Stephen Smale

Stephen Smale
Born	(1930-07-15) July 15, 1930 Flint, Michigan
Nationality	American
Alma mater	University of Michigan
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 Diffeomorphism
Awards	Wolf Prize (2007) National Medal of Science (1996) Chauvenet Prize (1988) Fields Medal (1966) Oswald Veblen Prize in Geometry (1966) Sloan Fellowship (1960)
Scientific career
Fields	Mathematics
Institutions	Toyota Technological Institute at Chicago City University of Hong Kong University of Chicago Columbia University University of California, Berkeley
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

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 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.

Education and career

Smale was born in Flint, Michigan and 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. When the department chair, Hildebrandt, threatened to kick Smale out, he began to take his studies more seriously. Smale finally earned his PhD 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 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." 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 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. 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 Berkeley where he has spent the main part of his career. He retired from 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.

From 2003 to 2012, Smale was a Professor at the Toyota Technological Institute at Chicago; starting August 1, 2009, he became a Distinguished University Professor at the City University of Hong Kong.

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

Books

• Smale, Steve (1980). The mathematics of time: essays on dynamical systems, economic processes, and related topics. New York-Berlin: Springer-Verlag. doi:10.1007/978-1-4613-8101-3. ISBN 0-387-90519-7. MR 0607330. Zbl 0451.58001.
• Blum, Lenore; Cucker, Felipe; Shub, Michael; Smale, Steve (1998). Complexity and real computation. With a foreword by Richard M. Karp. New York: Springer-Verlag. doi:10.1007/978-1-4612-0701-6. ISBN 0-387-98281-7. MR 1479636. Zbl 0948.68068.
• Hirsch, Morris W.; Smale, Stephen; Devaney, Robert L. (2013). Differential equations, dynamical systems, and an introduction to chaos (Third edition of 1974 original ed.). Amsterdam: Academic Press. doi:10.1016/C2009-0-61160-0. ISBN 978-0-12-382010-5. MR 3293130. Zbl 1239.37001.
• Cucker, F.; Wong, R., eds. (2000). The collected papers of Stephen Smale. In three volumes. Singapore: Singapore University Press. doi:10.1142/4424. ISBN 981-02-4307-3. MR 1781696. Zbl 0995.01005.

Important publications

• Smale, Stephen (1961). "Generalized Poincaré's conjecture in dimensions greater than four". Annals of Mathematics. 74 (2): 391–406. doi:10.2307/1970239. JSTOR 1970239. MR 0137124.
• Smale, S. (1965). "An infinite dimensional version of Sard's theorem". Amer. J. Math. 87 (4): 861–866. doi:10.2307/2373250. JSTOR 2373250.
• Smale, Stephen (1967). "Differentiable dynamical systems". Bulletin of the American Mathematical Society. 73 (6): 747–817. doi:10.1090/S0002-9904-1967-11798-1. MR 0228014.
• Blum, Lenore; Shub, Mike; Smale, Steve (1989). "On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines". Bull. Amer. Math. Soc. New Series. 21 (1): 1–46. doi:10.1090/S0273-0979-1989-15750-9.
• Shub, Michael; Smale, Stephen (1993). "Complexity of Bézout's Theorem I: Geometric Aspects". Journal of the American Mathematical Society. 6 (2). Providence, Rhode Island: American Mathematical Society: 459–501. doi:10.2307/2152805. JSTOR 2152805.
• Cucker, Felipe; Smale, Steve (2002). "On the mathematical foundations of learning". Bull. Amer. Math. Soc. New Series. 39 (1): 1–49. doi:10.1090/S0273-0979-01-00923-5.
• Cucker, Felipe; Smale, Steve (2007). "Emergent behavior in flocks". IEEE Trans. Autom. Control. 52 (5): 852–862. doi:10.1109/TAC.2007.895842. S2CID 206590734.*

See also

• 5-manifold
• Axiom A
• Geometric mechanics
• Homotopy principle
• Mean value problem

External links

• Stephen Smale at the Mathematics Genealogy Project
• O'Connor, John J.; Robertson, Edmund F., "Stephen Smale", MacTutor History of Mathematics Archive, University of St Andrews
• Weisstein, Eric W. "Smale's Problems". MathWorld.
• Robion Kirby, 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

• Steven Smale at the City University of Hong Kong
• Stephen Smale at the University of Chicago
• Steve Smale at the University of California, Berkeley