Stephen Smale

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Stephen Smale
Stephen Smale2.jpg
Born (1930-07-15) July 15, 1930 (age 93)
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
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
Homoclinic orbit
Smale's horseshoe
Smale's theorem
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
Notable awards Wolf Prize (2007)
National Medal of Science (1996)
Chauvenet Prize (1988)[1]
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[2] and spent more than three decades on the mathematics faculty of the University of California, Berkeley (1960–1961 and 1964–1995).

Education and career

Smale was born in Flint, Michigan and entered the University of Michigan in 1948.[3][4] 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. When the department chair, Hildebrandt, threatened to kick Smale out, he began to take his studies more seriously.[5] Smale finally earned his Ph.D. in 1957, under Raoul Bott.

Smale began his career as an instructor 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 started 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 (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.

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."[6] He has been politically active in various movements in the past, such as the Free Speech movement and the movement against the Vietnam War. 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.[7]

Since 2002 Smale has been 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.[8]

In 2007, Smale was awarded the Wolf Prize in mathematics.[9]


  • Lenore Blum, Felipe Cucker, Michael Shub, and Steve Smale. Complexity and real computation. With a foreword by Richard M. Karp. Springer-Verlag, New York, 1998. xvi+453 pp. ISBN 0-387-98281-7
  • Morris W. Hirsch, Stephen Smale, and Robert L. Devaney. Differential equations, dynamical systems, and an introduction to chaos. Third edition. Elsevier/Academic Press, Amsterdam, 2013. xiv+418 pp. ISBN 978-0-12-382010-5
  • The collected papers of Stephen Smale. Vol. 1–3. Edited by F. Cucker and R. Wong. Singapore University Press, Singapore; World Scientific Publishing Co., Inc., River Edge, NJ, 2000. Vol. 1: xxxiv+488 pp.; Vol. 2: pp. i–xii and 489–1031; Vol. 3: pp. i–xii and 1033–1677. ISBN 981-02-4307-3
  • Steve Smale. The mathematics of time. Essays on dynamical systems, economic processes, and related topics. Springer-Verlag, New York-Berlin, 1980. vi+151 pp. ISBN 0-387-90519-7

Important Publications

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See also


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  5. Video on YouTube
  6. 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.
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  8. Stephen Smale Vita. Accessed November 18, 2009.
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External links

Personal websites at universities