A Beautiful Mind (27 page)

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

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.”

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
x
²
+ Y
²
= 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 R
^{2k + 1}
and a connected component
W
of
V
so that
W
is a smooth manifold diffeomorphic to M.
²⁴
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

Nash’s paper on algebraic manifolds — the only one he was ever truly satisfied with, though it was not his deepest work
³³
— 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

16
MIT

B
Y THE END OF
J
UNE
, Nash was in Boston living in a cheap room on the Boston side of the Charles.
¹
Every morning he walked across the Harvard Bridge, over the yellow-gray river to east Cambridge where MIT s modern, aggressively utilitarian campus lay sprawled between the river and a swath of factories and warehouses. Even before he reached the far side, he could smell the factory smells, including the distinct odors of chocolate and soap mingling together from a Necco candy factory and a P&G detergent plant.
²
As he turned right onto Memorial Drive, he could see Building Two looming ahead, a featureless block of cement painted an “alarming brown,” just to the right of the new library, then under construction.
³
His office was on the third floor next to the stairwell in a corner suite assigned to several instructors, a spare, narrow room with a high ceiling, overlooking the river and the low Boston skyline beyond.
⁴

In 1951, before
Sputnik
and Vietnam, MIT was not exactly an intellectual backwater, but it was nothing like what it is today. The Lincoln Laboratory was famous for its wartime research, but its future academic superstars were still relatively unknown youngsters, and powerhouse departments for which it has since become known — economics, linguistics, computer science, mathematics — were either infants or gleams in some academic’s eye. It was, in spirit and in fact, still very much the nation’s leading engineering school, not a great research university.
⁵

An environment more antithetical to the hothouse atmosphere of Princeton is hard to imagine. MIT’s large scale and modern contours made it feel like the behemoth state universities of the Midwest. The military, as well as industry, loomed awfully large, so large that MIT’s armed, plainclothes campus security force existed solely for the purpose of guarding the half-dozen “classified” sites scattered around the campus and preventing those without proper security clearances and identification from wandering in. ROTC and courses in military science were required of all MIT’s two-thousand-plus undergraduate men.
⁶
The academic departments like mathematics and economics existed pretty much to cater to the engineering student — in Paul Samuelson’s words, “a pretty crude animal.”
⁷
All counted as “service departments,” gas stations where engineers pulled up to get their tanks filled with obligatory doses of fairly elementary mathematics, physics, and chemistry.
⁸
Economics, for example, had no graduate program at all until the war.
⁹
Physics had no Nobel Laureates on its faculty at the time.
¹⁰
Teaching loads were heavy — sixteen hours a week was not uncommon for senior faculty — and
were weighted toward large introductory courses like calculus, statistics, and linear algebra.
¹¹
Its faculty were younger, less well known, and less credentialed than Harvard’s, Yale’s, or Princeton’s.