Stephen Smale (born July 15, 1930) is an American mathematician from Flint, Michigan. His research concerns 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).
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.2 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 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".3 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.4
- Smale, Stephen (1961). "Generalized Poincaré's conjecture in dimensions greater than four". Annals of Mathematics 74 (2). pp. 391–406. doi:10.2307/1970239. MR 0137124.
- Smale, Stephen (1967). "Differentiable dynamical systems". Bulletin of the American Mathematical Society 73. pp. 747–817. doi:10.1090/S0002-9904-1967-11798-1. MR 0228014.
- 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.
- Smale, Steve (1985). "On the Efficiency of Algorithms in Analysis". Bulletin of the AMS, (N.S.) 13: 87–121.
- Video on YouTube
- 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.
- Stephen Smale Vita. Accessed November 18, 2009.
- Press release
- 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