Editing John Forbes Nash Jr. (section)

===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.}}