Stephen Smale
Encyclopedia
Steven Smale a.k.a. Steve Smale, 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).
in 1948. Initially, Smale 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 his first years, earning a C average as a graduate student. It was only when the department chair, Hildebrant, threatened to kick out Smale, that he began to work hard. 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; he later 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". 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 subpoena
ed 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.
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
.
In 2007, Smale was awarded the Wolf Prize in mathematics.
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
from Flint, Michigan
Flint, Michigan
Flint is a city in the U.S. state of Michigan and is located along the Flint River, northwest of Detroit. The U.S. Census Bureau reports the 2010 population to be placed at 102,434, making Flint the seventh largest city in Michigan. It is the county seat of Genesee County which lies in the...
. He was awarded the Fields Medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
in 1966, and spent more than three decades on the mathematics faculty of the University of California, Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
(1960–61 and 1964–1995).
Education and career
He entered the University of MichiganUniversity of Michigan
The University of Michigan is a public research university located in Ann Arbor, Michigan in the United States. It is the state's oldest university and the flagship campus of the University of Michigan...
in 1948. Initially, Smale was a good student, placing into an honors calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
sequence taught by Bob Thrall and earning himself A's. However, his sophomore
Sophomore
Sophomore is a term used in the United States to describe a student in the second year of study at high school or university.The word is also used as a synonym for "second", for the second album or EP released by a musician or group, the second movie of a director, or the second season of a...
and junior years were marred with mediocre grades, mostly Bs, Cs and even an F in nuclear physics
Nuclear physics
Nuclear physics is the field of physics that studies the building blocks and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those...
. However, with some luck, Smale was accepted as a graduate student at the University of Michigan's mathematics department. Yet again, Smale performed poorly his first years, earning a C average as a graduate student. It was only when the department chair, Hildebrant, threatened to kick out Smale, that he began to work hard. Smale finally earned his Ph.D.
Ph.D.
A Ph.D. is a Doctor of Philosophy, an academic degree.Ph.D. may also refer to:* Ph.D. , a 1980s British group*Piled Higher and Deeper, a web comic strip*PhD: Phantasy Degree, a Korean comic series* PhD Docbook renderer, an XML renderer...
in 1957, under Raoul Bott
Raoul Bott
Raoul Bott, FRS was a Hungarian mathematician known for numerous basic contributions to geometry in its broad sense...
.
Smale began his career as an instructor at the college at the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
. In 1958, he astounded the mathematical world with a proof of a sphere eversion
Smale's paradox
In differential topology, Smale's paradox states that it is possible to turn a sphere inside out in a three-dimensional space with possible self-intersections but without creating any crease, a process often called sphere eversion...
. He then cemented his reputation with a proof of the Poincaré conjecture
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
for all dimensions greater than or equal to 5; he later generalized the ideas in a 107 page paper that established the h-cobordism theorem.
After having made great strides in topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, he then turned to the study of dynamical systems, where he made significant advances as well. His first contribution is the Smale horseshoe
Horseshoe map
In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavior of the orbits of the van der Pol oscillator...
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
Morse theory
In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect...
into mathematical economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, as well as recent explorations of various theories of computation
Computation
Computation is defined as any type of calculation. Also defined as use of computer technology in Information processing.Computation is a process following a well-defined model understood and expressed in an algorithm, protocol, network topology, etc...
.
In 1998 he compiled a list of 18 problems in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
to be solved in the 21st century, known as Smale's problems
Smale's problems
Smale's problems refers to a list of eighteen unsolved problems in mathematics that was proposed by Steve Smale in 2000. Smale composed this list in reply to a request from Vladimir Arnold, then president of the International Mathematical Union, who asked several mathematicians to propose a list of...
. This list was compiled in the spirit of Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
's famous list of problems produced in 1900. In fact, Smale's list contains some of the original Hilbert problems, including the Riemann hypothesis
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
and the second half of Hilbert's sixteenth problem
Hilbert's sixteenth problem
Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, together with the other 22 problems....
, both of which are still unsolved. Other famous problems on his list include the Poincaré conjecture
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
, the P = NP problem, and the Navier-Stokes equations
Navier-Stokes existence and smoothness
The Navier–Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier–Stokes equations, one of the pillars of fluid mechanics . These equations describe the motion of a fluid in space. Solutions to the Navier–Stokes equations are used in many practical...
, all of which have been designated Millennium Prize Problems
Millennium Prize Problems
The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of September 2011, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize being awarded by the institute...
by the Clay Mathematics Institute
Clay Mathematics Institute
The Clay Mathematics Institute is a private, non-profit foundation, based in Cambridge, Massachusetts. The Institute is dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998...
.
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". This led to the withholding of his grant money from the NSF
National Science Foundation
The National Science Foundation is a United States government agency that supports fundamental research and education in all the non-medical fields of science and engineering. Its medical counterpart is the National Institutes of Health...
. He has been politically active in various movements in the past, such as the Free Speech movement
Free Speech Movement
The Free Speech Movement was a student protest which took place during the 1964–1965 academic year on the campus of the University of California, Berkeley under the informal leadership of students Mario Savio, Brian Turner, Bettina Aptheker, Steve Weissman, Art Goldberg, Jackie Goldberg, and...
. At one time he was subpoena
Subpoena
A subpoena is a writ by a government agency, most often a court, that has authority to compel testimony by a witness or production of evidence under a penalty for failure. There are two common types of subpoena:...
ed by the House Un-American Activities Committee
House Un-American Activities Committee
The House Committee on Un-American Activities or House Un-American Activities Committee was an investigative committee of the United States House of Representatives. In 1969, the House changed the committee's name to "House Committee on Internal Security"...
.
In 1960 Smale was appointed an associate professor of mathematics at the University of California, Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
, moving to a professorship at Columbia University
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
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
City University of Hong Kong
City University of Hong Kong is a comprehensive research university in Hong Kong. It was founded in 1984 as City Polytechnic of Hong Kong and became a fully accredited university in 1994. It has achieved fast growth in recent years and received international recognition for its academic achievements...
. 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.
Since 2002 Smale is a Professor at the Toyota Technological Institute at Chicago
Toyota Technological Institute at Chicago
In 2003, the Toyota Technological Institute of Nagoya, Japan opened the Toyota Technological Institute at Chicago, jointly with the University of Chicago. The stated purpose of the institute is to impact research and education in computer science....
; starting August 1, 2009, he is also a Distinguished University Professor at the City University of Hong Kong
City University of Hong Kong
City University of Hong Kong is a comprehensive research university in Hong Kong. It was founded in 1984 as City Polytechnic of Hong Kong and became a fully accredited university in 1994. It has achieved fast growth in recent years and received international recognition for its academic achievements...
.
In 2007, Smale was awarded the Wolf Prize in mathematics.
Important publications
- S. Smale, Generalized Poincaré's conjecture in dimensions greater than four, Annals of MathematicsAnnals of MathematicsThe Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...
, 2nd Ser., 74 (1961), no. 2, 391 – 406. (via JSTOR) - S. Smale, Differentiable dynamical systems, Bulletin of the American Mathematical SocietyBulletin of the American Mathematical SocietyThe Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society...
, 73 (1967), 747 – 817. (http://www.ams.org/bull/1967-73-06/S0002-9904-1967-11797-X/home.html) - 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.
External links
- Robion KirbyRobion KirbyRobion Cromwell Kirby is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology...
, Stephen Smale: The Mathematician Who Broke the Dimension Barrier, a book review of a biography in the Notices of the AMSAmerican Mathematical SocietyThe American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
.
Personal Website at Universities
- Steven Smale at the City University of Hong KongCity University of Hong KongCity University of Hong Kong is a comprehensive research university in Hong Kong. It was founded in 1984 as City Polytechnic of Hong Kong and became a fully accredited university in 1994. It has achieved fast growth in recent years and received international recognition for its academic achievements...
- Stephen Smale at the City University of Hong KongCity University of Hong KongCity University of Hong Kong is a comprehensive research university in Hong Kong. It was founded in 1984 as City Polytechnic of Hong Kong and became a fully accredited university in 1994. It has achieved fast growth in recent years and received international recognition for its academic achievements...
- Stephen Smale at the University of ChicagoUniversity of ChicagoThe University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
- Steve Smale at the University of California, BerkeleyUniversity of California, BerkeleyThe University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...