john nash phd thesis

John F. Nash, Jr.

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John Nash Jr., a legendary fixture of Princeton University’s Department of Mathematics renowned for his breakthrough work in mathematics and game theory as well as for his struggle with mental illness, died with his wife, Alicia, in an automobile accident May 23 in Monroe Township, New Jersey. He was 86, she was 82.

During the nearly 70 years that Nash was associated with the University, he was an ingenious doctoral student; a specter in Princeton’s Fine Hall whose brilliant academic career had been curtailed by his struggle with schizophrenia; then, finally, a quiet, courteous elder statesman of mathematics who still came to work every day and in the past 20 years had begun receiving the recognition many felt he long deserved. He had held the position of senior research mathematician at Princeton since 1995.

Nash was a private person who also had a strikingly public profile, especially for a mathematician. His life was dramatized in the 2001 film “A Beautiful Mind” in which he and Alicia Nash were portrayed by actors Russell Crowe and Jennifer Connelly. The film centered on his influential work in game theory, which was the subject of his 1950 Princeton doctoral thesis and the work for which he received the 1994 Nobel Prize in economics.

At heart, however, Nash was a devoted mathematician whose ability to see old problems from a new perspective resulted in some of his most astounding and influential work, friends and colleagues said.

At the time of their deaths, the Nashes were returning home from Oslo, Norway, where John had received the 2015 Abel Prize from the Norwegian Academy of Science and Letters, one of the most prestigious honors in mathematics. The prize recognized his seminal work in partial differential equations, which are used to describe the basic laws of scientific phenomena. For his fellow mathematicians, the Abel Prize was a long-overdue acknowledgment of his contributions to mathematics.

For Nash to receive his field’s highest honor only days before his death marked a final turn of the cycle of astounding achievement and jarring tragedy that seemed to characterize his life. “It was a tragic end to a very tragic life. Tragic, but at the same time a meaningful life,” said Sergiu Klainerman, Princeton’s Eugene Higgins Professor of Mathematics, who was close to John and Alicia Nash, and whose own work focuses on partial differential equation analysis.

“We all miss him,” Klainerman said. “It was not just the legend behind him. He was a very, very nice person to have around. He was very kind, very thoughtful, very considerate and humble. All that contributed to his legacy in the department. The fact that he was always present in the department, I think that by itself was very moving. It’s an example that stimulated people, especially students. He was an inspiring figure to have around, just being there and showing his dedication to mathematics.”

Princeton President Christopher L. Eisgruber said Sunday that the University community was “stunned and saddened by news of the untimely passing of John Nash and his wife and great champion, Alicia.”

“Both of them were very special members of the Princeton University community,” Eisgruber said. “John’s remarkable achievements inspired generations of mathematicians, economists and scientists who were influenced by his brilliant, groundbreaking work in game theory, and the story of his life with Alicia moved millions of readers and moviegoers who marveled at their courage in the face of daunting challenges.”

Although Nash did not teach or formally take on students, his continuous presence in the department over the past several decades, coupled with the almost epic triumphs and trials of his life, earned him respect and admiration, said David Gabai, the Hughes-Rogers Professor of Mathematics and department chair.

“John Nash, with his long history of achievements and his incredible battle with mental health problems, was hugely inspirational,” Gabai said. “It’s a huge loss not to have him around anymore.”

Gabai said the Nashes regularly attended department events such as receptions, special teas, and special dinners, and they also were very supportive of undergraduate education and regularly attended undergraduate events. Gabai, who was with the couple in Norway when John received the Abel Prize, likened their deaths to the department losing two family members.

Even in the 1970s when Nash, still struggling with mental illness, was an elusive presence known as the “Phantom of Fine Hall,” his reputation for bravely original thinking motivated aspiring mathematicians, said Gabai, who was a Princeton graduate student at the time. Nash’s creativity helped preserve the department’s emphasis on risk-taking and exploration, he said.

“In those days, he was very present, but rarely said anything and just wandered benignly through Fine Hall. Nevertheless, we all knew that the mathematics he did was really spectacular,” Gabai said. “It went beyond proving great results. He had a profound originality as if he somehow had insights into developing problems that no one had even thought about.

“I think he prided himself that he had his way of thinking about things,” Gabai continued. “He was such an extraordinary exemplar of the things that this department strives for. Beyond great originality, he demonstrated tremendous tenacity, courage and fearlessness.”

Since winning the Nobel Prize , Nash had entered a long period of renewed activity and confidence — which coincided with Nash’s greater control of his mental state — that allowed him to again put his creativity to work, Klainerman said. He met Nash upon joining the Princeton faculty in 1987, but his doctoral thesis had made use of a revolutionary method introduced by Nash in connection to the Nash embedding theorems, which the Norwegian Academy described as “among the most original results in geometric analysis of the twentieth century.”

“When he got the Nobel Prize, there was this incredible transformation,” Klainerman said. “Prior to that we didn’t realize he was becoming normal again. It was a very slow process. But after the prize he was like a different person. He was much more confident in himself.”

During their frequent talks in recent years, Nash would offer unique perspectives on numerous topics spanning mathematics and current events, Klainerman said. “Even though his mind wasn’t functioning as it did in his youth, you could tell that he had an interesting point of view on everything. He was always looking for a different angle than everybody else. He always had something interesting to say.”

Nash’s quick and distinctive mind still shone in his later years, said Michail Rassias, a visiting postdoctoral research associate in mathematics at Princeton who was working with Nash on the upcoming book, “Open Problems in Mathematics.” He and Nash had just finished the preface of their book before Nash left for Oslo. They agreed upon a quote from Albert Einstein that resonated with Nash (although Nash pointed out that Einstein was a physicist, not a mathematician, Rassias said): “Learn from yesterday, live for today, hope for tomorrow. The important thing is not to stop questioning.”

“Even at 86, his mind was still open,” Rassias said. “He still wanted to have new ideas. Of course, he couldn’t work like when he was 20, but he still had this spark, the soul of a young mathematician. The fact that he moved slowly and talked with a quiet voice had nothing to do with the enthusiasm with which he did mathematics. It was very inspirational.”

Sixty years younger than Nash, Rassias said his work with Nash began with a conversation in the Fine Hall commons room in September.

“I could tell there was mathematical chemistry between us and that led to this intense collaboration. He was very simple, very open to discussing ideas with new people if you said something that attracted his interest,” Rassias said. “Nash gave this impression that he was distant, but when you actually had the opportunity to talk to him he was not like that. He tended to walk alone, but if you got the courage to talk to him it would be very natural for him to talk to you.”

Rassias has been inspired by the enthusiasm and willingness with which a person of Nash’s stature dedicated months of his time to working with a young mathematician. It was an example Rassias hopes to emulate during his own career.

“Remembering what John Nash did for me, I will definitely try to give all my heart and soul to younger people in all steps of their careers,” Rassias said. “I also will try to keep my mind and enthusiasm for math alive to the end. That is something I will try to achieve like him.”

Born in Bluefield, West Virginia, in 1928, Nash received his doctorate in mathematics from Princeton in 1950 and his graduate and bachelor’s degrees from Carnegie Institute of Technology (now Carnegie Mellon University) in 1948.

His honors included the American Mathematical Society’s 1999 Leroy P. Steele Prize for Seminal Contribution to Research and the 1978 John von Neumann Theory Prize. Nash held membership in the National Academy of Sciences and in 2012 was an inaugural fellow of the American Mathematical Society.

Nash is survived by his sister, Martha Nash Legg, and sons John David Stier and John Charles Martin Nash. He had his younger son, John Nash, with Alicia shortly after their marriage in 1957, which ended in divorce in 1963. They remarried in 2001.

Despite their divorce, Alicia, who was born in El Salvador in 1933, endured the peaks and troughs of Nash’s life alongside him, Klainerman said. Their deaths at the same time after such a long life together of highs and lows seemed literary in its tragedy and romance, he said.

“They were a wonderful couple,” Klainerman said. “You could see that she cared very much about him, and she was protective of him. You could see that she cared a lot about his image and the way he felt. I felt it was very moving.

“Coming home from Oslo, he must have been extremely happy, and she must have been extremely happy for him,” he continued. “They went for the apotheosis of his career, and died in this terrible way on the way back. But they were together.”

-By Morgan Kelley, Princeton University Office of Communications

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Read John Nash’s Super Short PhD Thesis with 26 Pages & 2 Citations: The Beauty of Inventing a Field

in Math | June 1st, 2015 1 Comment

Last week John Nash , the Nobel Prize-winning mathematician, and subject of the blockbuster film A Beautiful Mind , passed away at the age of 86. He died in a taxi cab accident in New Jersey.

Days later, Cliff Pickover highlighted a curious factoid: When Nash wrote his Ph.D. thesis in 1950, “Non Cooperative Games” at Princeton University, the dissertation (you can read it online here) was brief. It ran only 26 pages. And more particularly, it was light on citations. Nash’s diss cited two texts: One was written by John von Neumann & Oskar Morgenstern, whose book, Theory of Games and Economic Behavior (1944), essentially created game theory and revolutionized the field of economics; the other cited text, “Equilibrium Points in n-Person Games,” was an article written by Nash himself. And it laid the foundation for his dissertation, another seminal work in the development of game theory, for which Nash won the Nobel Prize in Economic Sciences in 1994 .

The reward of inventing a new field, I guess, is having a slim bibliography.

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John Nash , in full John Forbes Nash, Jr. , (born June 13, 1928, Bluefield , West Virginia , U.S.—died May 23, 2015, near Monroe Township, New Jersey), American mathematician who was awarded the 1994 Nobel Prize for Economics for his landmark work, first begun in the 1950s, on the mathematics of game theory . He shared the prize with John C. Harsanyi and Reinhard Selten . In 2015 Nash won (with Louis Nirenberg ) the Abel Prize for his contributions to the study of partial differential equations .

Nash enrolled in chemical engineering at the Carnegie Institute of Technology (later Carnegie Mellon University ) in Pittsburgh before he switched to chemistry and then to mathematics, in which he finally received both bachelor’s and master’s degrees in 1948. Two years later, at age 22, he completed a doctorate at Princeton University . In 1951 he joined the faculty of the Massachusetts Institute of Technology (MIT), where he pursued research into partial differential equations. He resigned in the late 1950s after bouts of mental illness . He then began an informal association with Princeton, where he became a senior research mathematician in 1995.

While he was still in graduate school, Nash published (April 1950) his first paper, “The Bargaining Problem,” in the journal Econometrica . He expanded on his mathematical model for bargaining in his influential doctoral thesis, “ Non-Cooperative Games,” which appeared in September 1951 in the journal Annals of Mathematics . Nash thus established the mathematical principles of game theory, a branch of mathematics that examines the rivalries between competitors with mixed interests. Nash showed that for any finite game, all the players can arrive at an optimal outcome, known as the Nash equilibrium or the Nash solution , when considering the possible actions of the other players. Despite its practical limitations, the Nash equilibrium was widely applied by business strategists.

Nash’s research into differential equations at MIT led to his seminal paper “Real Algebraic Manifolds,” which was published in Annals of Mathematics in November 1952. His other influential work in mathematics included the Nash-Moser inverse function theorem, the Nash–De Giorgi theorem (a solution to David Hilbert ’s 19th problem, which Nash undertook at the suggestion of Nirenberg), and the Nash embedding (or imbedding) theorems, which the Norwegian Academy of Science and Letters described as “among the most original results in geometric analysis of the twentieth century”; the academy awarded Nash the Abel Prize. His other honours included the John von Neumann Theory Prize (1978) and the American Mathematical Society’s Leroy P. Steele Prize for a Seminal Contribution to Research (1999).

Nash’s research into game theory and his long struggle with paranoid schizophrenia became well known to the general public because of the Academy Award -winning motion picture A Beautiful Mind (2001), which was based on Sylvia Nasar’s 1998 biography of the same name. A more factually accurate exploration of Nash’s struggle with mental illness was offered by the public television documentary A Brilliant Madness (2002).

Someone could retype this in LaTeX... I would but I have only been using it for half a year now. Not as proficient as some.

It would be nice to see it in LaTeX. It's only 27 pages.

I did a presentation on this for my game theory class. A very neat use of a fixed point theorem.

But christ was that notation cumbersome.

27 pages?? God it was easy to get a PhD in math back then!

Only 2 citations too. A math thesis is a different beast.

People forget what a great economist he was, also.

People incorrectly call him an economist, he was/is a mathematician. I say this as someone who considers themself both a mathematician and an economist.

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COMMENTS

Nash, John (1928-2015) - Princeton University Library
Noted mathematician John Nash, Jr. (1928-2015) received his Ph.D. from Princeton University in 1950. The impact of his 27 page dissertation on the fields of mathematics and economics was tremendous. In 1951 he joined the faculty of the Massachusetts Institute of Technology in Cambridge.
John Nash's Super Short PhD Thesis: 26 Pages ... - Open Culture
When John Nash wrote “Non Cooperative Games,” his Ph.D. dissertation at Princeton in 1950, the text of his thesis ( read it online) was brief. It ran only 26 pages. And more particularly, it was light on citations.
John F. Nash, Jr. | Math - Princeton University
John Nash Jr., a legendary fixture of Princeton University’s Department of Mathematics renowned for his breakthrough work in mathematics and game theory as well as for his struggle with mental illness, died with his wife, Alicia, in an automobile accident May 23 in Monroe Township, New Jersey. He was 86, she was 82.
Read John Nash's Super Short PhD Thesis with 26 Pages & 2 ...
Read John Nash’s Super Short PhD Thesis with 26 Pages & 2 Citations: The Beauty of Inventing a Field in Math | June 1st, 2015 1 Comment 13.7k Last week John Nash , the Nobel Prize-winning mathematician, and subject of the blockbuster film A Beautiful Mind, passed away at the age of 86. He died in a taxi cab accident in New Jersey.
John Nash | Biography, Game Theory, Nobel Prize, & Facts
John Nash, in full John Forbes Nash, Jr., (born June 13, 1928, Bluefield, West Virginia, U.S.—died May 23, 2015, near Monroe Township, New Jersey), American mathematician who was awarded the 1994 Nobel Prize for Economics for his landmark work, first begun in the 1950s, on the mathematics of game theory.
John F Nash PhD - Game Theory
John F Nash PhD Project Details In this page you can find Nash’s PhD thesis: Original document Transcribed into tex/pdf * *: Thanks to Rebeca Duarte Miguel for this. Documents Download the documents related to this project here Latexed Thesis Original thesis Category General info
Non-Cooperative Games - Home | Princeton University Library
Non-Cooperative Games - Home | Princeton University Library
John Nash's PhD thesis (1950) [pdf] : r/math - reddit.com
John Nash's PhD thesis (1950) [pdf] princeton.edu This thread is archived New comments cannot be posted and votes cannot be cast 27 1 12 12 comments caks • 13 yr. ago Now imagine what he could have accomplished if they had LaTeX in his days... fburnaby • 13 yr. ago I've been reading many poorly scanned theses from the year 2002 recently...
John Nash's Ph.D. Thesis | PDF - Scribd
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