A Beautiful Mind (43 page)

Little is known about the existence, uniqueness and smoothness of solutions of the general equations of flow for a viscous, compressible, and heat conducting fluid. These are a non-linear parabolic system of equations. An interest in these questions led us to undertake this work. It became clear that nothing could be done about the continuum description of general fluid flow without the ability to handle non-linear parabolic equations and that this in turn required an
a priori
estimate of continuity.
²⁶

It was Louis Nirenberg, a short, myopic, and sweet-natured young protégé of Courant’s, who handed Nash a major unsolved problem in the then fairly new field of nonlinear theory.
²⁷
Nirenberg, also in his twenties, and already a formidable analyst, found Nash a bit strange. “He’d often seemed to have an internal smile, as if he was thinking of a private joke, as if he was laughing at a private joke that he never [told anyone about].”
²⁸
But he was extremely impressed with the technique Nash had invented for solving his embedding theorem and sensed that Nash might be the man to crack an extremely difficult outstanding problem that had been open since the late 1930s.

He recalled:

I worked in partial differential equations. I also worked in geometry. The problem had to do with certain kinds of inequalities associated with elliptic partial differential equations. The problem had been around in the field for some time and a number of people had worked on it. Someone had obtained such estimates much earlier, in the 1930s in two dimensions. But the problem was open for [almost] thirty years in higher dimensions.
²⁹

Nash began working on the problem almost as soon as Nirenberg suggested it, although he knocked on doors until he was satisfied that the problem was as important as Nirenberg claimed.
³⁰
Lax, who was one of those he consulted, commented recently: “In physics everybody knows the most important problems. They are well defined. Not so in mathematics. People are more introspective. For Nash, though, it had to be important in the opinion of others.”
³¹

Nash started coming to Nirenberg’s office to discuss his progress. But it was weeks before Nirenberg got any real sense that Nash was getting anywhere. “We
would meet often. Nash would say, ’I seem to need such and such an inequality. I think it’s true that…’” Very often, Nash’s speculations were far off the mark. “He was sort of groping. He gave that impression. I wasn’t very confident he was going to get through.”
³²

Nirenberg sent Nash around to talk to Lars Hörmander, a tall, steely Swede who was already one of the top scholars in the field. Precise, careful, and immensely knowledgeable, Hörmander knew Nash by reputation but reacted even more skeptically than Nirenberg. “Nash had learned from Nirenberg the importance of extending the Holder estimates known for second-order elliptic equations with two variables and irregular coefficients to higher dimensions,” Hörmander recalled in 1997.
³³
“He came to see me several times, ’What did I think of such and such an inequality?’ At first, his conjectures were obviously false. [They were] easy to disprove by known facts on constant coefficient operators. He was rather inexperienced in these matters. Nash did things from scratch without using standard techniques. He was always trying to extract problems… [from conversations with others]. He had not the patience to [study them].”

Nash continued to grope, but with more success. “After a couple more times,” said Hörmander, “he’d come up with things that were not so obviously wrong.”
³⁴

By the spring, Nash was able to obtain basic existence, uniqueness, and continuity theorems once again using novel methods of his own invention. He had a theory that difficult problems couldn’t be attacked frontally. He approached the problem in an ingeniously roundabout manner, first transforming the nonlinear equations into linear equations and then attacking these by nonlinear means. “It was a stroke of genius,” said Lax, who followed the progress of Nash’s research closely. “I’ve never seen that done. I’ve always kept it in mind, thinking, maybe it will work in another circumstance.”
³⁵

Nash’s new result got far more immediate attention than his embedding theorem. It convinced Nirenberg, too, that Nash was a genius.
³⁶
Hörmander’s mentor at the University of Lund, Lars Gårding, a world-class specialist in partial differential equations, immediately declared, “You have to be a genius to do that.”
³⁷

Courant made Nash a handsome job offer.
³⁸
Nash’s reaction was a curious one. Cathleen Synge Morawetz recalled a long conversation with Nash, who couldn’t make up his mind whether to accept the offer or to go back to MIT. “He said he opted to go to MIT because of the tax advantage” of living in Massachusetts as opposed to New York.
³⁹

Despite these successes, Nash was to look back on the year as one of cruel disappointment. In late spring, Nash discovered that a then-obscure young Italian, Ennio De Giorgi, had proven his continuity theorem a few months earlier. Paul Garabedian, a Stanford mathematician, was a naval attaché in London. It was an Office of Naval Research sinecure.
⁴⁰
In January 1957, Garabedian took a long car
trip around Europe and looked up young mathematicians. “I saw some oldtimers in Rome,” he recalled. “It was a scene. You’d talk mathematics for half an hour. Then you’d have lunch for three hours. Then a siesta. Then dinner. Nobody mentioned De Giorgi.” But in Naples, someone did, and Garabedian looked De Giorgi up on his way back through Rome. “He was this bedraggled, skinny little starved-looking guy. But I found out he’d written this paper.”

De Giorgi, who died in 1996, came from a very poor family in Lecce in southern Italy.
⁴¹
Later he would become an idol to the younger generation. He had no life outside mathematics, no family of his own or other close relationships, and, even later, literally lived in his office. Despite occupying the most prestigious mathematical chair in Italy, he lived a life of ascetic poverty, completely devoted to his research, teaching, and, as time went on, a growing preoccupation with mysticism that led him to attempt to prove the existence of God through mathematics.

De Giorgi’s paper had been published in the most obscure journal imaginable, the proceedings of a regional academy of sciences. Garabedian proceeded to report De Giorgi’s results in the Office of Naval Research’s European newsletter.

Nash’s own account, written after he had won the Nobel for his work in game theory, conveys the acute disappointment he felt:

I ran into some bad luck since, without my being sufficiently informed on what other people were doing in the area, it happened that I was working in parallel with Ennio De Giorgi of Pisa, Italy. And De Giorgi was first actually to achieve the ascent of the summit (of the figuratively described problem) at least for the particularly interesting case of “elliptic equations.”
⁴²

Nash’s view was perhaps overly subjective. Mathematics is not an intramural sport, and as important as being first is, how one gets to one’s destination is often as important as, if not more important than, the actual target. Nash’s work was almost universally regarded as a major breakthrough. But this was not how Nash saw it. Gian-Carlo Rota, a graduate student at Yale who spent that year at Courant, recalled in 1994: “When Nash learned about De Giorgi he was quite shocked. Some people even thought he cracked up because of that.”
⁴³
When De Giorgi came to Courant that summer and he and Nash met, Lax said later, “It was like Stanley meeting Livingstone.”
⁴⁴

Nash left the Institute for Advanced Study on a fractious note. In early July he apparently had a serious argument with Oppenheimer about quantum theory — serious enough, at any rate, to warrant a lengthy letter of apology from Nash to Oppenheimer written around July 10, 1957: “First, please let me apologize for my manner of speaking when we discussed quantum theory recently. This manner is unjustifiably aggressive.”
⁴⁵
After calling his own behavior unjustified, Nash nonetheless immediately justified it by calling “most physicists (also some mathematicians
who have studied Quantum Theory) … quite too dogmatic in their attitudes,” complaining of their tendency to treat “anyone with any sort of questioning attitude or a belief in ’hidden parameters’... as stupid or at best a quite ignorant person.”

Nash’s letter to Oppenheimer shows that before leaving New York, Nash had begun to think seriously of attempting to address Einstein’s famous critique of Heisenberg’s uncertainty principle:

Now I am making a concentrated study of Heisenberg’s original 1925 paper … This strikes me as a beautiful work and I am amazed at the great difference between expositions of “matrix mechanics,” a difference, which from my viewpoint, seems definitely in favor of the original.
⁴⁶

“I embarked on [a project] to revise quantum theory,” Nash said in his 1996 Madrid lecture. “It was not a priori absurd for a non-physicist. Einstein had criticized the indeterminacy of the quantum mechanics of Heisenberg.”
⁴⁷

He apparently had devoted what little time he spent at the Institute for Advanced Study that year talking with physicists and mathematicians about quantum theory. Whose brains he was picking is not clear: Freeman Dyson, Hans Lewy, and Abraham Pais were in residence at least one of the terms.
⁴⁸
Nash’s letter of apology to Oppenheimer provides the only record of what he was thinking at the time. Nash made his own agenda quite clear. “To me one of the best things about the Heisenberg paper is its restriction to the observable quantities,” he wrote, adding that “I want to find a different and more satisfying under-picture of a non-observable reality.”
⁴⁹

It was this attempt that Nash would blame, decades later in a lecture to psychiatrists, for triggering his mental illness — calling his attempt to resolve the contradictions in quantum theory, on which he embarked in the summer of 1957, “possibly overreaching and psychologically destabilizing.”
⁵⁰

31
The Bomb Factory

What’s the matter with being a loner and innovative? Isn’t that fine? But the
[lone genius] has the same wishes as other people. If he were back in high
school doing science projects, fine. But if he s too isolated and he’s
disappointed in something big, it’s frightening, and fright can precipitate

— P
AUL
H
OWARD,
M
c
L
ean
H
ospital

J
ÜRGEN
M
OSER
had joined the M I T faculty in the fall of 1957 and was living with his wife, Gertrude, and his stepson, Richy, in a tiny rented house to the west of Boston in Needham near Wellesley College. Needham was then more exurb than suburb, still predominantly rural, a lovely place for walking, boating, and stargazing, all of which Moser, a nature lover, was fond. That October and November, Moser would go outside every evening at dusk with eleven-year-old Richy, climb a great dirt mound behind their house, and wait for
Sputnik — a
tiny silvery dot reflecting the sun’s last rays — to pass slowly over Boston.
¹
Having calculated the satellite’s precise orbit, Moser always knew when it would appear on the horizon.

Very often, he would still be thinking of the afternoon’s conversation with Nash. Nash drove out to Needham often. Despite their very different temperaments, Nash and Moser had great respect for each other. Moser, who thought Nash’s implicit function theorem might be generalized and applied to celestial mechanics, was eager to learn more of Nash’s thinking. Nash, in turn, was interested in Moser’s ideas about nonlinear equations. Richard Emery recalled in 1996: “I remember Nash being very much a part of our life. He used to come to the house and talk with Jürgen. They would walk and talk together and spend time in the study. The intensity of it was unimaginable. There could be no interruptions. An interruption was an absolute sin, a violation most serious. It was met with real wrath. When Jürgen and Nash met, it was very intense. I always had to be quiet.”
²

Returning to Cambridge in late summer, Nash and Alicia found an apartment with some difficulty.
³
They each paid half the rent, for they had decided not to pool their funds.
⁴
Alicia got a job as a physics researcher at Technical Operations, one of the small high-tech companies that were springing up along Route 128.
⁵
She also enrolled in a course on quantum theory taught by J. C . Slater.

They quickly settled into the pleasant private and social rituals of a newly married academic couple. Alicia almost never cooked. She would meet Nash on the campus after work, they would eat out with one or more of Nash’s mathematics friends, and often spend the evening at a lecture, concert, or some social gathering.
⁶
Alicia made sure that they were always surrounded by amusing people, sometimes Nash’s old graduate-student friends, including Mattuck and Bricker, sometimes Emma Duchane and whomever Emma happened to be dating, and, increasingly, other young couples like themselves, including the Mosers, the Minskys, Hartley Rogers and his wife, Adrienne, and Gian-Carlo Rota and his wife, Terry.