "Calculus Made Simple" or "Calculus Made Easy," I have forgotten which, by Thompson. This text had at least two great virtues. It was very thin, and it proudly proclaimed (on its opening pages) that it was about to "save" Calculus from the obscurity into which it had been cast by the mathematicians. I have no idea what I would think of this book if I saw it now, but at least, for me, then, it conveyed the awe of calculus -- with no rigor at all -- and by golly, its title was correct.

I was also an avid reader of electronics books, and I picked up a certain amount of calculus just by reading them; and finally I went through a text by Kell which taught a version of Calculus directed to applications to aeronautical engineering; Kell's Calculus had a huge number of (rote, for the most part) exercises at the end of each chapter: I obsessively did every one of the problems in that book.

And then there was Warren Ambrose, a professor at MIT, who did research in functional analysis, but when I was at MIT he seemed particularly interested in algebraic topology -- and more specifically was very enthousiastic about the things that were going on in the Cartan seminar in Paris. Ambrose taught a graduate course in algebraic topology which I took (I had much help from Gus Solomon ). I suspect that Ambrose didn't know all that much algebraic topology at the time, but that fact somehow contributed to the greatness of the course he gave. For Ambrose, intuition was primary, and he would hold back on formal definitions long after anyone else would normally have presented them.

For example, he would say "OK, consider the group of chains on a topological space" and then he would write on the board

Only after having written this, would he begin to define the various formal variants of "pieces of space" (i.e., simplex, or singular simplex, etc. -- he also liked cubical chains). There would be an engaging mystery at the beginning of any of his explanations: a mystery that, so to speak, charged up the imaginative faculties, and made one ready, eager, for the eventual formalization. I was inspired by his classes.

And there was Kenkichi Iwasawa. Iwasawa was just the opposite of Ambrose, formal at every level. He would enter the room writing on the blackboard -- no informal interaction at all! Everything was presented absolutely perfectly: clean, clear, with wonderfully thought out notation. His problems were gems. There weren't very many of them, and they would often be thorny -- they would usually take a couple of weeks to figure out -- but they would really open up the subject for anyone who worked them through to the end.

Norman Steenrod was my adviser at first. Steenrod reminded me of Rodin's thinker (not in physique, but rather in the way in which he revelled in slow reflection). He made it clear that quickness was not essential to doing good mathematics; the important problems in mathematics, or at least the ones he valued, required having an appropriate view of the whole, and such overviews are gotten only after considerable experience, and long deliberation. Some of the most instructive moments to me in my first weeks in Princeton came from watching Steenrod, in his course, slowly ruminate (out loud) about the real meaning of "cohomology operation" and, with equal intensity, ponder the signficance of a sign in a formula. I very much liked it.

Steenrod also gave a lecture to incoming graduate students in which he said two things. He said : "In your undergraduate studies the mathematics that you have read has been primarily in textbooks. But now you are ready to read original articles -- articles are living mathematics, and textbooks are dead mathematics. You should read original articles, even if they're harder and not so well written." The other thing he said was quite important to me: he said that there was a natural tempo to "mathematical maturity": certain mathematical ideas take a certain amount of time to digest -- they deserve a certain amount of time -- and trying to rush this process is disrespectful of the ideas and might lead to a less full comprehension of them. So I took Steenrod's classes.

The other graduate students in Princeton at the time were terrific. We gave seminars every night -- on wild topics, whatever occurred to us. The graduate students I spent most time with were Jim Stasheff, John Stallings and Han Sah. We would go through various books together (books and not, in fact, articles, as Steenrod would have us do). For example, we read through Chevalley's books on algebraic functions of one variable, and on algebraic Lie groups. We also took delighted in finding strange "examples" -- the stranger to us, the better. These "examples" we cooked up were mainly in algebraic topology (e.g., CW-spaces equipped with various algebraic structures, given up to homotopy), but also in geometry, and pointset topology. In January, Steenrod suggested that I work on the differential geometry of manifolds with a Lorentz-type metric on their tangent bundles -- a space-time metric; it was a kind of open-ended problem. I thought a bit about this problem but got nowhere with it. At the end of the year I said that I wanted to go to Paris. Steenrod warned me that he thought I would not get anything done in Paris, and was strongly against my going. But I did.