Interview with John Milnor Martin Raussen and Christian Skau John W. Milnor is the recipient of the 2011 Abel Prize of the Norwegian Academy of Science and Letters. This in- terview, conducted by Martin Raussen and Christian Skau, took place in Oslo in conjunction with the Abel Prize celebration on May 25, 2011, and originally appeared in the September 2011 issue of the Newsletter of the European Mathematical Society. Raussen & Skau: Professor John Milnor, on behalf of Fenchel, and later Karol Borsuk, had proved the the Norwegian and Danish Mathematical Societies, we following statement: the total curvature of a closed would like to congratulate you for being selected as the curve in space is always at least 2π, with equality Abel Prize Laureate in 2011. only if the curve bounds a convex subset of some Milnor: Thank you very much. plane. Borsuk, a famous Polish topologist, had asked what one could say about total curvature if Student at Princeton University the curve was knotted? I thought about this for a R & S: What kindled your interest in mathematics and few days and came up with a proof that the total when did you discover that you had an extraordinary curvature is always greater than 4π. (I think I did aptitude for mathematics? a poor job explaining the proof in the published Milnor: I can place that quite clearly. The first time paper, but one has to learn how to explain math- that I developed a particular interest in mathematics ematics.) The Hungarian mathematician István was as a freshman at Princeton University. I had been Fáry had produced a similar proof at more or rather socially maladjusted and did not have too many less the same time, but this was still a wonderful friends, but when I came to Princeton, I found myself introduction to mathematics. very much at home in the atmosphere of the math- R & S: That was quite an achievement! When ematics common room. People were chatting about you started your studies at Princeton in 1948, you mathematics, playing games, and one could come by met John Nash, three years your senior, who was at any time and just relax. I found the lectures very a Ph.D. student. John Nash is well known through interesting. I felt more at home there than I ever had the book and movie A Beautiful Mind. Did you have before and I have stayed with mathematics ever since. any interaction with him? And how was it to be a Princeton student? R & S: You were named a Putnam Fellow as one of Milnor: As I said, I spent a great deal of time the top scorers of the Putnam competition in math- in the common room, and so did Nash. He was a ematics in 1949 and 1950. Did you like solving math- very interesting character and full of ideas. He also ematics problems and puzzles? used to wander in the corridors whistling things Milnor: I think I always approached mathematics like Bach, which I had never really heard before—a as interesting problems to be solved, so I certainly strange way to be introduced to classical music! found that congenial. I saw quite a bit of him over those years and I R & S: Your first important paper was accepted al- also became interested in game theory, in which ready in 1949 and published in 1950 in the prestigious he was an important contributor. He was a very journal Annals of Mathematics. You were only 18 interesting person. years of age at the time and this is rather exceptional. R & S: At Princeton, you played Kriegspiel, Go, The title of the paper was “On the Total Curvature and a game called Nash? of Knots”. Could you tell us how you got the idea for Milnor: That is true. Kriegspiel is a game of that paper? chess in which the two players are back-to-back Milnor: I was taking a course in differential geom- and do not see each other’s boards. There is a etry under Albert Tucker. We learned that Werner referee who tells whether the moves are legal or not. It is very easy for the referee to make a Martin Raussen is associate professor of mathematics at mistake, and it often happened that we could Aalborg University, Denmark. His email address is raussen@ not finish because he got confused. In that case math.aau.dk. we said that the referee won the game! It was a Christian Skau is professor of mathematics at the Norwegian marvelous game. University of Science and Technology, Trondheim, Norway. The game of Go was also very popular there. My His email address is [email protected]. first professor, Ralph Fox, was an expert in Go. So DOI: http://dx.doi.org/10.1090/noti803 I learned something of it from him and also from 400 NOTICES OF THE AMS VOLUME 59, NUMBER 3 many other people who played. The game that we famous unsolved problem until called Nash had actually been developed earlier in 1957, when Papakyriakopoulus Denmark by Piet Hein, but Nash invented it inde- developed completely new meth- pendently. This game, also called Hex, is based on ods and managed to give a proof topology. It is very interesting from a mathemati- of “Dehn’s Lemma” and related cal point of view. It is not hard to prove that the theorems. first player will always win if he plays correctly, That was a big step in math- but there is no constructive proof. In fact, when ematics and an example of a case you play, it often happens that the first player in which someone working in isola- does not win. tion made tremendous progress. R & S: You even published some papers on game There are relatively few examples theory with John Nash? of that. Andrew Wiles’s proof of Milnor: We often talked about game theory, Fermat’s last theorem is also an Photo by Knut Falch. but there was only one joint paper. Together with example of someone who had been John Milnor C. Kalish and E. D. Nering, we carried out an ex- working by himself and surprised periment with a group of people playing a many- everyone when he came up with the proof. Another person game. This experiment convinced me that example is Grigori Perelman in Russia who was work- many-person game theory is not just a subject of ing very much by himself and produced a proof of mathematics. It is also about social interactions the Poincaré hypothesis. These are isolated examples. and things far beyond mathematics, so I lost my Usually mathematicians work in a much more social enthusiasm for studying it mathematically. context, communicating ideas to each other. In fact, One paper written on my own described a theo- ideas often travel from country to country very rap- retical model for the game of Go. This was further idly. We are very fortunate that mathematics is usually developed by Olof Hanner and much later by Ber- totally divorced from political situations. Even at the lekamp and Wolfe. (John Conway’s construction of height of the Cold War, we received information from “surreal numbers” is closely related.) the Soviet Union and people in the Soviet Union were eagerly reading papers from outside. Mathematics Knot Theory was much more open than most scientific subjects. R & S: You wrote your Ph.D. thesis under the su- R & S: As a footnote to what you said: Max Dehn pervision of Ralph Fox; the title of the thesis was was a student of David Hilbert and he solved Hilbert’s “Isotopy of links”. Did you get the idea to work on third problem about three-dimensional polyhedra of this topic yourself? And what was the impact of equal volume, showing that you cannot always split this work? them up into congruent polyhedra. No wonder people Milnor: Fox was an expert in knot theory, so I trusted his proof because of his name. learned a great deal about knots and links from Milnor: It’s a cautionary tale because we tend to be- him. There were many people in the department lieve in mathematics that when something is proved, then that were active in this area, although there it stays proved. Cases like Dehn’s Lemma, where a were also other people at the department that false proof was accepted for many years, are very rare. considered it a low-class subject and not very inter- esting. I think it’s strange that, although it wasn’t Manifolds considered a very central subject then, today it’s a R & S: For several years after your Ph.D. your research subject which is very much alive and active. concentrated on the theory of manifolds. Could you As one example, I often saw a quiet Greek explain what a manifold is and why manifolds are gentleman Christos Papakyriakopoulus around the important? common room, but I never got to know him very Milnor: In low dimensions manifolds are things well. I had no idea he was doing important work, that are easily visualized. A curve in space is an ex- but Fox had managed to find money to support ample of a one-dimensional manifold; the surfaces him for many years, while he did research more of a sphere and of a doughnut are examples of two- or less by himself. He finally succeeded in solving dimensional manifolds. But for mathematicians a very important problem in knot theory which, the dimensions one and two are just the beginning; perhaps, was the beginning of a rebirth of the things get more interesting in higher dimensions.