Stephen Smale

{{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 = American

| field = Mathematics

| work_institution = Toyota Technological Institute at Chicago
City University of Hong Kong
University of Chicago
Columbia University
University of California, Berkeley

| education = University of Michigan (BS, PhD)

| doctoral_advisor = 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
César Camacho
Robert L. Devaney
John Guckenheimer
Morris Hirsch
Nancy Kopell
Jacob Palis
Themistocles M. Rassias
James Renegar
Siavash Shahshahani
Mike Shub

| known_for = Generalized Poincaré conjecture
Handle decomposition
h-cobordism theorem
Homoclinic orbit
Horseshoe map
Smale conjecture
Smale's problems
Morse–Smale system
Morse–Smale diffeomorphism
Palais–Smale compactness condition
Blum–Shub–Smale machine
Smale–Williams attractor
Morse–Palais lemma
Regular homotopy
Sard's theorem
Sphere eversion
Structural stability
Whitehead torsion
Diffeomorphism

| awards = Wolf Prize (2007)
National Medal of Science (1996)
Chauvenet Prize (1988){{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}}
Fields Medal (1966)
Oswald Veblen Prize in Geometry (1966)
Sloan Fellowship (1960)

}}

Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966{{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}} 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.{{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}}

Education and career

Smale was born in Flint, Michigan and entered the University of Michigan in 1948.{{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}}{{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}} Initially, he was a good student, placing into an honors calculus sequence taught by 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 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, Hildebrandt, threatened to kick Smale out, he began to take his studies more seriously.{{YouTube|LmpPUjOeMGI}} 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."He discovered the famous Smale horseshoe map on a beach in Leme, Rio de Janeiro. See: S. Smale (1996), [http://math.berkeley.edu/~smale/biblio/chaos.ps Chaos: Finding a Horseshoe on the Beaches of Rio].{{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 }} 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.{{cite book |last1=Schrecker |first1=Ellen |title=The Lost Promise: American Universities in the 1960s |date=2021 |publisher=University of Chicago Press |page=121}} In 1966, having travelled to Moscow under an 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.{{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}} At one time he was subpoenaed{{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 }} by the House Un-American Activities Committee.

In 1960, Smale received a Sloan Research Fellowship and was appointed to the Berkeley mathematics faculty, moving to a professorship at 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.{{cite web|url=http://www.lithographie.org/bookshop/the_smale_collection.htm|title=Lithographie LTD|website=www.lithographie.org}}

From 2003 to 2012, Smale was a professor at the Toyota Technological Institute at Chicago;{{cite web|url=https://www.ttic.edu/faculty-alumni/|title=Faculty Alumni|website=ttic.edu}} starting August 1, 2009, he became a Distinguished University Professor at the City University of Hong Kong.[http://ttic.uchicago.edu/~smale/vita.html Stephen Smale Vita.] Accessed November 18, 2009.

In 1988, Smale was the recipient of the Chauvenet Prize of the MAA. In 2007, Smale was awarded the Wolf Prize in mathematics.{{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}}

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,{{cite journal|mr=0701256|last1=Hatcher|first1=Allen E.|author-link1=Allen Hatcher|title=A proof of the Smale conjecture, Diff(S³) ≃ 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}} as well as to other topological types.{{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}}

In another early work, he studied the immersions of the two-dimensional sphere into Euclidean space.{{sfnm|1a1=Smale|1y=1959a}} By relating immersion theory to the algebraic topology of Stiefel manifolds, 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 manifolds.{{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}} Along with John Nash's work on isometric immersions, the Hirsch–Smale immersion theory was highly influential in Mikhael Gromov's early work on development of the h-principle, which abstracted and applied their ideas to contexts other than that of immersions.{{cite book|last1=Gromov|first1=Mikhael|title=Partial differential relations|series=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)}}

In the study of dynamical systems, 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 of the underlying space to the dimensions of the (un)stable manifolds. 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.{{cite book|mr=0190942|last1=Milnor|first1=John|title=Lectures on the h-cobordism theorem|others=Notes by 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 }} 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 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}}

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 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 Grigori Perelman), the P = NP problem, and the 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

{{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 }}*

References

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 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