John Forbes Nash Jr.

• 1. Introduction

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  John Forbes Nash Jr., an American mathematician, once mailed letters to embassies in Washington, D.C., announcing that he was founding a government. He signed them John Nash, Emperor of Antarctica, a title he believed he was in line to inherit.

  Between 1945 and 1996, the same man published only 23 scientific papers. Many of them are now treated as landmarks. His ideas reach across game theory, real algebraic geometry, differential geometry, and the study of partial differential equations, and they are used in economics and other sciences.

  This documentary asks how a mind capable of that work came to write those letters. It asks what the hospitals, the treatments, and the long years of delusion cost him. It also asks how he found his way back, and what the world made of his story once it was told in print and on film.

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• 2. Bluefield to Princeton

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  Bluefield, West Virginia, is where John Forbes Nash Jr. was born on the 13th of June 1928. His father and namesake worked as an electrical engineer for the Appalachian Electric Power Company. His mother, Margaret Virginia Martin Nash, had been a schoolteacher before her marriage. A younger sister, Martha, was born on the 16th of November 1930.

  At nearby Bluefield College, now Bluefield University, Nash took advanced mathematics courses during his final year of high school. His parents arranged this, and they looked for other ways to stretch his schooling. Books from his parents and grandparents had already been a main source of his learning.

  A full George Westinghouse Scholarship took Nash to the Carnegie Institute of Technology, which later became Carnegie Mellon University. He began in chemical engineering, then moved to chemistry. A teacher named John Lighton Synge advised him to try mathematics, and that is where he stayed. In 1948 he graduated with both a bachelor's and a master's degree in mathematics.

  Richard Duffin, Nash's adviser and a former Carnegie Tech professor, supported his application to Princeton University. Duffin's letter stated, "He is a mathematical genius." Harvard University, the University of Chicago, and the University of Michigan had accepted Nash too. Yet Solomon Lefschetz, chairman of Princeton's mathematics department, offered him the John S. Kennedy fellowship. That persuaded Nash that Princeton valued him most, and Princeton was also close to his family in Bluefield.

  At Princeton, Nash began the work on equilibrium theory that would later carry his name.

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• 3. The Graduate Student

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  Just 28 pages long, Nash's doctoral dissertation of 1950 dealt with noncooperative games. Albert W. Tucker supervised it. The thesis contained the definition and properties of what became the Nash equilibrium, a crucial concept for that field. A version appeared in the Annals of Mathematics a year later. In the early 1950s he kept working on related ideas, including the theory of cooperative games and the Nash bargaining solution. Both are now considered central to game theory and its uses in various sciences.

  A different discovery came in 1949, while Nash was still a graduate student, in the field of real algebraic geometry. He announced the theorem in a contributed paper at the International Congress of Mathematicians in 1950, before he had worked out the details of the proof. By October 1951 the work was finished, and he submitted it to the Annals of Mathematics.

  Since the 1930s, mathematicians had known something about closed smooth manifolds. Each one is diffeomorphic to the zero set of some collection of smooth functions on Euclidean space. Nash proved that those functions can always be taken to be polynomials. Many found this surprising, because smooth functions and smooth manifolds are usually far more flexible than polynomials.

  His proof introduced the Nash function and the Nash manifold, both widely studied since. Michael Artin and Barry Mazur later applied the theorem to dynamical systems, combining Nash's polynomial approximation with Bézout's theorem.

  Within a few years, Nash's attention would shift toward differential geometry.

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• 4. The Embedding Theorems

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  Warren Ambrose, a differential geometer, told the postdoctoral fellow at MIT about a famous conjecture. The question was whether every Riemannian manifold is isometric to a submanifold of Euclidean space. Nash was eager for high-profile problems, and he took it on. His results are now called the Nash embedding theorems. Mikhael Gromov has called the second one "one of the main achievements of mathematics of the 20th century."

  The first theorem dates to 1953. It shows that any Riemannian manifold can be isometrically embedded in a Euclidean space by a continuously differentiable mapping. The construction allows the codimension to be very small. So small, in fact, that in many cases a highly differentiable embedding of the same kind cannot exist. Nicolaas Kuiper soon found even smaller codimensions, and the improved result is often called the Nash-Kuiper theorem. Because it lives in the setting of low differentiability, the result sits somewhat outside the mainstream of differential geometry, where high differentiability matters.

  In fluid mechanics, Camillo De Lellis and László Székelyhidi adapted the logic of Nash's proof to construct turbulent solutions of the Euler equations. In the 1970s, Gromov developed Nash's ideas into a general framework called convex integration. Stefan Müller and Vladimír Šverák used it to build counterexamples to generalized forms of Hilbert's nineteenth problem in the calculus of variations.

  Around a year and a half of intensive work went into the second embedding theorem, which concerns smoothly differentiable embeddings. Nash had found the task unexpectedly difficult. The ideas involved are largely separate from those of the first theorem. At the heart of the proof is an implicit function theorem for isometric embeddings. The usual versions fail here because of a technical obstacle called loss of regularity. Nash overcame it by deforming an embedding along an ordinary differential equation, with extra regularity injected continually along the way. This is regarded as a fundamentally novel technique in mathematical analysis.

  The Leroy P. Steele Prize for Seminal Contribution to Research honored his 1956 paper on the problem in 1999. It cited his "most original idea" for the loss of regularity issue as "one of the great achievements in mathematical analysis in this century." Gromov said one must be "a novice in analysis or a genius like Nash" to believe such a thing could be true. He added that what Nash achieved is "as impossible as the story of his life."

  Jürgen Moser extended Nash's ideas to other problems, notably in celestial mechanics, so the resulting implicit function theorem is called the Nash-Moser theorem. Gromov, Richard Hamilton, Lars Hörmander, Jacob Schwartz, and Eduard Zehnder are among those who extended and generalized it. Schwartz later said Nash's ideas were "not just novel, but very mysterious," and that it was hard to "get to the bottom of it."

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• 5. A Race With De Giorgi

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  At the Courant Institute in New York City, Louis Nirenberg told Nash about a well-known conjecture on elliptic partial differential equations. In 1938, Charles Morrey had proved a basic regularity result for functions of two independent variables. Similar results for more variables had proved elusive. After long discussions with Nirenberg and Lars Hörmander, Nash extended Morrey's results to functions of more than two variables. He also extended them to parabolic partial differential equations. As in Morrey's work, the continuity of the solutions is controlled uniformly, without assuming any differentiability of the equation's coefficients. Along the way Nash found the Nash inequality, whose proof he credited to Elias Stein.

  Paul Garabedian, newly back from Italy, soon told Nash that the then-unknown Ennio De Giorgi had reached nearly identical results for elliptic equations. The two men's methods had little to do with each other. Nash's were somewhat more powerful, since they covered parabolic equations as well. A few years later, inspired by De Giorgi, Jürgen Moser found a different approach to the same results. The body of work is now called the De Giorgi-Nash theorem or the De Giorgi-Nash-Moser theory. It is distinct from the Nash-Moser theorem.

  De Giorgi and Moser relied mainly on a judicious choice of test functions in the weak formulation of the equations. Nash instead analyzed the heat kernel. The first approach spread through the work of Olga Ladyzhenskaya, James Serrin, and Neil Trudinger, among others. Eugene Fabes and Daniel Stroock later revisited Nash's approach. They used it to re-derive and extend results first obtained with the techniques of De Giorgi and Moser.

  Almost 60 years had passed since Hilbert's nineteenth problem was posed. It concerned the smoothness of minimizers of many functionals in the calculus of variations. Those minimizers solve elliptic partial differential equations, which put the problem within reach of the De Giorgi-Nash theory. Nash won instant recognition, and Peter Lax called the work a "stroke of genius."

  The Fields Medal of 1958 went to Klaus Roth and René Thom. Nash later speculated that without De Giorgi's simultaneous discovery he would have received it. The committee's reasoning is not fully known, and it was not purely about mathematical merit. Archival research shows that Nash placed third in the vote.

  The following year, 1959, brought the first signs of an illness that would shadow him for decades.

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• 6. The Illness

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  Men in red ties, Nash came to believe, belonged to a "crypto-communist party" that was secretly plotting against him. His illness first showed itself as paranoia, and his wife later described his behavior as erratic. Early in 1959, he gave an American Mathematical Society lecture at Columbia University. He had meant to present a proof of the Riemann hypothesis. The talk was so incoherent that colleagues in the audience knew at once that something was wrong.

  In April 1959, McLean Hospital admitted Nash for one month. Doctors diagnosed schizophrenia, citing paranoid and persecutory delusions, hallucinations, and growing withdrawal from other people. In 1961 he entered the New Jersey State Hospital at Trenton. Over the next nine years he moved in and out of psychiatric hospitals, where he received antipsychotic medications and insulin shock therapy. He later wrote that he took prescribed medication only under pressure.

  Nash dated the start of what he called "mental disturbances" to the early months of 1959, when his wife was pregnant. He described a slide from scientific rationality into delusional thinking. He saw himself as a messenger with a special function, surrounded by supporters, opponents, and hidden schemers. He searched for signs of divine revelation. At the height of the psychosis he sometimes referred to himself in the third person as "Johann von Nassau."

  Nash once said, "I wouldn't have had good scientific ideas if I had thought more normally." He tied his delusions partly to unhappiness, a desire to be recognized, and his characteristic way of thinking.

  Nash said he began hearing voices in 1964, and he later worked at consciously rejecting them. In 1994 he described his hospital stays in New Jersey. Each lasted five to eight months, was always involuntary, and always involved him arguing legally for release. When he had been confined long enough, he would renounce his delusional hypotheses and return to mathematics. He called these interludes "enforced rationality." They still produced respectable research. Heisuke Hironaka named one idea "the Nash blowing-up transformation," and Nash also listed a paper titled "Arc Structure of Singularities."

  By the late 1960s he had relapsed into what he called "influenced" thinking, though his behavior stayed relatively moderate. That let him mostly avoid hospitals and psychiatrists. Slowly he began to reject those lines of thought. The first to go, he said, was politically oriented thinking, which he judged "a hopeless waste of intellectual effort." In 1995 he said that nearly 30 years of illness had kept him from realizing his full potential.

  After 1970, Nash took no medication and was never committed to a hospital again. His recovery was gradual. He spent his days in the Princeton mathematics department, where his eccentricities were tolerated even when his condition was poor. Alicia Lardé, then his former wife, encouraged this, and she credits his recovery to "a quiet life" with social support.

  By the mid-1980s he was well enough to return to academic work.

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• 7. Return and Recognition

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  In the late 1980s, Nash began using email to reach working mathematicians. Some of them realized that he was the John Nash and that his new work had value. Part of that circle then contacted the Bank of Sweden's Nobel award committee. They vouched for his mental health and his ability to receive the award.

  In 1994, Nash shared the Nobel Memorial Prize in Economic Sciences with John Harsanyi and Reinhard Selten. The citation praised their pioneering analysis of equilibria in the theory of non-cooperative games. Nash was honored for game theory work he had done as a Princeton graduate student. Earlier, in 1978, he and Carlton Lemke had received the John von Neumann Theory Prize for outstanding contributions to the theory of games.

  Partial agency was among the topics of Nash's later game theory work, which showed that he still preferred to choose his own path and problems. He also offered hypotheses about mental illness. Nash compared failing to think in an acceptable way, or not fitting a usual social function, to being "on strike" in economic terms. In evolutionary psychology, he argued that apparently nonstandard behaviors or roles might bring benefits.

  Nash also proposed a standard he called "Ideal Money." He criticized Keynesian monetary ideas that let a central bank carry out monetary policy. His alternative was pegged to an "industrial consumption price index" and was more stable than what he called "bad money." He noted that his thinking paralleled that of the economist Friedrich Hayek.

  In 2011, the National Security Agency declassified letters that Nash wrote in the 1950s. In them he proposed a new encryption-decryption machine. The letters show that he had anticipated many ideas of modern cryptography, which rest on computational hardness.

  Honorary degrees came from Carnegie Mellon University in 1999, the University of Naples Federico II in 2003, the University of Antwerp in 2007, and the City University of Hong Kong in 2011. Two West Virginia colleges also honored him, the University of Charleston in 2003 and West Virginia University Tech in 2006. He spoke at events such as the Warwick Economics Summit in 2005. The American Philosophical Society elected him in 2006, and in 2012 he became a fellow of the American Mathematical Society.

  On the 19th of May 2015, King Harald V of Norway presented Nash and Louis Nirenberg with the 2015 Abel Prize at a ceremony in Oslo. The citation praised their "striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to Geometric analysis." Nash had only a few days left to live.

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• 8. Private Life and Legacy

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  In 1951, the Massachusetts Institute of Technology hired Nash as a C. L. E. Moore instructor in mathematics. About a year later he began a relationship with Eleanor Stier, a nurse he met while he was a patient. They had a son, John David Stier. Nash left her when she told him she was pregnant. He was said to have abandoned her because he thought her social status beneath his.

  Santa Monica, California, was the setting for Nash's arrest in 1954, when he was in his 20s. The arrest came in a sting operation targeting gay men, on a charge of indecent exposure. The charges were dropped. Even so, he lost his top-secret security clearance and was fired from the RAND Corporation, where he had been a consultant.

  Alicia Lardé Lopez-Harrison was a naturalized U.S. citizen from El Salvador and a physics graduate of MIT. She and Nash married in February 1957. Nash was an atheist, yet the ceremony took place in an Episcopal church. MIT gave him a tenured position in 1958, and his first signs of mental illness soon followed. He resigned in the spring of 1959. Their son, John Charles Martin Nash, was born a few months later. The boy went unnamed for a year, because Alicia felt Nash should have a say in the choice.

  In 1963, the strain of his illness led Nash and Lardé to divorce. After his final hospital discharge in 1970, he lived in her house as a boarder. The stability seemed to help him. Princeton let him audit classes, and he was eventually allowed to teach again. In the 1990s the two resumed their relationship, and they remarried in 2001. Their son John Charles Martin was diagnosed with schizophrenia in high school and did not graduate. He later earned a PhD in mathematics from Rutgers University.

  In the 1970s, students at Princeton knew Nash as "The Phantom of Fine Hall," after the university's mathematics center. He was a shadowy figure who scribbled arcane equations on blackboards in the middle of the night. Rebecca Goldstein's 1983 novel The Mind-Body Problem, which is set at Princeton, refers to him.

  Sylvia Nasar published her biography A Beautiful Mind in 1998. A film of the same name followed in 2001, directed by Ron Howard, with Russell Crowe as Nash and Jennifer Connelly as Alicia. It won four Academy Awards, including Best Picture. Crowe won the Golden Globe and the BAFTA for best actor, and he was nominated for an Oscar. Nash said the film wrongly implied that he took atypical antipsychotics. He blamed a screenwriter worried that the film might encourage people with mental illness to stop their medication. Before the 2002 Oscars, critics also faulted it for leaving out Eleanor Stier and her son.

  On the 23rd of May 2015, Nash and his wife died in a car crash on the New Jersey Turnpike in Monroe Township, New Jersey. They were returning home from receiving the Abel Prize in Norway. Their taxi had picked them up at Newark Airport. The driver lost control, and the cab struck a guardrail. Neither passenger was wearing a seatbelt, and both were ejected and killed. Nash was survived by two sons, John Charles Martin Nash, who lived with his parents, and his elder child, John Stier.

  Obituaries appeared in scientific and popular media around the world. Alongside its obituary, The New York Times published a second article of Nash's own reflections on his life and achievements, assembled from media and other published sources.

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