View source for Biography:Stephen Smale

{{short description|American mathematician (born 1930)}}
{{Infobox scientist
| image             = Stephen Smale2.jpg
| image_size        = 
| name              = Stephen Smale
| caption           = Smale in 2008
| birth_name        = Stephen Smale
| birth_date        = {{birth date and age|1930|07|15|mf=yes}}
| birth_place       = Flint, Michigan, U.S.
| death_date        = 
| death_place       = 
| nationality       = {{wipe|United States}}
| field             = [[Mathematics]]
| work_institution  = [[Organization:Toyota Technological Institute at Chicago|Toyota Technological Institute at Chicago]]<br />City University of Hong Kong<br />[[Organization:University of Chicago|University of Chicago]]<br />[[Organization:Columbia University|Columbia University]]<br />[[Organization:University of California, Berkeley|University of California, Berkeley]]
| education         = [[Organization:University of Michigan|University of Michigan]] (BS, [[Social:Doctor of Philosophy|PhD]])
| doctoral_advisor  = [[Biography:Raoul Bott|Raoul Bott]]
| thesis_year       = 1957
| thesis_url        = https://www.proquest.com/docview/301954268?pq-origsite=gscholar&fromopenview=true
| thesis_title      = Regular Curves on Riemannian Manifolds
| doctoral_students = Rufus Bowen<br />César Camacho<br />Robert L. Devaney<br />John Guckenheimer<br />[[Biography:Morris Hirsch|Morris Hirsch]]<br />Nancy Kopell<br />[[Biography:Jacob Palis|Jacob Palis]]<br />[[Biography:Themistocles M. Rassias|Themistocles M. Rassias]]<br />[[Biography:James Renegar|James Renegar]]<br >Siavash Shahshahani<br />Mike Shub
| known_for         = [[Generalized Poincaré conjecture]]<br>[[Handle decomposition]]<br>[[Homoclinic orbit]]<br>Smale's horseshoe<br>Smale's theorem<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 (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|topology]], [[Dynamical system|dynamical system]]s and [[Finance:Mathematical economics|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 [[Organization:University of California, Berkeley|University of California, Berkeley]] (1960–1961 and 1964–1995), where he currently is Professor Emeritus, with research interests in algorithms, [[Numerical analysis|numerical analysis]] and [[Global analysis|global analysis]].<ref>{{cite web|url= https://math.berkeley.edu/people/faculty/stephen-smale|title= Stephen Smale|author=<!--Not stated--> |website= University of California, Berkeley|access-date= 27 November 2021}}</ref>

==Education and career==
Smale was born in Flint, Michigan and entered the [[Organization:University of Michigan|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|calculus]] sequence taught by [[Biography:Robert M. Thrall|Bob Thrall]] and earning himself A's.  However, his sophomore and junior years were marred with mediocre grades, mostly Bs, Cs and even an F in [[Physics:Nuclear physics|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, [[Biography: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 [[Biography:Raoul Bott|Raoul Bott]], beginning his career as an instructor at the [[Organization:University of Chicago|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.  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. In 1966, having travelled to Moscow under an [[Organization:National Science Foundation|NSF]] grant to accept the Fields Medal, he held a press conference there to denounce the American position in Vietnam, 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 subpoenaed<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 [[Organization:University of California, Berkeley|Berkeley]] mathematics faculty, moving to a professorship at [[Organization: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 [[Organization:Toyota Technological Institute at Chicago|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 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 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.|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|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|sphere eversion]]. He also extended his results to higher-dimensional spheres,{{sfnm|1a1=Smale|1y=1959b}} and his doctoral student [[Biography:Morris Hirsch|Morris Hirsch]] extended his work to immersions of general smooth manifolds.<ref>{{cite journal|mr=0119214|last1=Hirsch|first1=Morris W.|title=Immersions of manifolds|journal=[[Organization:Transactions of the American Mathematical Society|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 Nash's work on isometric immersions, the Hirsch–Smale immersion theory was highly influential in [[Biography: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, 3. Folge|volume=9|publisher=[[Physics:Springer-Verlag|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|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 inequalities relating the [[Cohomology|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|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 [[Biography:Laurent C. Siebenmann|L. Siebenmann]] and J. Sondow|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|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 identified the [[Horseshoe map|Smale horseshoe]], 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 [[Finance:Economics|economics]], as well as recent explorations of various theories of [[Computation|computation]].

In 1998 he compiled a list of 18 problems in [[Mathematics|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 [[Biography:David Hilbert|Hilbert]]'s 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 [[Biography:Grigori Perelman|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=[[Physics:Springer-Verlag|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 [[Biography:Richard M. Karp|Richard M. Karp]]|publisher=[[Physics:Springer-Verlag|Springer-Verlag]]|location=New York|year=1998|isbn=0-387-98281-7|doi=10.1007/978-1-4612-0701-6|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=[[Company:Academic Press|Academic Press]]|location=Amsterdam|year=2013|isbn=978-0-12-382010-5|zbl=1239.37001|doi=10.1016/C2009-0-61160-0|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|}}

== Important publications ==
{{refbegin}}
* {{cite journal|mr=0104227|last1=Smale|first1=Stephen|title=A classification of immersions of the two-sphere|journal=[[Organization:Transactions of the American Mathematical Society|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=[[Organization:Proceedings of the American Mathematical Society|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=[[Organization:Bulletin of the American Mathematical Society|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=[[Organization:Bulletin of the American Mathematical Society|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|  jstor=2152805| title= Complexity of Bézout's Theorem I: Geometric Aspects| journal= [[Organization:Journal of the American Mathematical Society|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-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|5-manifold]]
*[[Axiom A]]
*[[Physics:Geometric mechanics|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}}
* [[Biography:Robion Kirby|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 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 [[Organization:University of Chicago|University of Chicago]]
* [http://math.berkeley.edu/~smale/ Steve Smale] at the [[Organization:University of California, Berkeley|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}}

{{DEFAULTSORT:Smale, Stephen}}
[[Category:American computer scientists]]
[[Category:General equilibrium theorists]]
[[Category:Mathematical economists]]
[[Category:Numerical analysts]]
[[Category:Theoretical computer scientists]]
[[Category:Topologists]]

{{Sourceattribution|Stephen Smale}}