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Stephen Smale

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Title: Stephen Smale  
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Subject: Themistocles M. Rassias, Fields Medal, Wolf Prize in Mathematics, John Milnor, John G. Thompson
Collection: 1930 Births, 20Th-Century American Mathematicians, 21St-Century American Mathematicians, American Atheists, American Computer Scientists, Columbia University Faculty, Dynamical Systems Theorists, Fields Medalists, General Equilibrium Theorists, Living People, Mathematical Economists, Members of the Brazilian Academy of Sciences, Members of the United States National Academy of Sciences, National Medal of Science Laureates, Numerical Analysts, People from Flint, Michigan, Recipients of the Great Cross of the National Order of Scientific Merit (Brazil), Theoretical Computer Scientists, Topologists, University of California, Berkeley Faculty, University of Chicago Faculty, University of Michigan Alumni, Wolf Prize in Mathematics Laureates
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Stephen Smale

Stephen Smale
Born (1930-07-15) July 15, 1930
Flint, Michigan
Nationality American
Fields Mathematics
Institutions Toyota Technological Institute at Chicago
City University of Hong Kong
University of Chicago
Columbia University
University of California, Berkeley
Alma mater University of Michigan
Doctoral advisor Raoul Bott
Doctoral students Rufus Bowen
John Guckenheimer
Morris Hirsch
Nancy Kopell
Jacob Palis
Themistocles M. Rassias
Siavash Shahshahani
Known for Generalized Poincaré conjecture
Handle decomposition
Homoclinic orbit
Smale's horseshoe
Smale's paradox
Smale's theorem
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
Structural stability
Whitehead torsion
Notable awards Wolf Prize (2007)
National Medal of Science (1996)
Chauvenet Prize (1988)
Fields Medal (1966)
Oswald Veblen Prize in Geometry (1966)
Sloan Fellowship (1960)

Stephen Smale (born July 15, 1930) is an American mathematician from Flint, Michigan. He was awarded the Fields Medal in 1966, and spent more than three decades on the mathematics faculty of the University of California, Berkeley (1960–61 and 1964–1995).


  • Education and career 1
  • Important publications 2
  • References 3
  • External links 4

Education and career

Smale entered the University of Michigan in 1948. 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. However, 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. It was only when the department chair, Hildebrandt, threatened to kick out Smale, that he began to work hard.[1] Smale finally earned his Ph.D. in 1957, under Raoul Bott.

Smale began his career as an instructor at the college at the University of Chicago. In 1958, he astounded the mathematical world with a proof of a sphere eversion. He then cemented his reputation with a proof of the Poincaré conjecture for all dimensions greater than or equal to 5, published in 1961; in 1962 he generalized the ideas in a 107 page paper that established the h-cobordism theorem.

After having made great strides in topology, he then turned to the study of dynamical systems, where he made significant advances as well. His first contribution is the Smale horseshoe that jumpstarted significant research in dynamical systems. 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. 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, the P = NP problem, and the Navier–Stokes equations, all of which have been designated Millennium Prize Problems by the Clay Mathematics Institute.

Earlier 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".[2] This led to the withholding of his grant money from the NSF. He has been politically active in various movements in the past, such as the Free Speech movement. At one time he was subpoenaed by the House Un-American Activities Committee.

In 1960 Smale was appointed an associate professor of mathematics at the University of California, Berkeley, moving to a professorship at Columbia University the following year. In 1964 he returned to a professorship at UC Berkeley where he has spent the main part of his career. He retired from UC 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.[3]

Since 2002 Smale is a Professor at the Toyota Technological Institute at Chicago; starting August 1, 2009, he is also a Distinguished University Professor at the City University of Hong Kong.[4]

In 2007, Smale was awarded the Wolf Prize in mathematics.[5]

Important publications

  • S. Smale, Generalized Poincaré's conjecture in dimensions greater than four, Annals of Mathematics, 2nd Ser., 74 (1961), no. 2, 391 – 406. (via JSTOR)
  • S. Smale, Differentiable dynamical systems, Bulletin of the American Mathematical Society, 73 (1967), 747 – 817. ([1])
  • F. Cucker & R Wong, The Collected Papers of Stephen Smale, ISBN 978-981-02-4307-4
  • L. Blum, F. Cucker, M. Shub and S.Smale, Complexity and Real Computation, ISBN 0-387-98281-7.


  1. ^ Video on YouTube
  2. ^ He discovered the famous Smale horseshoe map on a beach in Leme, Rio de Janeiro. See: S. Smale (1996), Chaos: Finding a Horseshoe on the Beaches of Rio.
  3. ^
  4. ^ Stephen Smale Vita. Accessed November 18, 2009.
  5. ^ Press release

External links

Personal websites at universities
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