Editing John Forbes Nash Jr. (section)

== Research contributions ==
[[File:John f nash 20061102 3.jpg|thumb|upright|Nash in November 2006 at a [[game theory]] conference in [[Cologne]], Germany]]

Nash did not publish extensively, although many of his papers are considered landmarks in their fields.<ref>{{cite journal | first = John | last = Milnor | author-link = John Milnor |year = 1998 | title = John Nash and 'A Beautiful Mind' | url = https://www.ams.org/notices/199810/milnor.pdf| journal = [[Notices of the American Mathematical Society]] | volume = 25 | issue = 10|pages = 1329–1332 }}</ref> As a graduate student at Princeton, he made foundational contributions to [[game theory]] and [[real algebraic geometry]]. As a postdoctoral fellow at [[MIT]], Nash turned to [[differential geometry]]. Although the results of Nash's work on differential geometry are phrased in a geometrical language, the work is almost entirely to do with the [[mathematical analysis]] of [[partial differential equations]].<ref name="steele">{{cite journal|title=1999 Steele Prizes|journal=[[Notices of the American Mathematical Society]]|date=April 1999|pages=457–462|volume=46|issue=4|url=https://www.ams.org/notices/199904/comm-steele-prz.pdf |archive-url=https://web.archive.org/web/20000829221437/http://www.ams.org/notices/199904/comm-steele-prz.pdf |archive-date=2000-08-29 |url-status=live}}</ref> After proving his two [[Nash embedding theorems|isometric embedding theorem]]s, Nash turned to research dealing directly with partial differential equations, where he discovered and proved the De Giorgi–Nash theorem, thereby resolving one form of [[Hilbert's nineteenth problem]].

In 2011, the [[National Security Agency]] declassified letters written by Nash in the 1950s, in which he had proposed a new [[encryption]]–decryption machine.<ref>{{cite web |url=https://www.nsa.gov/Press-Room/Press-Releases-Statements/Press-Release-View/Article/1630570/national-cryptologic-museum-opens-new-exhibit-on-dr-john-nash/ |publisher=[[National Security Agency]] |title=2012 Press Release – National Cryptologic Museum Opens New Exhibit on Dr. John Nash |access-date=July 30, 2022}}</ref> The letters show that Nash had anticipated many concepts of modern [[cryptography]], which are based on [[Computational hardness assumption|computational hardness]].<ref>{{cite web |url=http://agtb.wordpress.com/2012/02/17/john-nashs-letter-to-the-nsa/ |title=John Nash's Letter to the NSA; Turing's Invisible Hand |access-date=February 25, 2012|date=February 17, 2012 }}</ref>

=== Game theory ===
Nash earned a PhD in 1950 with a 28-page dissertation on [[Non-cooperative game theory|non-cooperative games]].<ref name="JohnNash_PhD">{{cite web |last1=Nash |first1=John F. |author-link=John Forbes Nash Jr. |title=Non-Cooperative Games |work=PhD thesis |publisher=Princeton University |date=May 1950 |url=https://www.princeton.edu/mudd/news/faq/topics/Non-Cooperative_Games_Nash.pdf |access-date=May 24, 2015 |archive-url=https://web.archive.org/web/20150420144847/http://www.princeton.edu/mudd/news/faq/topics/Non-Cooperative_Games_Nash.pdf |archive-date=April 20, 2015  }}</ref><ref>{{cite book |first=Martin J. |last=Osborne |date=2004 |title=An Introduction to Game Theory |url=https://archive.org/details/introductiontoga00osbo |url-access=limited |publisher=[[Oxford University Press]] |location=Oxford, England |page=[https://archive.org/details/introductiontoga00osbo/page/n34 23] |isbn=0-19-512895-8}}</ref> The thesis, written under the supervision of doctoral advisor [[Albert W. Tucker]], contained the definition and properties of the [[Nash equilibrium]], a crucial concept in non-cooperative games. A version of his thesis was published a year later in the [[Annals of Mathematics]].{{sfnm|1a1=Nash|1y=1951}} In the early 1950s, Nash carried out research on a number of related concepts in game theory, including the theory of [[cooperative game theory|cooperative games]].{{sfnm|1a1=Nash|1y=1950a|2a1=Nash|2y=1950b|3a1=Nash|3y=1953}} For his work, Nash was one of the recipients of the [[Nobel Memorial Prize in Economic Sciences]] in 1994.

=== Real algebraic geometry ===
In 1949, while still a graduate student, Nash found a new result in the mathematical field of [[real algebraic geometry]].{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 15}} He announced his theorem in a contributed paper at the [[International Congress of Mathematicians]] in 1950, although he had not yet worked out the details of its proof.{{sfnm|1a1=Nash|1y=1952a}} Nash's theorem was finalized by October 1951, when Nash submitted his work to the [[Annals of Mathematics]].{{sfnm|1a1=Nash|1y=1952b}} It had been well-known since the 1930s that every [[closed manifold|closed]] [[smooth manifold]] is [[diffeomorphic]] to the [[zero set]] of some collection of [[smooth function]]s on [[Euclidean space]]. In his work, Nash proved that those smooth functions can be taken to be [[polynomial]]s.<ref name="bochnak">{{cite book|mr=1659509|last1=Bochnak|first1=Jacek|last2=Coste|first2=Michel|last3=Roy|first3=Marie-Françoise|title=Real algebraic geometry|edition=Translated and revised from 1987 French original|series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete|Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge]]|volume=36|publisher=[[Springer-Verlag]]|location=Berlin|year=1998|isbn=3-540-64663-9|zbl=0912.14023|doi=10.1007/978-3-662-03718-8|s2cid=118839789 |author-link3=Marie-Françoise Roy}}</ref> This was widely regarded as a surprising result,{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 15}} since the class of smooth functions and smooth manifolds is usually far more flexible than the class of polynomials. Nash's proof introduced the concepts now known as [[Nash function]] and [[Nash manifold]], which have since been widely studied in real algebraic geometry.<ref name="bochnak" /><ref>{{cite book|mr=0904479|last1=Shiota|first1=Masahiro|title=Nash Manifolds |series=[[Lecture Notes in Mathematics]]|volume=1269|publisher=[[Springer-Verlag]]|location=Berlin|year=1987|isbn=3-540-18102-4|doi=10.1007/BFb0078571|zbl=0629.58002}}</ref> Nash's theorem itself was famously applied by [[Michael Artin]] and [[Barry Mazur]] to the study of [[dynamical system]]s, by combining Nash's polynomial approximation together with [[Bézout's theorem]].<ref>{{cite journal|mr=0176482|last1=Artin|first1=M.|last2=Mazur|first2=B.|title=On periodic points|journal=[[Annals of Mathematics]]|series=Second Series|volume=81|year=1965|pages=82–99|issue=1|doi=10.2307/1970384|jstor=1970384 |zbl=0127.13401|author-link1=Michael Artin|author-link2=Barry Mazur}}</ref><ref>{{cite journal|first1=Mikhaïl|last1=Gromov|url=https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/1024.pdf|title=On the entropy of holomorphic maps|journal=[[L'Enseignement mathématique|L'Enseignement Mathématique. Revue Internationale]]|series=2e Série|volume=49|year=2003|issue=3–4|pages=217–235|mr=2026895|zbl=1080.37051|author-link1=Mikhael Gromov (mathematician)}}</ref>

===Differential geometry===
During his postdoctoral position at [[MIT]], Nash was eager to find high-profile mathematical problems to study.{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 20}} From [[Warren Ambrose]], a [[differential geometry|differential geometer]], he learned about the conjecture that any [[Riemannian manifold]] is [[isometry|isometric]] to a [[submanifold]] of [[Euclidean space]]. Nash's results proving the conjecture are now known as the [[Nash embedding theorem]]s, the second of which [[Mikhael Gromov (mathematician)|Mikhael Gromov]] has called "one of the main achievements of mathematics of the twentieth century".<ref name="Nash2015">{{cite conference|mr=3470099|book-title=Open problems in mathematics|editor-first1=John Forbes Jr.|editor-last1=Nash|editor-first2=Michael Th.|editor-last2=Rassias|publisher=[[Springer, Cham]]|year=2016|isbn=978-3-319-32160-8|doi=10.1007/978-3-319-32162-2|first1=Misha|last1=Gromov|author-link1=Mikhael Gromov (mathematician)|title=Introduction John Nash: theorems and ideas|arxiv=1506.05408}}</ref>

Nash's first embedding theorem was found in 1953.{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 20}} He found that any Riemannian manifold can be isometrically embedded in a Euclidean space by a [[continuously differentiable]] mapping.{{sfnm|1a1=Nash|1y=1954}} Nash's construction allows the [[codimension]] of the embedding to be very small, with the effect that in many cases it is logically impossible that a highly-differentiable isometric embedding exists. (Based on Nash's techniques, [[Nicolaas Kuiper]] soon found even smaller codimensions, with the improved result often known as the ''Nash–Kuiper theorem''.) As such, Nash's embeddings are limited to the setting of low differentiability. For this reason, Nash's result is somewhat outside the mainstream in the field of [[differential geometry]], where high differentiability is significant in much of the usual analysis.<ref>{{cite book|last1=Eliashberg|first1=Y.|last2=Mishachev|first2=N.|title=Introduction to the h-principle|series=[[Graduate Studies in Mathematics]]|volume=48|publisher=[[American Mathematical Society]]|location=Providence, RI|year=2002|isbn=0-8218-3227-1|mr=1909245|author-link1=Yakov Eliashberg|doi=10.1090/gsm/048}}</ref><ref name="pdr">{{cite book|last1=Gromov|first1=Mikhael|title=Partial differential relations|series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3)|volume=9|publisher=[[Springer-Verlag]]|location=Berlin|year=1986|isbn=3-540-12177-3|mr=0864505|author-link1=Mikhael Gromov (mathematician)|doi=10.1007/978-3-662-02267-2}}</ref>

However, the logic of Nash's work has been found to be useful in many other contexts in [[mathematical analysis]]. Starting with work of [[Camillo De Lellis]] and László Székelyhidi, the ideas of Nash's proof were applied for various constructions of turbulent solutions of the [[Euler equations]] in [[fluid mechanics]].<ref>{{cite journal|last1=De Lellis|first1=Camillo|last2=Székelyhidi|first2=László Jr.|title=Dissipative continuous Euler flows|journal=[[Inventiones Mathematicae]]|volume=193|year=2013|issue=2|pages=377–407|mr=3090182|author-link1=Camillo De Lellis|doi=10.1007/s00222-012-0429-9| arxiv=1202.1751 | bibcode=2013InMat.193..377D | s2cid=2693636 }}</ref><ref>{{cite journal|last1=Isett|first1=Philip|title=A proof of Onsager's conjecture|journal=[[Annals of Mathematics]]|series=Second Series|year=2018|volume=188|issue=3|pages=871–963|mr=3866888|doi=10.4007/annals.2018.188.3.4|s2cid=119267892|url=https://authors.library.caltech.edu/87369/|arxiv=1608.08301|access-date=October 11, 2022|archive-date=October 11, 2022|archive-url=https://web.archive.org/web/20221011050610/https://authors.library.caltech.edu/87369/|url-status=dead}}</ref> In the 1970s, [[Mikhael Gromov (mathematician)|Mikhael Gromov]] developed Nash's ideas into the general framework of ''convex integration'',<ref name="pdr" /> which has been (among other uses) applied by [[Stefan Müller (mathematician)|Stefan Müller]] and [[Vladimír Šverák]] to construct counterexamples to generalized forms of [[Hilbert's nineteenth problem]] in the [[calculus of variations]].<ref>{{cite journal|last1=Müller|first1=S.|last2=Šverák|first2=V.|title=Convex integration for Lipschitz mappings and counterexamples to regularity|journal=[[Annals of Mathematics]]|series=Second Series|volume=157|year=2003|issue=3|pages=715–742|mr=1983780|author-link1=Stefan Müller (mathematician)|author-link2=Vladimir Šverák|doi=10.4007/annals.2003.157.715| s2cid=55855605 |doi-access=free|arxiv=math/0402287}}</ref>

Nash found the construction of smoothly differentiable isometric embeddings to be unexpectedly difficult.{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 20}} However, after around a year and a half of intensive work, his efforts succeeded, thereby proving the second Nash embedding theorem.{{sfnm|1a1=Nash|1y=1956}} The ideas involved in proving this second theorem are largely separate from those used in proving the first. The fundamental aspect of the proof is an [[implicit function theorem]] for isometric embeddings. The usual formulations of the implicit function theorem are inapplicable, for technical reasons related to the ''loss of regularity'' phenomena. Nash's resolution of this issue, given by deforming an isometric embedding by an [[ordinary differential equation]] along which extra regularity is continually injected, is regarded as a fundamentally novel technique in [[mathematical analysis]].<ref name="hamilton82">{{cite journal|first=Richard S.|last=Hamilton|mr=0656198|title=The inverse function theorem of Nash and Moser|journal=[[Bulletin of the American Mathematical Society]] |series=New Series |volume=7|year=1982|issue=1|pages=65–222|doi-access=free|doi=10.1090/s0273-0979-1982-15004-2|zbl=0499.58003|author-link1=Richard S. Hamilton}}</ref> Nash's paper was awarded the [[Leroy P. Steele Prize for Seminal Contribution to Research]] in 1999, where his "most original idea" in the resolution of the ''loss of regularity'' issue was cited as "one of the great achievements in mathematical analysis in this century".<ref name="steele" /> According to Gromov:<ref name="Nash2015" />
{{blockquote|You must be a novice in analysis or a genius like Nash to believe anything like that can be ever true and/or to have a single nontrivial application.}}

Due to [[Jürgen Moser]]'s extension of Nash's ideas for application to other problems (notably in [[celestial mechanics]]), the resulting implicit function theorem is known as the [[Nash–Moser theorem]]. It has been extended and generalized by a number of other authors, among them Gromov, [[Richard S. Hamilton|Richard Hamilton]], [[Lars Hörmander]], [[Jacob T. Schwartz|Jacob Schwartz]], and [[Eduard Zehnder]].<ref name="pdr" /><ref name="hamilton82" /> Nash himself analyzed the problem in the context of [[analytic function]]s.{{sfnm|1a1=Nash|1y=1966}} Schwartz later commented that Nash's ideas were "not just novel, but very mysterious," and that it was very hard to "get to the bottom of it."{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 20}} According to Gromov:<ref name="Nash2015" />
{{blockquote|Nash was solving classical mathematical problems, difficult problems, something that nobody else was able to do, not even to imagine how to do it. ...&nbsp; what Nash discovered in the course of his constructions of isometric embeddings is far from 'classical'&nbsp;– it is something that brings about a dramatic alteration of our understanding of the basic logic of analysis and differential geometry. Judging from the classical perspective, what Nash has achieved in his papers is as impossible as the story of his life&nbsp;... [H]is work on isometric immersions&nbsp;... opened a new world of mathematics that stretches in front of our eyes in yet unknown directions and still waits to be explored.}}

===Partial differential equations===
While spending time at the [[Courant Institute]] in New York City, [[Louis Nirenberg]] informed Nash of a well-known conjecture in the field of [[elliptic partial differential equation]]s.{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 30}} In 1938, [[Charles Morrey]] had proved a fundamental [[elliptic regularity]] result for functions of two independent variables, but analogous results for functions of more than two variables had proved elusive. After extensive discussions with Nirenberg and [[Lars Hörmander]], Nash was able to extend Morrey's results, not only to functions of more than two variables, but also to the context of [[parabolic partial differential equation]]s.{{sfnm|1a1=Nash|1y=1957|2a1=Nash|2y=1958}} In his work, as in Morrey's, uniform control over the continuity of the solutions to such equations is achieved, without assuming any level of differentiability on the coefficients of the equation. The [[Sobolev inequality|Nash inequality]] was a particular result found in the course of his work (the proof of which Nash attributed to [[Elias Stein]]), which has been found useful in other contexts.<ref name="davies">{{cite book|mr=0990239|last1=Davies|first1=E. B.|title=Heat kernels and spectral theory|series=Cambridge Tracts in Mathematics|volume=92|publisher=[[Cambridge University Press]]|location=Cambridge|year=1989|isbn=0-521-36136-2|doi=10.1017/CBO9780511566158|author-link1=E. Brian Davies}}</ref><ref>{{cite book|mr=2569498|last1=Grigor'yan|first1=Alexander|title=Heat kernel and analysis on manifolds|series=AMS/IP Studies in Advanced Mathematics|volume=47|publisher=[[American Mathematical Society]]|location=Providence, RI|year=2009|isbn=978-0-8218-4935-4|doi=10.1090/amsip/047}}</ref><ref>{{cite book|mr=1840042|last1=Kigami|first1=Jun|title=Analysis on fractals|series=Cambridge Tracts in Mathematics|volume=143|publisher=[[Cambridge University Press]]|location=Cambridge|year=2001|isbn=0-521-79321-1}}</ref><ref>{{cite book|mr=1817225 |last1=Lieb|first1=Elliott H.|last2=Loss|first2=Michael|title=Analysis|edition=Second edition of 1997 original|series=[[Graduate Studies in Mathematics]]|volume=14|publisher=[[American Mathematical Society]]|location=Providence, RI|year=2001|isbn=0-8218-2783-9|author-link1=Elliott Lieb|author-link2=Michael Loss}}</ref>

Soon after, Nash learned from [[Paul Garabedian]], recently returned from Italy, that the then-unknown [[Ennio De Giorgi]] had found nearly identical results for elliptic partial differential equations.{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 30}} De Giorgi and Nash's methods had little to do with one another, although Nash's were somewhat more powerful in applying to both elliptic and parabolic equations. A few years later, inspired by De Giorgi's method, [[Jürgen Moser]] found a different approach to the same results, and the resulting body of work is now known as the De Giorgi–Nash theorem or the De Giorgi–Nash–Moser theory (which is distinct from the [[Nash–Moser theorem]]). De Giorgi and Moser's methods became particularly influential over the next several years, through their developments in the works of [[Olga Ladyzhenskaya]], [[James Serrin]], and [[Neil Trudinger]], among others.<ref>{{cite book|mr=1814364|last1=Gilbarg|first1=David|last2=Trudinger|first2=Neil S.|title=Elliptic partial differential equations of second order|edition=Reprint of the second|series=Classics in Mathematics|publisher=[[Springer-Verlag]]|location=Berlin|year=2001|isbn=3-540-41160-7|doi=10.1007/978-3-642-61798-0|author-link1=David Gilbarg|author-link2=Neil Trudinger}}</ref><ref>{{cite book|mr=1465184|last1=Lieberman|first1=Gary M.|title=Second order parabolic differential equations|publisher=[[World Scientific|World Scientific Publishing Co., Inc.]]|location=River Edge, NJ|year=1996|isbn=981-02-2883-X|doi=10.1142/3302}}</ref> Their work, based primarily on the judicious choice of [[test function]]s in the [[weak solution|weak formulation]] of partial differential equations, is in strong contrast to Nash's work, which is based on analysis of the [[heat kernel]]. Nash's approach to the De Giorgi–Nash theory was later revisited by [[Eugene Fabes]] and [[Daniel Stroock]], initiating the re-derivation and extension of the results originally obtained from De Giorgi and Moser's techniques.<ref name="davies" /><ref>{{cite journal|mr=0855753|last1=Fabes|first1=E. B.|last2=Stroock|first2=D. W.|title=A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash|journal=[[Archive for Rational Mechanics and Analysis]]|volume=96|year=1986|issue=4|pages=327–338|doi=10.1007/BF00251802 |bibcode=1986ArRMA..96..327F |s2cid=189774501 |author-link2=Daniel Stroock}}</ref>

From the fact that minimizers to many functionals in the [[calculus of variations]] solve elliptic partial differential equations, [[Hilbert's nineteenth problem]] (on the smoothness of these minimizers), conjectured almost sixty years prior, was directly amenable to the De Giorgi–Nash theory. Nash received instant recognition for his work, with [[Peter Lax]] describing it as a "stroke of genius".{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 30}} Nash would later speculate that had it not been for De Giorgi's simultaneous discovery, he would have been a recipient of the prestigious [[Fields Medal]] in 1958.<ref name="Nash1995" /> Although the medal committee's reasoning is not fully known, and was not purely based on questions of mathematical merit,{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 31}} archival research has shown that Nash placed third in the committee's vote for the medal, after the two mathematicians ([[Klaus Roth]] and [[René Thom]]) who were awarded the medal that year.<ref>{{cite journal|last1=Barany|first1=Michael|title=The Fields Medal should return to its roots|journal=[[Nature (journal)|Nature]]|volume=553|date=January 18, 2018|issue=7688 |pages=271–273|doi=10.1038/d41586-018-00513-8|bibcode=2018Natur.553..271B |doi-access=free}}</ref>