In Perspective: John Nash & the 2015 Abel Prize in Mathematics

The fatal crash that claimed the lives of the mathematician John Forbes Nash, Jr., and his wife, Alicia, cast a long shadow over what had been, until then, a grand celebration of the careers and accomplishments of two giants of 20th century mathematics.

On May 19, just four days before the crash, Nash, who spent his career at Princeton and MIT, and his fellow celebrant, Louis Nirenberg of the Courant Institute of Mathematical Sciences at New York University, received the Abel Prize in mathematics. Nash was undoubtedly one of the best-known mathematicians of his generation—the subject of the 2001 film, A Beautiful Mind, not to mention the book by Sylvia Nasar that gave its title to the film. Nirenberg, unknown to the public, is highly respected among mathematicians not only for his numerous contributions to theory, but also for guiding dozens of doctoral students to useful research topics.

The Abel Prize is Norway’s answer to the Swedish Academy’s Nobel prizes—perhaps best known among mathematicians for the glaring omission of mathematics. (Perhaps not so incidentally, the festivities surrounding the awards began the day after Norway’s national holiday, which celebrates the beginnings of its independence from Sweden in 1814.)

And compared with its Swedish counterpart, the Abel award ceremony was surprisingly unpretentious. After a mere half hour devoted to music, speeches and a film about the laureates, King Harald V of Norway, wearing dark gray business attire, strode across the stage and handed each laureate his award. The king said not a word. Each laureate thanked the relevant Norwegians; the musicians performed once more; and the audience stood in silence while the king and his retinue departed. End of ceremony.

But the event with royalty was only the centerpiece of a week-long math festival proclaimed with colorful banners in the streets of Oslo, Norway’s capital city, and starring the two mathematicians. And in spite of their advanced ages—Nash was a frail-looking 86, and Nirenberg is a vigorous 91-year old, albeit confined to a wheelchair—both men met their social obligations to their hosts with astonishingly good humor.

Those obligations went far beyond simply accepting a block of glass from the king. The day before the award ceremony, they placed a ceremonial wreath on a monument in the palace park to Nils Henrik Abel, the namesake of the prize and one of Norway’s most distinguished mathematicians. The day after the ceremony, each one gave a public lecture at the University of Oslo. They also navigated a punishing schedule of official lunches and dinners, public and private interviews, attendance at other people’s public lectures, and travel within the country to meet with and inspire students. No doubt the honor bestowed by the Abel Prize, not to mention the share of $800,000 that goes with it, were enticement enough for both men to endure international travel. But after a week of this, both were surely relieved to be heading home.

The details of the accident are well known. The Nashes’ flight from Norway arrived at Newark Airport more than an hour ahead of schedule. They decided, reasonably enough, not to wait for the car service they had previously arranged. Instead, they hailed a taxi for the ride to Princeton Junction, via Exit 8A of the New Jersey Turnpike. Not far from the exit, the driver lost control on a lane change and slammed into a wall. Both Nashes were thrown from the vehicle. Much has been made of the fact that neither Nash was wearing a seatbelt, but of course no one knows whether two elderly people would have survived the forces without severely crippling injuries even if they had been secured to the back seat. The driver was injured, though he has survived. He had been operating a cab for no more than two weeks.

But the Nashes’ tragedy has unfortunately refracted all the light away from the work that deserved to be recognized. The Abel Prize committee cited both mathematicians for their “striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis.”

As John Rognes, the chair of the committee, noted in remarks at the awards ceremony, many phenomena in nature, as well as in economics, cannot be described by standard calculus—the kind taught to advanced high school students or freshmen in college. To describe the motions of falling bodies, from apples to planets, Newton appealed to our intuitions that all these phenomena happen smoothly as time passes. That meant it was possible to define (and calculate with) the instantaneous velocity of each motion.

More generally, instantaneous rates of change became the key in describing virtually every concept of classical (Newtonian) mechanics: force, momentum, energy, and the like. Ordinary differential calculus provides the technical tools for deriving instantaneous rates of change for a wide range of circumstances—so long as those changes evolve smoothly with time.

But what about the instantaneous rate of change of a pollen grain undergoing Brownian motion in water, the turbulence of fluid flow, or the zigzag prices of a stock? Since those changes are anything but smooth, what sense does it make to describe them by instaneous rates?
In mathematical terms, these many small changes lead to infinite sums of algebraic expressions, from which, as Abel himself wrote, “You can get whatever result you want when you use them”—an untenable situation.

To address the problem, Rognes went on, mathematicians “introduced generalized notions of rates of change and weakened notions of solutions” that describe how “jerky,” discontinuous phenomena evolve with time. When are the generalized rates of change actual, the weakened solutions real? Nash and Nirenberg answer those questions with estimates, or inequalities, that provide the guidance needed to force the infinite sums into giving unambiguous results.

This work “has greatly widened the range of scientific phenomena that can be meaningfully modelled by mathematical methods.”

Peter Brown was the US science journalist selected by the World Federation of Science Journalists to observe and report on the proceedings in Oslo. He is the former editor in chief of The Sciences and Natural History magazines.

Here’s what I wrote for Science (ScienceInsider website):

As Sylvia Nasar describes him in her biography that gave its title to the movie, Nash was a brash young graduate student at Princeton who made little attempt to hide his contempt for others whose gifts were not as extravagant as his own. His utter confidence in his own abilities was no less irritating for being well-grounded. “I’m a genius,” he proclaimed on more than one occasion.

At one public appearance last week, Nash was still referring to himself matter-of-factly as “a genius”—though at least he qualified it: “It’s a popular word,” he said, “and I’m not quite sure what it means.”

“It’s correct that [as a schoolboy] I didn’t shine in math,” he told the writer and broadcaster Vivienne Parry, who interviewed him at a reception following the awards ceremony. “But that’s sort of like saying that Einstein didn’t shine in school in Germany.”

Nash still showed flashes of humor whenever he sensed a bubble of hot air rising nearby, along with a wicked desire to prick it. Asked about whether mathematics is more art than science, Nash replied with a story about the mid-20th-century mathematician Emil Artin. “He liked to say it was an art,” Nash deadpanned. “Maybe it was because his name was Artin.”

“And how do you know,” Parry asked, “when you’re in the midst of a problem, that you’re on the right track?” “Well, you don’t know,” was Nash’s reply, to much appreciative laughter.

But perhaps his most revealing comments to Parry were his views about his own mental illness—views that recall the mid-20th-century theories of Thomas Szasz.

“In many cases,” Nash insisted, “[mental health] has a voluntary element. When I was mentally disturbed, I went on strike. I wasn’t available to do my regular work. At MIT [where he was a member of the mathematics faculty in the 1950s], I was supposed to have functions at various levels: administrator, computer research, teaching and so on. I just didn’t want to continue all the work.

“When people are well, they’re behaving as we desire. When they’re unwell, they’re not doing their work—maybe any work. There are some subtleties about mental health, sanity, and insanity, and how you look at it.”

In other words, however bizarre Nash appeared to others during his illness, from the “inside,” as he reconstructed it, he was in control.

And looking back on the events of the past few days, it’s hard to imagine, had Nash been able to control them, too, how he could have planned a more dramatic exit.

Peter Brown,

Former editor in chief of The Sciences and Natural History magazines. US science journalist selected by the World Federation of Science Journalists to observe and report on the proceedings in Oslo.