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John Nash’s Super Short PhD Thesis: 26 Pages & 2 Citations

in Math | July 9th, 2018 2 Comments

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. Nash’s diss cited two texts: John von Neumann & Oskar Morgenstern’s Theory of Games and Economic Behavior (1944), which 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 is having a slim bibliography.

Note: An earlier version of this post appeared on our site in June, 2015.

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Published: 24 June 2015

John Forbes Nash (1928–2015)

Martin A. Nowak 1

Nature volume 522 , page 420 ( 2015 ) Cite this article

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Mathematics and computing

Master of games and equations.

John Forbes Nash, an exalted mathematician whose life took dramatic turns between genius, mental illness and celebrity status, made major contributions to game theory, geometry and the field of partial differential equations.

Nash, who died on 23 May, was born in Bluefield, West Virginia, in 1928. His father was an electrical engineer and his mother a schoolteacher. In 1945, after excelling in mathematics at high school, he attended the Carnegie Institute of Technology (now Carnegie Mellon University) in Pittsburgh, Pennsylvania. At first he studied chemical engineering, but soon after enrolling he switched to chemistry and then to maths.

In Nash's final year, one of his professors wrote a recommendation letter for the 19-year-old supporting his application to graduate school. It simply stated: “He is a mathematical genius.” In 1948, Nash was accepted by Harvard University in Cambridge, Massachusetts, and by Princeton University in New Jersey. He chose Princeton.

As a PhD student, Nash proved the existence of the equilibrium that now carries his name. His 1950 paper 'Equilibrium points in n -person games', contains about 330 words, two references and not one equation (J. F. Nash Jr Proc. Natl Acad. Sci. USA 36 , 48–49; 1950). One of the citations is the 1944 book Theory of Games and Economic Behavior — in which mathematician John von Neumann and economist Oskar Morgenstern introduce game theory, a mathematical approach for studying strategic and economic decisions.

The Nash equilibrium is a position in a game from which none of the players can change their strategy to improve their pay-off. Imagine a game with two players (yourself and another person) and two strategies, A and B. If you both choose A, your pay-off is 2. If you choose A and your opponent chooses B, you score 0. If you choose B and the other player chooses A, your pay-off is 3. If you both choose B, you score 1. The same applies to your opponent.

In this example, the Nash equilibrium occurs when both players choose B. If both players choose B, their pay-off is 1; if either player switches to A, their pay-off falls to 0. In other words, neither player can independently switch their strategy and improve their pay-off. Observe that if both players select A, there is no Nash equilibrium because you could improve your pay-off by switching to B.

Calculating the Nash equilibrium can be a formidable task in a complex game. There is also the uncertainty over whether the person you are playing against is sufficiently rational to play the equilibrium strategy. If both players are rational and their rationality is common knowledge, they would play it. But experiments often reveal that people are not rational. Regardless of whether people actually play the Nash equilibrium in social or economic interactions, working out what it is (or what the Nash equilibria are) is the first step to analysing any game.

Although dismissed at the time by von Neumann as a triviality, the Nash equilibrium has been used to analyse all sorts of competitive situations. As well as being key to decision-making in economics and politics, the idea is important in biology. Here, the nearly equivalent concept, formulated by evolutionary biologist John Maynard Smith in the 1970s is called an evolutionarily stable strategy (ESS). If all members of a population adopt an ESS, then natural selection prevents a rare mutant from spreading.

On completing his PhD, Nash joined the Massachusetts Institute of Technology (MIT) in Cambridge in 1951. He worked — first as an instructor and later as a professor — in the mathematics faculty until he resigned in 1959. It was while he was at MIT that Nash met and married Alicia Lopez-Harrison de Lardé, a physics student there.

Among mathematicians, Nash is best known for his work in real algebraic geometry and nonlinear partial differential equations. He was not afraid to tackle the hardest problems in the field, and he succeeded. In 1957, he — in parallel with Italian mathematician Ennio de Giorgi — solved Hilbert's nineteenth problem involving partial differential equations.

It was during a talk in 1959 on what is seen to be one of the hardest problems in maths — the Riemann hypothesis — that the audience realized that there was something wrong with Nash. His talk was incomprehensible.

He was diagnosed with paranoid schizophrenia that year. Over the next two decades, Nash was in and out of hospitals. He underwent therapy, and for a while left the United States and sought asylum in Switzerland in an attempt to escape his imagined tormentors. For many years he wandered around the Princeton campus. Throughout this period, Alicia, who divorced Nash in 1963, oversaw much of his care.

In the late 1980s, Nash reappeared in academic circles, and in 1994 he was awarded the Nobel Memorial Prize in Economic Sciences for his work on game theory. The Nobel and the 2001 film A Beautiful Mind , based on journalist Sylvia Nasar's book of the same name, which recounted Nash's struggles, propelled him into the limelight.

In May this year, Nash received the Abel prize from the Norwegian Academy of Science and Letters for his work on partial differential equations. On the way back from the celebration in Norway, John and Alicia (who had remarried in 2001) were killed in a car accident in a taxi on the New Jersey turnpike. John was 86.

I met John in 1998 at the Institute for Advanced Study in Princeton. Over the years, I gave several talks there that he attended. One summer's day, when the usual sitting arrangements for lunch were disrupted by the closure of the main kitchen, I noticed John, the physicist Edward Witten and Andrew Wiles, the British mathematician who proved Fermat's last theorem, sitting down together at a small table. I wondered which of them would start the conversation. None of them did. I seem to remember that they ate their meal in silence.

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The Masterpieces of John Forbes Nash Jr.

First Online: 24 February 2019

Cite this chapter

Camillo De Lellis 3 , 4

Part of the book series: The Abel Prize ((AP))

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In this set of notes I follow Nash’s four groundbreaking works on real algebraic manifolds, on isometric embeddings of Riemannian manifolds and on the continuity of solutions to parabolic equations. My aim has been to stay as close as possible to Nash’s original arguments, but at the same time present them with a more modern language and notation. Occasionally I have also provided detailed proofs of the points that Nash leaves to the reader.

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Analysis on Noncompact Manifolds and Index Theory: Fredholm Conditions and Pseudodifferential Operators

What is the Bochner Technique and Where is it Applied

Fred Gehring’s Mathematics

In a short autobiographical note, cf. [ 80 , Ch. 2], Nash states that he made his important discovery while completing his PhD at Princeton. In his own words “…I was fortunate enough, besides developing the idea which led to “NonCooperative Games”, also to make a nice discovery relating manifolds and real algebraic varieties. So, I was prepared actually for the possibility that the game theory work would not be regarded as acceptable as a thesis in the mathematics department and then that I could realize the objective of a Ph.D. thesis with the other results.”

Here we are using the nontrivial fact that in a connected real analytic manifold any pair of points can be joined by a real analytic arc. One simple argument goes as follows: use first Whitney’s theorem to assume, without loss of generality, that Σ is a real analytic submanifold of $\mathbb R^N$ . Fix two points p and q and use the existence of a real analytic projection in a neighborhood of Σ to reduce our claim to the existence of a real analytic arc connecting any two points inside a connected open subset of the Euclidean space. Finally use the Weierstrass polynomial approximation theorem to show the latter claim.

The projection of an algebraic subvariety is not always an algebraic subvariety: here as well we are taking advantage of the genericity of the projection.

Many thanks to Riccardo Ghiloni for suggesting this argument, which follows closely the proof of [ 55 , Lem. 3.2].

Observe that in this context the closure in the Euclidean topology coincides with the Zariski closure.

Closed manifolds can be C 1 isometrically immersed in lower dimension: already at the time of Nash’s paper this could be shown in $\mathbb R^{2n-1}$ (for n > 1!) using Whitney’s immersion theorem. Nowadays we can use Cohen’s solution of the immersion conjecture to lower the dimension to n − a ( n ), where a ( n ) is the number of 1’s in the binary expansion of n , cf. [ 17 ].

This is what Nash calls “a stage”, cf. [ 73 , p. 391].

Although the term is nowadays rather common, it was not introduced by Nash, neither in [ 73 ] nor in the subsequent paper [ 75 ].

In his paper Nash claims indeed a much larger K ( n ), cf. [ 73 , bottom of p. 386].

The argument of Nash is slightly different, since it covers the space of positive definite matrices with appropriate simplices.

Nash cites Steenrod’s classical book, [ 94 ].

Nash writes Also they could be obtained by orthogonal propagation , cf. [ 73 , top of p. 387].

The term free was not coined by Nash, but introduced later in the literature by Gromov.

It must be observed that Nash employs this fact without explicitly stating it and he does not prove it neither he gives a reference. He uses it twice, once in the proof of Theorem 29 and once in the proof of Proposition 35 , and although in the first case one could appeal to a more elementary argument, I could not see an easier way in the second.

Indeed Nash does not give any argument and just refers to a similar reasoning that he uses in Proposition 35 below.

Nash suggests an alternative argument which avoids the discussion of the dimensions of $\mathcal {C} (p, L)$ and $\mathcal {B}$ . One can apply his result on real algebraic varieties to find an embedding v which realizes v ( Σ ) as a real algebraic submanifold, cf. Theorem 1 . Then any set of coefficients $A^r_{ij}$ which is algebraically independent over the minimal field $\mathbb F$ of definition of v ( Σ ) (see Proposition 12 ) belongs to the complement of $\mathcal {B}$ . Since $\mathbb F$ is finitely generated over the rationals (see Proposition 12 ), it has countable cardinality and the conclusion follows easily.

In Nash’s paper the operator is called S θ , where θ corresponds to ε −1 . Since it is nowadays rather unusual to parametrize a family of convolutions as Nash does, I have switched to a more modern convention.

Nash does not take advantage of this simple remark and introduces instead a rather unusual notation to keep track of all the estimates for the intermediate norms in the bounds corresponding to ( 59 ), ( 60 ) and ( 61 ).

In fact, De Giorgi’s statement is stronger, since in his theorem ∥ v ∥ ∞ in ( 125 ) is replaced by the L 2 norm of v (note that the power of r should be suitably adjusted: the reader can easily guess the correct exponent using the invariance of the statement under the transformation u r ( x ) = u ( rx )).

Indeed, it was known that the first partial derivatives of the minimizer satisfy a uniformly elliptic partial differential equation with measurable coefficients. De Giorgi’s stronger version of Theorem 50 would then directly imply the desired Hölder estimate. Nash’s version was also sufficient, because a theorem of Stampacchia guaranteed the local boundedness of the first partial derivatives, cf. [ 93 ].

Nash does not provide any argument nor reference, he only briefly mentions that Theorem 48 follows from Theorem 51 using a regularization scheme and the maximum principle. Note that a derivation of the latter under the weak regularity assumptions of Theorem 48 is, however, not entirely trivial: in Sect. 5.8 we give an alternative argument based on a suitable energy estimate.

In order to simplify the notation we omit the domain of integration when it is the entire space.

The first two equations are the first two equations from [ 77 , p. 487, (1)] whereas the third should correspond to [ 77 , p. 488, (1c)]. The latter is derived by Nash from the third equation in [ 77 , p. 487, (1)], which in turn corresponds to the classical conservation law for the entropy, see, for instance, [ 61 , (49.5)]. The third equation of [ 77 , p. 487, (1)] contains two typos, which disappear in [ 77 , p. 488, (1c)]. The latter however contains another error: Nash has η and ζ in place of $\frac {\eta }{\rho T S_T}$ and $\frac {\zeta }{\rho T S_T}$ , but it is easy to see that this would not be consistent with the way he describes its derivation.

Nash’s error has no real consequence for the rest of the note, since he treats the coefficients in front of $\mathcal {S} (v)_{ij} \mathcal {S} (v)_{ij}$ and (div v ) 2 as arbitrary real analytic functions of ρ and T and the same holds for $\frac {\eta }{\rho T S_T}$ and $\frac {\zeta }{\rho T S_T}$ under the assumption S T ≠ 0. The latter inequality is needed in any case even to treat Nash’s “wrong” equation for T .

Indeed Nash does not mention the positivity of S T , although this is certainly required by his argument when he reduces the existence of solutions of ( 240 ) to the existence of a solutions of a suitable parabolic system, cf. [ 77 , (6) and (7)]: the equation in T is parabolic if and only if $\frac {\varkappa }{\rho T S_T}$ is positive.

I also have the impression that his argument does not really need the positivity of S and p , although these are quite natural assumptions from the thermodynamical point of view.

In the modern literature it is customary to take an equivalent definition of X through formal power series; we refer to [ 58 ] for the latter and for several important subtleties related to variants of the Nash arc space.

In fact, Nash claims the proposition with any algebraic subset W of V in place of V s but, although the proposition does hold for W = V s , it turns out to be false for a general algebraic subset W ; cf. [ 21 , Ex. 3.7] for a simple explicit counterexample.

S. Akbulut and H. King. On approximating submanifolds by algebraic sets and a solution to the Nash conjecture. Invent. Math. , 107(1):87–98, 1992.

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Acknowledgements

I am very grateful to Helge and Ragni for entrusting to me this portion of the Nash volume, a wonderful occasion to deepen my understanding of the mathematics of a true genius, who has had a tremendous influence in my own work.

Most of the manuscript has been written while I was visiting the CMSA at Harvard and I wish to thank Shing-Tung Yau and the staff at CMSA for giving me the opportunity to carry on my work in such a stimulating environment.

Several friends and colleagues have offered me kind and invaluable help with various portions of this note. In particular I wish to thank Davide Vittone for giving me several precious suggestions with the Sects. 3 and 4 and reading very carefully all the manuscript; Gabriele Di Cerbo, Riccardo Ghiloni and János Kollár for clarifying several important points concerning Sect. 2 and pointing out a few embarassing mistakes; Tommaso de Fernex and János Kollar for kindly reviewing a first rather approximate version of Sect. 6.4 ; Eduard Feireisl for his suggestions on Sect. 6.3 ; Cedric Villani for allowing me to steal a couple of paragraphs from his beautiful review of [ 76 ] in the Nash memorial article [ 26 ]; Francois Costantino for helping me with a delicate topological issue; Jonas Hirsch and Govind Menon for proofreading several portions of the manuscript; Helge Holden for going through all the manuscript with extreme care.

This work has been supported by the grant agreement 154903 of the Swiss National Foundation.

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De Lellis, C. (2019). The Masterpieces of John Forbes Nash Jr.. In: Holden, H., Piene, R. (eds) The Abel Prize 2013-2017. The Abel Prize. Springer, Cham. https://doi.org/10.1007/978-3-319-99028-6_19

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This is the best letter of recommendation ever

by Dylan Matthews

John Nash — the Princeton game theorist who shared the 1994 Nobel Prize in economics and was portrayed by Russell Crowe in A Beautiful Mind — died in a car crash on May 23, along with his wife, Alicia. In light of his passing, Princeton has digitized his academic file and released it to the public; you can see the full PDF here .

Probably the best document it contains is a recommendation letter by Richard Duffin, Nash's undergraduate advisor at the Carnegie Institute of Technology (now Carnegie Mellon), to Solomon Lefschetz, a math professor at Princeton, where Nash was applying to grad school:

The letter was successful. Nash went on to get his PhD at Princeton. His dissertation, "Non-cooperative games," introduced a concept that would become known as the "Nash equilibrium," a crucial concept in game theory. Khan Academy explains the idea, and how it relates to the famous prisoner's dilemma game, here:

Nash's dissertation has been cited about 7,189 times , according to Google Scholar, though that excludes many uses of the term "Nash equilibrium" that don't cite the actual paper. "If Nash got a dollar for every time someone wrote or said 'Nash equilibrium,'" Princeton economist Avinash Dixit once wrote, "he would be a rich man."

Nash submitted his thesis in May 1950, when he was 21 years old. It was a mere 27 pages long, which includes an acknowledgements section and a very short bibliography. The entire dissertation cites only two sources: game theory's founders John von Neumann and Oskar Morgenstern, and John Nash.

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Incredible story of john nash and his short phd thesis.

At 86, Nash (center) continued to inspire and work with younger researchers such as Michail Rassias (left), a Princeton visiting postdoctoral research associate in mathematics who was working on an upcoming book with Nash. Although 60 years Nash’s junior, Rassias admired Nash’s quick thinking and curiosity. This photo, taken during a special departmental reception in honor of Nash’s winning the Abel Prize, includes Yakov Sinai (right), a Princeton professor of mathematics, who was awarded the 2014 Abel Prize. (Photo by Danielle Alio, Office of Communications)

This is a story about John Nash, the famed mathematician who inspired the movie 'A Beautiful Mind,' and his 27-page-long PhD thesis. He altered our understanding of the world in a mere 27 pages and shaped a life that, while cut short, was long in its reach and profound in its influence.

The year is 1950. John Nash, a young graduate student at Princeton, submits his PhD dissertation. It’s not a voluminous work stretching into hundreds of pages, as one might expect from groundbreaking research. Instead, it consists of a mere 27 pages. Titled “ Non-Cooperative Games ,” this unassuming document would go on to revolutionize the field of economics and game theory. As such it eventually earned Nash the Nobel Prize in Economic Sciences in 1994.

A minimalist approach where less is more

Nash’s thesis was unique not just in its length, but also in its reliance on citations. Only two sources were cited in the entire document. The first was the 1944 seminal work “ Theory of Games and Economic Behavior ” by John von Neumann and Oskar Morgenstern, the book that essentially laid the foundation for game theory.

The second was an article titled “ Equilibrium Points in n-Person Games ” — authored by none other than John Nash himself. This deliberate selection hinted at Nash’s confidence in his original ideas and his audacity to let them stand almost solely on their own merit.

In academic circles, the importance of a thesis is sometimes measured by its length and the number of citations it carries. Nash defied this unwritten rule in a spectacular manner. It was as if he was saying, “ Here are my ideas; they stand on their own .”

The essence of Nash’s brilliance

What made Nash’s dissertation extraordinary was its concept of equilibrium in “ non-cooperative ” games. A significant departure from traditional theories that primarily focused on “ cooperative ” games where participants can make binding agreements.

His work argued that in any game involving two or more players, each with their set of strategies, there exists an equilibrium where each player’s strategy is optimal given the strategies chosen by everyone else.

This would come to be known as the “ Nash Equilibrium. ”

Nash’s theories extended well beyond economics. They found applications in various domains, including biology, political science, and even philosophy. They provided a robust analytical framework to understand how individual decisions in complex systems could ultimately result in a stable, balanced outcome.

Nash’s impact was far-reaching and profound

The magnitude of the impact of Nash’s work can be best understood by the accolades it received. In 1994, nearly five decades after he wrote his thesis, Nash was awarded the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel – Nobel prize for Economics . This prestigious recognition signified the immense influence and lasting relevance of his work.

Part of the Abstract section of John Nash's PhD paper.

While the 26 pages may have seemed trivial at the time, the depth and breadth of the ideas contained within were far from it. Many researchers and scholars have since built upon Nash’s principles, and his equilibrium concepts have become a standard teaching module in economics courses worldwide.

Paving the way for modern game theory

Before Nash, game theory was an interesting but limited field. His dissertation blew open the doors to various new interpretations and applications. For instance, Nash’s ideas are now routinely applied in everything from market economics to election strategies. You can find them in negotiations, and even evolutionary biology.

His theories are applied in fields as varied as artificial intelligence, where algorithms often use Nash equilibrium concepts for decision-making. They are visible to international politics, where countries apply non-cooperative game theories in diplomatic negotiations and strategy planning. His work even finds applications in evolutionary biology, as mentioned earlier, explaining survival strategies among different species.

A life fraught with challenges — genius, vulnerability and the enigma of recovery

John Nash’s mental health battles are commonly thought to have been a lifelong struggle. However, the truth, as explained by his biographer Sylvia Nasar, paints a more nuanced picture .

Nash experienced the onset of schizophrenia at age 30 and underwent treatment only up to age 40. After that point, he had what he described as “ aging out of it. ” Nash speculated that hormonal changes played a role in his recovery, a hypothesis that aligns with the principles of cognitive and behavioral therapies that help patients manage delusions rather than will them away.

Nash’s case was exceptional even among those with chronic schizophrenia. He was among the small fraction who experience spontaneous recoveries. Generally, only 8 to 10% of patience experience such recovery.

While his intellectual brilliance was undoubted, another underappreciated aspect of his life was the role his wife Alicia played in his recovery. As Nasar notes, Nash had a stable home and family to return to, providing a sanctuary of sorts that undoubtedly played a role in his well-being.

Hopes and recognition and the Nobel as a crowning achievement and a final cure

The Nobel Prize in 1994 wasn’t just a triumph of intellectual prowess; it served a dual role.

For Nash, the award was a recognition not just from the academic community but also a form of reintegration into a society, The same society that had perhaps distanced itself from him due to his mental illness.

According to Nasar, the Nobel “completed the cure” in the sense that Nash could re-enter a community that, despite years of estrangement, still held significance for him and to which he could contribute anew.

This external validation, while late in coming, offered a sense of closure and belonging. And that cannot be underestimated in understanding Nash’s multifaceted life journey.

The unshakable support of his family and the acknowledgment from a community that had once shunned him combined to provide a form of holistic recovery that is as awe-inspiring as it is rare. Nash’s story isn’t merely one of intellectual triumph but is equally a narrative about the resilience of the human spirit, underscored by the roles of love, belonging, and recognition.

A tale of two narratives — Hollywood vs. Reality

Nash’s life was depicted in the 2001 film “A Beautiful Mind,” inspired by Nash’s life. While the movie captured the essence of Nash’s genius and his battle with schizophrenia, it took several creative liberties.

One such discrepancy is the film’s portrayal of visual hallucinations, which Nash himself did not experience. Moreover, the movie chose to focus on the more dramatic aspects of his condition. At the same time it omitted significant elements of his personal life. For example, his divorce and eventual remarriage to Alicia.

What the film did accurately convey, however, was Nash’s resilience and the unwavering support he received from Alicia. It managed to humanize a man often perceived merely as a towering intellectual figure. It brought to light the everyday battles he fought against his own mind.

A final chapter

The story of John Nash and his wife Alicia took a heartbreaking turn on May 23, 2015. Just shortly after Nash had received the Abel Prize for his contributions to mathematics , the couple died in a car crash while returning home from the airport. It was a shocking and sudden end. Not just to a life marked by incredible highs and lows, but also to a love story that had withstood the test of time, illness, and even separation.

The news of their deaths reverberated across academic circles and beyond, offering a grim reminder of the unpredictability of life. At the time of his passing, Nash was still active in his work, never one to rest on his laurels. His death, therefore, served as a stark full stop to a career that had recently received another accolade, and a life journey that had been marked by extraordinary resilience.

The couple’s sudden departure was an unfortunate punctuation to an enduring narrative of love, perseverance, and intellectual curiosity. As much as Nash’s theories revolutionized economics and mathematics, his life story with Alicia provided a real-world testament to the power of human endurance and connection.

Though they left the world abruptly, the indelible marks both John and Alicia Nash made—on science, on perceptions of mental health, and on each other’s lives—endure as a testament to their shared journey. They may have departed, but their story continues to resonate, serving as a complex but ultimately inspiring legacy.

A lifetime of legacy in 27 pages and a life that stretched its bounds

In the grand arc of John Nash’s life, which was full and long, living to the age of 86, one can see echoes of the succinct but powerful document that made him famous.

His 27-page PhD thesis, “Non Cooperative Games,” was remarkable for distilling complex theories into an understandable, transformative format. Similarly, Nash’s long life was dense with accomplishments and challenges that transcended the years he lived.

While his life did stretch over eight decades, each chapter was marked by a density of experience. From the highs of academic achievements to the lows of mental health struggles, from the love and support of his family to the ultimate, tragic conclusion — each period of his life was as impactful as those concise 27 pages that first put him on the map.

The abrupt ending to Nash’s life was a cruel stroke of fate. However, in no way it diminishes the longevity or significance of his journey. If his thesis taught us that profound ideas don’t need a lot of space to make an impact, his life demonstrated that the impact of human existence isn’t solely measured by its length, but by its depth and the ripples it creates in the world around it.

John Nash did just that. Altered our understanding of the world in a mere 27 pages and shaped a life that, while cut short, was long in its reach and profound in its influence.

Table of Contents of John Nash's PhD paper.

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John F. Nash, Jr.

In memoriam, life & work.

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.”

A memorial service for Nash will be planned at the University in the fall.

John F Nash PhD

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In this page you can find Nash’s PhD thesis:

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*: Thanks to Rebeca Duarte Miguel for this, and to Jeek Midford for spotting some spelling mistakes.

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John F. Nash, Jr.

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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.

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

-By Morgan Kelley, Princeton University Office of Communications

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John Nash's Super Short PhD Thesis: 26 Pages & 2 Citations
John Nash's Super Short PhD Thesis: 26 Pages & 2 Citations. in Math | July 9th, 2018 2 Comments. 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.
Nash, John (1928-2015)
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. His battle with schizophrenia began around 1958, and the ...
Does the length of a PhD thesis matter?
6. John Nash's thesis was 26 pages long with only two references and he later won a Nobel prize. What matters is scientific quality not quantity, if your ideas are superior nobody will object the length (mine was less than 100 pages). Math is really special in this respect.
John Forbes Nash Jr.
John Forbes Nash, Jr. (June 13, 1928 - May 23, 2015), known and published as John Nash, was an American mathematician who made fundamental contributions to game theory, real algebraic geometry, ... Nash earned a PhD in 1950 with a 28-page dissertation on non-cooperative games. ...
PDF The Work of John Nash in Game Theory
In the short period of 1950 - 53, John Nash published four brilliant papers ([35], [37], [38], [39]), in which he made at least three fundamentally important contributions to game theory: (1) He introduced the distinction between cooperative and non-cooperative games.
John Forbes Nash (1928-2015)
As a PhD student, Nash proved the existence of the equilibrium that now carries his name. His 1950 paper 'Equilibrium points in n-person games', contains about 330 words, two references and not ...
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1950 he earned a PhD de-gree with his celebrated work on noncooperative games,whichwonhimthe ... as a thesis at the Princeton mathematics department. ... John Nachbar and Jonathan Weinstein Nash Equilibrium Game theory is a mathematicalframeworkfor analyzing
The masterpieces of John Forbes Nash Jr.
John Nash has written very few papers: if for each mathematician in the 20th century we were to divide the depth, originality, and impact of the corresponding production by ... These notes leave aside Nash's celebrated PhD thesis on game theory and focus on the remaining four fundamental papers that have started an equal number of revolutions in
The Masterpieces of John Forbes Nash Jr.
These notes leave aside Nash's celebrated PhD thesis on game theory and focus on the remaining four fundamental papers that have started an equal number of revolutions in their respective topics, namely the 1952 note on real algebraic varieties, the 1954 paper on C 1 isometric embeddings, the 1956 subsequent work on smooth isometric ...
This is the best letter of recommendation ever
Nash went on to get his PhD at Princeton. His dissertation, "Non-cooperative games," introduced a concept that would become known as the "Nash equilibrium," a crucial concept in game theory.
Incredible story of John Nash and his short PhD thesis
This is a story about John Nash, the famed mathematician who inspired the movie 'A Beautiful Mind,' and his 27-page-long PhD thesis. He altered our understanding of the world in a mere 27 pages and shaped a life that, while cut short, was long in its reach and profound in its influence. The year is 1950.
Life & Work
John Nash Jr., a legendary fixture ... 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 ...
‪John Forbes Nash‬
The work of John F. Nash Jr. in game theory. HW Kuhn, JC Harsanyi, R Selten, JW Weibull, E van Damme. 25. 1994. The agencies method for modeling coalitions and cooperation in games. JF Nash Jr. International Game Theory Review 10 (04), 539-564. , 2008. 20.
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, and to Jeek Midford for spotting some spelling mistakes.
Dr Paul Coxon on Twitter: "John Nash's PhD thesis was 27 pages long
"John Nash's PhD thesis was 27 pages long & only had 2 references, 1 of which was his own paper http://t.co/soWT3aXLC8"
PDF Nash
Title: Nash.pdf Created Date: 12/11/2001 4:05:15 PM
John Nash's PhD thesis (1950) [pdf] : r/math
A math thesis is a different beast. So this was not a math thesis? No it was, he was saying what is required for a PhD in mathematics differs quite a bit to that of a lot of other fields, this being a prime example where a lot of it was new to the field. People forget what a great economist he was, also.
John Nash
John Nash (born June 13, 1928, Bluefield, West Virginia, U.S.—died May 23, 2015, near Monroe Township, New Jersey) was an 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 ...
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John Nash Phd Thesis Length - Free download as PDF File (.pdf), Text File (.txt) or read online for free. Scribd is the world's largest social reading and publishing site.
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John F. Nash, Jr.
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 ...

John Nash’s Super Short PhD Thesis: 26 Pages & 2 Citations

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