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Devaney]]<br />[[John Guckenheimer]]<br />[[Morris Hirsch]]<br />[[Nancy Kopell]]<br />[[Jacob Palis]]<br />[[Themistocles M. Rassias]]<br />[[James Renegar]]<br >[[Siavash Shahshahani]]<br />[[Michael Shub|Mike Shub]] | known_for = [[Generalized Poincaré conjecture]]<br>[[Handle decomposition]]<br>[[h-cobordism|h-cobordism theorem]]<br>[[Homoclinic orbit]]<br>[[Horseshoe map]]<br>[[Smale conjecture]]<br>[[Smale's problems]]<br>[[Morse–Smale system]]<br>[[Morse–Smale diffeomorphism]]<br>[[Palais–Smale compactness condition]]<br>[[Blum–Shub–Smale machine]]<br>[[Smale–Williams attractor]]<br>[[Morse–Palais lemma]]<br>[[Regular homotopy]]<br>[[Sard's theorem]]<br>[[Sphere eversion]]<br>[[Structural stability]]<br>[[Whitehead torsion]]<br>[[Diffeomorphism]] | awards = [[Wolf Prize in Mathematics|Wolf Prize]] (2007)<br />[[National Medal of Science]] (1996)<br />[[Chauvenet Prize]] (1988)<ref name="ChauvenetSmale1988">{{cite journal|author=Smale, Steve|title=On the Efficiency of Algorithms in Analysis|journal=Bulletin of the American Mathematical Society |series=New Series|volume=13|issue=2|year=1985|pages=87–121|url=http://www.maa.org/programs/maa-awards/writing-awards/on-the-efficiency-of-algorithms-in-analysis|doi=10.1090/S0273-0979-1985-15391-1|doi-access=free}}</ref> <br />[[Fields Medal]] (1966)<br />[[Oswald Veblen Prize in Geometry]] (1966)<br /> [[Sloan Fellowship]] (1960) }} '''Stephen Smale''' (born July 15, 1930) is an American mathematician, known for his research in [[topology]], [[dynamical system]]s and [[mathematical economics]]. He was awarded the [[Fields Medal]] in 1966<ref>{{cite news|url=https://www.nytimes.com/2014/08/10/opinion/sunday/how-math-got-its-nobel-.html?_r=0|title=How Math Got Its 'Nobel'|date=8 August 2014|newspaper=The New York Times|access-date=21 October 2016}}</ref> and spent more than three decades on the mathematics faculty of the [[University of California, Berkeley]] (1960–1961 and 1964–1995), where he currently is Professor Emeritus, with research interests in [[algorithms]], [[numerical analysis]] and [[global analysis]].<ref>{{cite web|url= https://math.berkeley.edu/people/faculty/stephen-smale|title= Stephen Smale|author= |website= University of California, Berkeley|access-date= 27 November 2021}}</ref> ==Education and career== Smale was born in [[Flint, Michigan]] and entered the [[University of Michigan]] in 1948.<ref>{{Cite book|url=https://books.google.com/books?id=HcmcAAAAQBAJ&pg=PA225|title=The Britannica Guide to Geometry|date=2010|publisher=Britannica Educational Publishing|isbn=9781615302178|editor=William L. Hosch|page=225}}</ref><ref>{{Cite book|url=https://books.google.com/books?id=C1cLnAm7gcwC&pg=PA11|title=Steven Smale: The Mathematician Who Broke the Dimension Barrier|last=Batterson|first=Steve|date=2000|publisher=American Mathematical Soc.|isbn=9780821826966|page=11}}</ref> Initially, he was a good student, placing into an honors [[calculus]] sequence taught by [[Robert M. Thrall|Bob Thrall]] and earning himself A's. However, his [[Sophomore year|sophomore]] and junior years were marred with mediocre grades, mostly Bs, Cs and even an F in [[nuclear physics]]. Smale obtained his Bachelor of Science degree in 1952. Despite his grades, with some luck, Smale was accepted as a graduate student at the University of Michigan's mathematics department. Yet again, Smale performed poorly in his first years, earning a C average as a graduate student. When the department chair, [[Theophil Henry Hildebrandt|Hildebrandt]], threatened to kick Smale out, he began to take his studies more seriously.<ref>{{YouTube|LmpPUjOeMGI}}</ref> Smale finally earned his [[PhD]] in 1957, under [[Raoul Bott]], beginning his career as an instructor at the [[University of Chicago]]. Early in his career, Smale was involved in controversy over remarks he made regarding his work habits while proving the higher-dimensional Poincaré conjecture. He said that his best work had been done "on the beaches of Rio."<ref>He discovered the famous [[Smale horseshoe map]] on a beach in [[Leme (Rio de Janeiro)|Leme]], [[Rio de Janeiro]]. See: S.&nbsp;Smale (1996), [http://math.berkeley.edu/~smale/biblio/chaos.ps Chaos: Finding a Horseshoe on the Beaches of Rio].</ref><ref>{{Cite journal |author= CS Aravinda |year= 2018 |title= ICM 2018: On the beaches of Rio de Janeiro |url= https://bhavana.org.in/icm-2018-beaches-rio-de-janeiro |journal= Bhāvanā |volume= 2 |issue= 3 |access-date= 8 October 2022 }}</ref> He has been politically active in various movements in the past, such as the [[Free Speech movement]] and member of the [[Fair Play for Cuba Committee]].<ref>{{cite book |last1=Schrecker |first1=Ellen |title=The Lost Promise: American Universities in the 1960s |date=2021 |publisher=University of Chicago Press |page=121}}</ref> In 1966, having travelled to Moscow under an [[National Science Foundation|NSF]] grant to accept the Fields Medal, he held a press conference there to denounce the [[Vietnam War|American position in Vietnam]], [[Hungary–Soviet Union relations|Soviet intervention in Hungary]] and Soviet maltreatment of intellectuals. After his return to the US, he was unable to renew the grant.<ref>{{cite news | newspaper = The Harvard Crimson | url = https://www.thecrimson.com/article/1967/10/5/math-professors-question-denial-of-smale/ | title = Math Professors Question Denial Of Smale Grant | date = 5 October 1967 | author = Andrew Jamison | access-date = 13 February 2022}}</ref> At one time he was [[subpoena]]ed<ref name="Greenberg pp. 130–133">{{cite journal | last=Greenberg | first=D. S. | title=The Smale Case: NSF and Berkeley Pass Through a Case of Jitters | journal=Science | publisher=American Association for the Advancement of Science (AAAS) | volume=154 | issue=3745 | date=1966-10-07 | issn=0036-8075 | doi=10.1126/science.154.3745.130 | pages=130–133| pmid=17740098 | bibcode=1966Sci...154..130G }}</ref> by the [[House Un-American Activities Committee]]. In 1960, Smale received a [[Sloan Research Fellowship]] and was appointed to the [[University of California, Berkeley|Berkeley]] mathematics faculty, moving to a professorship at [[Columbia University|Columbia]] the following year. In 1964 he returned to a professorship at Berkeley, where he has spent the main part of his career. He became a professor emeritus at Berkeley in 1995 and took up a post as professor at the [[City University of Hong Kong]]. He also amassed over the years one of the finest private mineral collections in existence. Many of Smale's mineral specimens can be seen in the book ''The Smale Collection: Beauty in Natural Crystals''.<ref>{{cite web|url=http://www.lithographie.org/bookshop/the_smale_collection.htm|title=Lithographie LTD|website=www.lithographie.org}}</ref> From 2003 to 2012, Smale was a professor at the [[Toyota Technological Institute at Chicago]];<ref>{{cite web|url=https://www.ttic.edu/faculty-alumni/|title=Faculty Alumni|website=ttic.edu}}</ref> starting August 1, 2009, he became a Distinguished University Professor at the [[City University of Hong Kong]].<ref>[http://ttic.uchicago.edu/~smale/vita.html Stephen Smale Vita.] Accessed November 18, 2009.</ref> In 1988, Smale was the recipient of the [[Chauvenet Prize]]<ref name="ChauvenetSmale1988"/> of the [[Mathematical Association of America|MAA]]. In 2007, Smale was awarded the [[Wolf Prize]] in mathematics.<ref>{{cite web|url=http://www.huji.ac.il/cgi-bin/dovrut/dovrut_search_eng_dev.pl?mesge116895485932688760|title=The Hebrew University of Jerusalem - Division of Marketing & Communication|website=www.huji.ac.il|access-date=2007-02-04|archive-date=2016-03-03|archive-url=https://web.archive.org/web/20160303194235/http://www.huji.ac.il/cgi-bin/dovrut/dovrut_search_eng_dev.pl?mesge116895485932688760|url-status=dead}}</ref> ==Research== Smale proved that the [[diffeomorphism group|oriented diffeomorphism group]] of the two-dimensional sphere has the same [[homotopy type]] as the [[special orthogonal group]] of {{math|3 × 3}} matrices.{{sfnm|1a1=Smale|1y=1959c}} Smale's theorem has been reproved and extended a few times, notably to higher dimensions in the form of the [[Smale conjecture]],<ref>{{cite journal|mr=0701256|last1=Hatcher|first1=Allen E.|author-link1=Allen Hatcher|title=A proof of the Smale conjecture, Diff(''S''<sup>3</sup>) ≃ O(4)|journal=[[Annals of Mathematics]]|series=Second Series|volume=117|year=1983|issue=3|pages=553–607|doi=10.2307/2007035|jstor=2007035 |zbl=0531.57028}}</ref> as well as to other topological types.<ref>{{cite journal|mr=0276999|last1=Earle|first1=Clifford J.|last2=Eells|first2=James|title=A fibre bundle description of Teichmüller theory|journal=[[Journal of Differential Geometry]]|volume=3|year=1969|issue=1–2|pages=19–43|doi=10.4310/jdg/1214428816|doi-access=free|author-link2=James Eells|author-link1=Clifford John Earle Jr.|zbl=0185.32901}}</ref> In another early work, he studied the [[Immersion (mathematics)|immersion]]s of the two-dimensional sphere into Euclidean space.{{sfnm|1a1=Smale|1y=1959a}} By relating immersion theory to the [[algebraic topology]] of [[Stiefel manifold]]s, he was able to fully clarify when two immersions can be deformed into one another through a family of immersions. Directly from his results it followed that the standard immersion of the sphere into three-dimensional space can be deformed (through immersions) into its negation, which is now known as [[sphere eversion]]. He also extended his results to higher-dimensional spheres,{{sfnm|1a1=Smale|1y=1959b}} and his doctoral student [[Morris Hirsch]] extended his work to immersions of general [[smooth manifold]]s.<ref>{{cite journal|mr=0119214|last1=Hirsch|first1=Morris W.|author-link1=Morris Hirsch|title=Immersions of manifolds|journal=[[Transactions of the American Mathematical Society]]|volume=93|year=1959|pages=242–276|zbl=0113.17202|issue=2|doi=10.1090/S0002-9947-1959-0119214-4|doi-access=free}}</ref> Along with [[John Forbes Nash|John Nash]]'s work on [[isometric immersion]]s, the Hirsch–Smale immersion theory was highly influential in [[Mikhael Gromov (mathematician)|Mikhael Gromov]]'s early work on development of the [[h-principle]], which abstracted and applied their ideas to contexts other than that of immersions.<ref>{{cite book|last1=Gromov|first1=Mikhael|title=Partial differential relations|series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete|Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge]]|volume=9|publisher=[[Springer-Verlag]]|location=Berlin|year=1986|isbn=3-540-12177-3|mr=0864505|doi=10.1007/978-3-662-02267-2|zbl=0651.53001|author-link1=Mikhael Gromov (mathematician)}}</ref> In the study of [[dynamical system]]s, Smale introduced what is now known as a [[Morse–Smale system]].{{sfnm|1a1=Smale|1y=1960}} For these dynamical systems, Smale was able to prove [[Morse inequality|Morse inequalities]] relating the [[cohomology]] of the underlying space to the dimensions of the [[stable manifold|(un)stable manifold]]s. Part of the significance of these results is from Smale's theorem asserting that the [[gradient flow]] of any [[Morse function]] can be arbitrarily well approximated by a Morse–Smale system without closed orbits.{{sfnm|1a1=Smale|1y=1961a}} Using these tools, Smale was able to construct ''self-indexing'' Morse functions, where the value of the function equals its [[Morse index]] at any critical point.<ref name="milnor">{{cite book|mr=0190942|last1=Milnor|first1=John|title=Lectures on the h-cobordism theorem|others=Notes by [[Laurent C. Siebenmann|L. Siebenmann]] and J. Sondow|author-link1=John Milnor|publisher=[[Princeton University Press]]|location=Princeton, NJ|year=1965|zbl=0161.20302|doi=10.1515/9781400878055|isbn=9781400878055 }}</ref> Using these self-indexing Morse functions as a key tool, Smale resolved the [[generalized Poincaré conjecture]] in every dimension greater than four.{{sfnm|1a1=Smale|1y=1961b}} Building on these works, he also established the more powerful [[h-cobordism|h-cobordism theorem]] the following year, together with the full classification of [[simply-connected]] smooth five-dimensional manifolds.{{sfnm|1a1=Smale|1y=1962a|2a1=Smale|2y=1962b}}<ref name="milnor" /> Smale also introduced the [[horseshoe map]], inspiring much subsequent research. He also outlined a research program carried out by many others. Smale is also known for injecting [[Morse theory]] into mathematical [[economics]], as well as recent explorations of various theories of [[computation]]. In 1998 he compiled a list of 18 problems in [[mathematics]] to be solved in the 21st century, known as [[Smale's problems]].{{sfnm|1a1=Smale|1y=1998|2a1=Smale|2y=2000}} This list was compiled in the spirit of [[David Hilbert|Hilbert]]'s [[Hilbert's problems|famous list of problems]] produced in 1900. In fact, Smale's list contains some of the original Hilbert problems, including the [[Riemann hypothesis]] and the second half of [[Hilbert's sixteenth problem]], both of which are still unsolved. Other famous problems on his list include the [[Poincaré conjecture]] (now a theorem, proved by [[Grigori Perelman]]), the [[P = NP problem]], and the [[Navier–Stokes existence and smoothness|Navier–Stokes equations]], all of which have been designated [[Millennium Prize Problems]] by the [[Clay Mathematics Institute]]. == Books == * {{cite book|mr=0607330|last1=Smale|first1=Steve|title=The mathematics of time: essays on dynamical systems, economic processes, and related topics|publisher=[[Springer-Verlag]]|location=New York-Berlin|year=1980|isbn=0-387-90519-7|zbl=0451.58001|doi=10.1007/978-1-4613-8101-3}} * {{cite book|mr=1479636|last1=Blum|first1=Lenore|last2=Cucker|first2=Felipe|last3=Shub|first3=Michael|last4=Smale|first4=Steve|title=Complexity and real computation|others=With a foreword by [[Richard M. Karp]]|publisher=[[Springer-Verlag]]|location=New York|year=1998|isbn=0-387-98281-7|doi=10.1007/978-1-4612-0701-6|author-link1=Lenore Blum|author-link2=Felipe Cucker|author-link3=Michael Shub|zbl=0948.68068|s2cid=12510680 }} * {{cite book|mr=3293130|last1=Hirsch|first1=Morris W.|last2=Smale|first2=Stephen|last3=Devaney|first3=Robert L.|title=Differential equations, dynamical systems, and an introduction to chaos|edition=Third edition of 1974 original|publisher=[[Academic Press]]|location=Amsterdam|year=2013|isbn=978-0-12-382010-5|zbl=1239.37001|doi=10.1016/C2009-0-61160-0|author-link3=Robert L. Devaney|author-link1=Morris Hirsch}} * {{cite encyclopedia|title=The collected papers of Stephen Smale|others=In three volumes|editor-first1=F.|editor-last1=Cucker|editor-first2=R.|editor-last2=Wong|publisher=[[Singapore University Press]]|location=Singapore|year=2000|isbn=981-02-4307-3|mr=1781696|zbl=0995.01005|doi=10.1142/4424|editor-link1=Felipe Cucker}} == Important publications == {{refbegin}} * {{cite journal|mr=0104227|last1=Smale|first1=Stephen|title=A classification of immersions of the two-sphere|journal=[[Transactions of the American Mathematical Society]]|volume=90|issue=2|year=1959a|pages=281–290|doi-access=free|doi=10.1090/S0002-9947-1959-0104227-9|zbl=0089.18102}} * {{cite journal|mr=0105117|last1=Smale|first1=Stephen|title=The classification of immersions of spheres in Euclidean spaces|journal=[[Annals of Mathematics]]|series=Second Series|volume=69|year=1959b|pages=327–344|doi=10.2307/1970186|zbl=0089.18201|issue=2|jstor=1970186 }} * {{cite journal|mr=0112149|last1=Smale|first1=Stephen|title=Diffeomorphisms of the 2-sphere|journal=[[Proceedings of the American Mathematical Society]]|volume=10|year=1959c|pages=621–626|doi=10.1090/S0002-9939-1959-0112149-8|doi-access=free|zbl=0118.39103|issue=4}} * {{cite journal|mr=0117745|last1=Smale|first1=Stephen|title=Morse inequalities for a dynamical system|journal=[[Bulletin of the American Mathematical Society]]|volume=66|issue=1|year=1960|pages=43–49|doi=10.1090/S0002-9904-1960-10386-2|doi-access=free|zbl=0100.29701}} * {{cite journal|mr=0133139|last1=Smale|first1=Stephen|title=On gradient dynamical systems|journal=[[Annals of Mathematics]]|series=Second Series|volume=74|year=1961a|pages=199–206|issue=1|doi=10.2307/1970311|jstor=1970311 |zbl=0136.43702}} * {{cite journal | first=Stephen | last=Smale | title=Generalized Poincaré's conjecture in dimensions greater than four | journal=[[Annals of Mathematics]]|series=Second Series | volume=74 | year=1961b | pages=391–406 | mr=0137124 | doi=10.2307/1970239 | issue=2| jstor=1970239 | zbl=0099.39202 }} * {{cite journal|mr=0153022|last1=Smale|first1=S.|title=On the structure of manifolds|journal=[[American Journal of Mathematics]]|volume=84|year=1962a|pages=387–399|doi=10.2307/2372978|issue=3|jstor=2372978 |zbl=0109.41103}} * {{cite journal|mr=0141133|last1=Smale|first1=Stephen|title=On the structure of 5-manifolds|journal=[[Annals of Mathematics]]|series=Second Series|volume=75|year=1962b|pages=38–46|doi=10.2307/1970417|issue=1|jstor=1970417 |zbl=0101.16103}} *{{cite journal |last1=Smale |first1=S. |title=An infinite dimensional version of Sard's theorem |journal=Amer. J. Math. |date=1965 |volume=87 |issue=4 |pages=861–866|doi=10.2307/2373250 |jstor=2373250 }} * {{cite journal | first=Stephen | last=Smale | title=Differentiable dynamical systems | mr=0228014 | journal=[[Bulletin of the American Mathematical Society]] | volume=73 | year=1967 | issue=6 | pages=747–817 | doi=10.1090/S0002-9904-1967-11798-1|doi-access=free}} *{{cite journal |last1=Blum |first1=Lenore |last2=Shub |first2=Mike |last3=Smale |first3=Steve |title=On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines |journal=Bull. Amer. Math. Soc. |series=New Series |date=1989 |volume=21 |issue=1 |pages=1–46|doi=10.1090/S0273-0979-1989-15750-9 |doi-access=free }} *{{cite journal| first1=Michael| last1= Shub| first2=Stephen| last2= Smale| authorlink2=Stephen Smale| jstor=2152805| title= Complexity of Bézout's Theorem I: Geometric Aspects| journal= [[Journal of the American Mathematical Society]] | publisher=[[American Mathematical Society]]| location=Providence, Rhode Island| volume =6| issue= 2| date= 1993| pages=459–501| doi=10.2307/2152805}} * {{cite journal|mr=1631413|last1=Smale|first1=Steve|title=Mathematical problems for the next century|journal=[[The Mathematical Intelligencer]]|volume=20|year=1998|issue=2|pages=7–15|doi=10.1007/BF03025291|zbl=0947.01011|s2cid=1331144 }} *{{cite conference|mr=1754783|last1=Smale|first1=Steve|title=Mathematical problems for the next century|book-title=Mathematics: frontiers and perspectives|pages=271–294|publisher=[[American Mathematical Society]]|location=Providence, RI|year=2000|zbl=1031.00005|isbn=0-8218-2070-2|editor-first1=V.|editor-last1=Arnold|editor-first2=M.|editor-last2=Atiyah|editor-first3=P.|editor-last3=Lax|editor-first4=B.|editor-last4=Mazur|editor-link1=Vladimir Arnold|editor-link2=Michael Atiyah|editor-link3=Peter Lax|editor-link4=Barry Mazur}} *{{cite journal |last1=Cucker |first1=Felipe |last2=Smale |first2=Steve |title=On the mathematical foundations of learning |journal=Bull. Amer. Math. Soc. |series=New Series |date=2002 |volume=39 |issue=1 |pages=1–49|doi=10.1090/S0273-0979-01-00923-5 |doi-access=free |url=http://math.bu.edu/people/mkon/ma717/cuckersmale.pdf }} * {{cite journal |last1=Cucker |first1=Felipe |last2=Smale |first2=Steve |title=Emergent behavior in flocks |journal=IEEE Trans. Autom. Control |date=2007 |volume=52 |issue=5 |pages=852–862|doi=10.1109/TAC.2007.895842 |s2cid=206590734 }}* {{refend}} ==See also== *[[5-manifold]] *[[Axiom A]] *[[Geometric mechanics]] *[[Homotopy principle]] *[[Mean value problem]] == References == {{Reflist}} == External links == * {{cite web|url=https://scholar.google.com/citations?user=rKyTmpkAAAAJ&hl=en&oi=ao=|title=Stephen Smale|website=[[Google Scholar]]}} * {{MathGenealogy |id=5086}} * {{MacTutor Biography|id=Smale}} * {{MathWorld|urlname=SmalesProblems|title=Smale's Problems}} * [[Robion Kirby]], ''[http://www.ams.org/notices/200011/rev-kirby.pdf Stephen Smale: The Mathematician Who Broke the Dimension Barrier]'', a book review of a biography in the Notices of the [[American Mathematical Society|AMS]]. ;Personal websites at universities * [http://www.ee.cityu.edu.hk/~cccn/smale.htm Steven Smale] at the [[City University of Hong Kong]] * [http://ttic.uchicago.edu/~smale/vita.html Stephen Smale] at the [[University of Chicago]] * [http://math.berkeley.edu/~smale/ Steve Smale] at the [[University of California, Berkeley]] {{Fields medalists}} {{Wolf Prize in Mathematics}} {{Winners of the National Medal of Science|math-stat-comp}} {{Chauvenet Prize recipients}} {{Veblen Prize recipients}} {{John von Neumann Lecturers}} {{Authority control}} {{DEFAULTSORT:Smale, Stephen}} [[Category:1930 births]] [[Category:Living people]] [[Category:20th-century American mathematicians]] [[Category:21st-century American mathematicians]] [[Category:American atheists]] [[Category:American computer scientists]] [[Category:Columbia University faculty]] [[Category:Dynamical systems theorists]] [[Category:Fields Medalists]] [[Category:General equilibrium theorists]] [[Category:Institute for Advanced Study visiting scholars]] [[Category:Mathematical economists]] [[Category:Members of the Brazilian Academy of Sciences]] [[Category:Members of the United States National Academy of Sciences]] [[Category:National Medal of Science laureates]] [[Category:Numerical analysts]] [[Category:People from Flint, Michigan]] [[Category:Recipients of the Great Cross of the National Order of Scientific Merit (Brazil)]] [[Category:American theoretical computer scientists]] [[Category:American topologists]] [[Category:University of California, Berkeley College of Letters and Science faculty]] [[Category:University of Chicago faculty]] [[Category:University of Michigan alumni]] [[Category:Wolf Prize in Mathematics laureates]] [[Category:Sloan Research Fellows]] [[Category:Fellows of the Econometric Society]] [[Category:Mathematicians from Michigan]] </textarea><div class="templatesUsed"><div class="mw-templatesUsedExplanation"><p><span id="templatesused">Pages transcluded onto the current version of this page<span class="posteditwindowhelplinks"> (<a href="/wiki/Help:Transclusion" title="Help:Transclusion">help</a>)</span>:</span> </p></div><ul> <li><a href="/wiki/Template:Authority_control" title="Template:Authority 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