Princeton, 1950–51
STRANGE AS IT MAY NOW SEEM, the dissertation that would one day win Nash a Nobel wasn’t highly regarded enough to assure him an offer from a top academic department. Game theory did not inspire much interest or respect among the mathematical elite, von Neumann’s prestige notwithstanding. Indeed, Nash’s mentors at Carnegie and Princeton were vaguely disappointed in him; they had expected the youngster who had re-proved theorems of Brouwer and Gauss to tackle a really deep problem in an abstract field like topology.1 Even his biggest fan, Tucker, had concluded that while Nash could “hold his own in pure mathematics,” it was not “his real strength.”2
Having successfully sidestepped the threat of the draft, Nash now began working on a paper that he hoped would win him recognition as a pure mathematician.3 The problem concerned geometric objects called manifolds, which were of great interest to mathematicians at that time. Manifolds were a new way of looking at the world, so much so that even defining them sometimes tripped up eminent mathematicians. At Princeton, Salomon Bochner, one of the leading analysts of his day and a fine lecturer, used to walk into his graduate classes, start to give a definition of a manifold, get hopelessly bogged down, and finally give up, saying with an exasperated air, before moving on, “Well, you all know what a manifold is.”4
In one dimension, a manifold may be a straight line, in two dimensions a plane, or the surface of a cube, a balloon, or a doughnut. The defining feature of a manifold is that, from the vantage point of any spot on such an object, the immediate vicinity looks like perfectly regular and normal Euclidean space. Think of yourself shrunk to the size of a pinpoint, sitting on the surface of a doughnut. Look around you, and it seems that you’re sitting on a flat disk. Go down one dimension and sit on a curve, and the stretch nearby looks like a straight line. Should you be perched on a three-dimensional manifold, however esoteric, your immediate neighborhood would look like the interior of a ball. In other words, how the object appears from afar may be quite different from the way it appears to your nearsighted eye.
By 1950, topologists were having a field day with manifolds, redefining every object in sight topologically. The diversity and sheer number of manifolds is such that today, although all two-dimensional objects have been defined topologically, not all three- and four-dimensional objects — of which there is literally an infinite assortment—have been so precisely described. Manifolds turn up in a wide variety of physical problems, including some in cosmology, where they are often very hard to cope with. The notoriously difficult three-body problem proposed by King Oskar II of Sweden and Norway in 1885 for a mathematical competition in which Poincaré took part, which entails predicting the orbits of any three heavenly bodies — such as the sun, moon, and earth — is one in which manifolds figure largely.5
Nash became fascinated with the subject of manifolds at Carnegie.6 But it is likely that his ideas did not crystallize until after he came to Princeton and began having regular conversations with Steenrod. In his Nobel autobiography, Nash says that, right around the time that he got his equilibrium result for n-person games, that is, in the fall of 1949, he also made “a nice discovery relating to manifolds and real algebraic varieties.”7 This is the result that he had considered writing up as a dissertation after von Neumann’s cool reaction to his ideas about equilibrium for games with many players.
The discovery came long before Nash had worked out the laborious steps of the actual proof. Nash always worked backward in his head. He would mull over a problem and, at some point, have a flash of insight, an intuition, a vision of the solution he was seeking. These insights typically came early on, as was the case, for example, with the bargaining problem, sometimes years before he was able, through prolonged effort, to work out a series of logical steps that would lead one to his conclusion. Other great mathematicians — Riemann, Poincaré, Wiener — have also worked in this way.8 One mathematician, describing the way he thought Nash’s mind worked, said: “He was the kind of mathematician for whom the geometric, visual insight was the strongest part of his talent. He would see a mathematical situation as a picture in his mind. Whatever a mathematician does has to be justified by a rigorous proof. But that’s not how the solution presents itself to him. Instead, it’s a bunch of intuitive threads that have to be woven together. And some of the early ones present themselves visually.”9
With Steenrod’s encouragement,10 Nash gave a short talk on his theorem at the International Congress of Mathematicians in Cambridge in September 1950.11 Judging from the published abstract, however, Nash was still missing essential elements of his proof. Nash planned to complete it at Princeton. Unfortunately for Nash, Steenrod was on leave in France.12 Lefschetz, who undoubtedly was pressing Nash to have the paper ready before the annual job market got under way in February, urged Nash to go to Donald Spencer, the visiting professor who had been on Nash’s generals committee and had just been hired away from Stanford, and to use Spencer as a sounding board for completing the paper.13
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As a visiting professor, Spencer occupied a tiny office squeezed between Artin’s huge corner office and an equally grand study belonging to William Feller. Spencer, as Lefschetz wrote to the dean of faculty, was “probably the most attractive mathematician in America at that moment,” as well as “one of the most versatile American born mathematicians.”14 A doctor’s son, Spencer grew up in Colorado and was admitted to Harvard, where he intended to study medicine. Instead, he wound up at MIT studying theoretical aerodynamics and then at Cambridge, England, where he became a student of J. E. Littlewood, Hardy’s great coauthor.15 Spencer did brilliant work in complex analysis, a branch of pure mathematics that has widespread engineering applications.16 He was a much sought-after collaborator, his most celebrated collaboration being with the Japanese mathematician Kunihiko Kodaira, a Fields medalist.17 Spencer himself won the Bôcher Prize.18 Although he primarily worked in highly theoretical fields, he nonetheless had some applied interests, namely hydrodynamics.19
A lively, voluble man, Spencer was “sometimes daunting in his reckless energy.”20 His appetite for difficult problems was boundless, his powers of concentration impressive. He could drink enormous quantities of alcohol — five martinis out of “bird bath” glasses — and still talk circles around other mathematicians.21 A man whose natural exuberance hid a darker tendency toward depression and introspection, Spencer’s appetite for abstraction was accompanied by an extraordinary empathy for colleagues who were in trouble.22
He did not, however, suffer fools gladly. The first draft of Nash’s paper gave Spencer little confidence that the younger mathematician was up to the task he’d set for himself. “I didn’t know what he was going to do, really. But I didn’t think he was going to get anywhere.”23 But for months, Nash showed up at Spencer’s door once or twice a week. Each time he would lecture Spencer on his problem for an hour or two. Nash would stand at the blackboard, writing down equations and expounding his points. Spencer would sit and listen and then shoot holes in Nash’s arguments.
Spencer’s initial skepticism slowly gave way to respect. He was impressed by the calm, professional way that Nash responded to his most outrageous challenges and his fussiest objections. “He wasn’t defensive. He was absorbed in his work. He responded thoughtfully.” He also liked Nash for not being a whiner. Nash never talked about himself, Spencer recalled. “Unlike other students who felt underappreciated,” he said, “Nash never complained.” The more he listened to Nash, moreover, the more Spencer appreciated the sheer originality of the problem. “It was not a problem that somebody gave Nash. People didn’t give Nash problems. He was highly original. Nobody else could have thought of this problem.”
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Many breakthroughs in mathematics come from seeing unsuspected relationships between objects that seem intractable and ones that mathematicians have already got their arms around.
Nash had in mind a very broad category of manifolds, all manifolds that are compact (meaning that they are bounded and do not run off into infinity the way a plane does, but are self-enclosed like a sphere) and smooth (meaning that they have no sharp bends or corners, as there are, for example, on the surface of a cube). His “nice discovery,” essentially, was that these objects were more manageable than they appeared at first glance because they were in fact closely related to a simpler class of objects called real algebraic varieties, something previously unsuspected.
Algebraic varieties are, like manifolds, also geometric objects, but they are objects defined by a locus of points described by one or more algebraic equations. Thus x2 + y2 = 1 represents a circle in the plane, while xy = 1 represents a hyperbola. Nash’s theorem states the following: Given any smooth compact k-dimensional manifold M, there exists a real algebraic variety V in R2k+ 1 and a connected component W of V so that W is a smooth manifold diffeomorphic to M24. In plain English, Nash is asserting that for any manifold it is possible to find an algebraic variety one of whose parts corresponds in some essential way to the original object. To do this, he goes on to say, one has to go to higher dimensions.
Nash’s result was a big surprise, as the mathematicians who nominated Nash for membership in the National Academy of Sciences in 1996 were to write: “It had been assumed that smooth manifolds were much more general objects than varieties.”25 Today, Nash’s result still impresses mathematicians as “beautiful” and “striking” — quite apart from any applicability. “Just to conceive of the theorem was remarkable,” said Michael Artin, professor of mathematics at MIT.26 Artin and Barry Mazur, a mathematician at Harvard, used Nash’s result in a 1965 paper to estimate periodic points of a dynamical system.27
Just as biologists want to find many species distinguished by only minor differences to trace evolutionary patterns, mathematicians seek to fill in the gaps in the continuum between bare topological spaces at one end and very elaborate structures like algebraic varieties at the other. Finding a missing link in this great chain — as Nash did with this result — opened up new avenues for solving problems. “If you wanted to solve a problem in topology, as Mike and I did,” said Mazur recently, “you could climb one rung of the ladder and use techniques from algebraic geometry.”28
What impressed Steenrod and Spencer, and later on, mathematicians of Artin and Mazur’s generation, was Nash’s audacity. First, the notion that every manifold could be described by a polynomial equation is a larger-than-life thought, if only because the immense number and sheer variety of manifolds would seem to make it inherently unlikely that all could be described in so relatively simple a fashion. Second, believing that one could prove such a thing also involves daring, even hubris. The result Nash was aiming for would have seemed “too strong” and therefore improbable and unprovable. Other mathematicians before Nash had spotted relationships between some manifolds and some algebraic varieties, but had treated these correspondences very narrowly, as highly special and unusual cases.29
By early winter, Spencer and Nash were satisfied that the result was solid and that the various parts of the lengthy proof were correct. Although Nash did not get around to submitting a final draft of his paper to the Annals of Mathematics until October 1951,30 Steenrod, in any case, vouched for the results that February, referring to “a piece of research which he has nearly completed, and with which I am well acquainted since he used me as a sounding board.”31 Spencer thought game theory was so boring that he never bothered to ask Nash in the course of that whole year what it was that he had proved in his thesis.32
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Nash’s paper on algebraic manifolds — the only one he was ever truly satisfied with, though it was not his deepest work33 — established Nash as a pure mathematician of the first rank. It did not, however, save him from a blow that fell that winter.
Nash hoped for an offer from the Princeton mathematics department. Although the department’s stated policy was not to hire its own students, it did not, as a matter of practice, pass up ones of exceptional promise. Lefschetz and Tucker very likely dropped hints that an offer was a real possibility. Although most of the faculty other than Tucker neither understood nor displayed any interest in his thesis topic, they were aware that it had been greeted with respect by economists.34
In January, Tucker and Lefschetz made a formal proposal that Nash be offered an assistant professorship.35 Bochner and Steenrod were strongly in favor, although Steenrod, of course, was not present at the discussion. The proposal, however, was doomed to failure. No appointment could be made without unanimous support in a department as small as Princeton’s, and at least three members of the faculty, including Emil Artin, voiced strong opposition. Artin simply did not feel that he could live with Nash, whom he regarded as aggressive, abrasive, and arrogant, in such a small department.36 Artin, who supervised the honors calculus program in which Nash taught for a term, also complained that Nash couldn’t teach or get along with students.37
So the appointment wasn’t offered. It was a bitter moment. The thought must have occurred to Nash that he was being rejected less on the basis of his work than on the basis of his personality. It was an even greater blow because the same faculty made it clear that it hoped that John Milnor, only a junior by this time, would one day become part of the Princeton faculty.38
The job market, while not as bad as in the Depression, was nonetheless rather bleak, the Korean War having cut into university enrollments. Having been turned down by Princeton, Nash knew he would be lucky to get a temporary instructorship in a respectable department.
Both MIT and Chicago, it turns out, were interested in hiring Nash as an instructor.39 Bochner had the ear of William Ted Martin, the new chairman of the MIT mathematics department, and strongly urged Martin to offer Nash an instructorship.40 Bochner urged Martin to ignore the gossip about Nash’s supposedly difficult personality. Tucker, meanwhile, was pushing Chicago to do the same.41 When MIT offered Nash a C. L. E. Moore instructorship, Nash, who liked the idea of living in Cambridge, accepted.42