Early in the film A Beautiful Mind, Russell Crowe, playing the brilliant young mathematician John Forbes Nash, strides into a classroom at the Massachusetts Institute of Technology to teach his first undergraduate class in vector calculus. Nash makes no effort to hide his resentment of the students and of his teaching duties, hurling the assigned textbook into a wastebasket. Then he writes a series of equations on the blackboard and announces that the rest of the course will be devoted solely to solving the problem they represent—a task, he says, that will take some of them “all your natural lives.”

It’s a pivotal moment. The student who later rises to the challenge—unsuccessfully —is Nash’s future wife Alicia (played by Jennifer Connelly), who will nurse him through three decades of mental illness and share his triumph when he wins the Nobel Prize. Like many key scenes in the film, though, the one that launches their journey together mixes fact and fiction. Though Alicia Larde did take John Nash’s advanced calculus class, he never threw out such a challenge.

So when director Ron Howard needed a mathematical problem for the scene, he could not just pluck one out of someone’s 50-year-old class notes. Instead, he asked Dave Bayer to make it up—to invent the math that Nash would have written if such a scene had occurred. Some mathematicians or math historians might have balked, but for Bayer—an algebraic geometer at Barnard College in New York City, who was moonlighting as the film’s mathematical consultant—it was all in a day’s work. “For me, movies are dream sequences,” Bayer says. “But even the wildest dream sequences are anchored in reality.”

Bayer’s task was to forge the anchors. It is a job, Bayer says, that Howard and his team took very seriously: “Audiences can tell when the mathematics is real, and they want it to be real.”

Bayer came to the film by a circuitous route. In 2000, he had written a review of the Broadway play Proof for Notices of the American Mathematical Society. As a movie and theater aficionado, he wanted to draw his colleagues’ attention to a play that treated mathematics seriously. The review found its way to Howard, who liked what he read, interviewed Bayer, and hired him on the spot. Once Bayer was on board, Howard outlined his main concerns about the role of mathematics in the film: How could such an intensely internal subject be captured visually? Could mathematics show Nash’s descent into mental illness and his slow emergence?

Answering those questions took months of intense effort. “An amazing amount of work goes into every second of film,” says Bayer. His work included writing every one of the countless formulas and computations that cover blackboards and windows throughout the film, apart from a few that Crowe wrote on camera. Bayer also consulted on set design and props.

The classroom scene was the pièce de résistance, Bayer says. He approached it as if he were the actor playing Nash, by putting himself into the character’s shoes. “This is someone who really doesn’t want to teach the mundane details, who will home in on what’s really interesting,” he says.

The problem Bayer finally chose (see photo) was a more complicated version of a classical physics problem: determining whether a static electric field (the F in lines 1 and 2) necessarily has a potential function (indicated by g). If the “electric field” is allowed to be infinite or simply nonexistent at certain points (collectively indicated by X), the question becomes physically unrealistic but mathematically very rich. The answer depends not only on the geometry of the set X but also on one’s assumptions about the field F—as the fictional Nash explains to Alicia rather brusquely when she offers her stab at a solution.

As an unexpected bonus, Bayer wound up on camera himself. Near the end of the movie, he appears as one of Nash’s fellow professors who approach him in the Princeton faculty club and lay their pens on his table as tribute. The “pen ceremony” scene is fiction, but it is one of the most moving scenes in the movie and, as Bayer says, it is a beautiful example of the way a good director creates emotional truth.

Although some critics grumble that A Beautiful Mind exaggerates the competitive atmosphere of postwar Princeton and leaves out important parts of Nash’s life and work, the mathematics in the film has come through peer review with flying colors.

And the best-informed critic of all seems to be satisfied. John Nash, who has seen A Beautiful Mind several times, wrote to Bayer that he appreciated the “bona fide sophistication” of the math in the movie—although he added that in the film’s portrayal of his later work, the fictional Nash seems to know some things that “the real Nash (me)” never did.

—Dana Mackenzie ’79

Adapted from the Feb. 1 edition of Science with permission. Dana Mackenzie, a writer in Santa Cruz, Calif., took Professor David Rosen’s Complex Analysis class with Dave Bayer in fall 1976. “That class had the most awesome group of students of any that I took at Swarthmore,” he said.