Fields Medal
The obverse of the Fields Medal
Awarded for	Outstanding contributions in Mathematics
Presented by	International Mathematical Union (IMU)
Country	Hosted every four years, at a varying location
Reward	C$15,000
First awarded	1936
Last awarded	2010
Official website	Official site

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 (IMU), a meeting that takes place every four years. The colloquial name is in honour of Canadian mathematician John Charles Fields.^[1] Fields was instrumental in establishing the award, designing the medal itself, and funding the monetary component.^[1] The Fields Medal is often viewed as the greatest honour a mathematician can receive.^[2][3] It comes with a monetary award, which since 2006 is C$15,000.^[4][5] The medal was first awarded in 1936 to Finnish mathematician Lars Ahlfors and American mathematician Jesse Douglas, and it has been awarded every four years since 1950. Its purpose is to give recognition and support to younger mathematical researchers who have made major contributions.

Conditions of the award[link]

The Fields Medal is often described as the "Nobel Prize of Mathematics" for being traditionally regarded as the most prestigious award in the field of mathematics,^[6] however in contrast to the Nobel Prize, the Fields Medal is awarded only every four years. The Medal also has an age limit: a recipient's 40th birthday must not occur before January 1 of the year in which the Fields Medal is awarded. As a result some great mathematicians have missed it by having done their best work (or having had their work recognized) too late in life. The 40-year rule is based on Fields' desire that

... while it was in recognition of work already done, it was at the same time intended to be an encouragement for further achievement on the part of the recipients and a stimulus to renewed effort on the part of others.

The monetary award is much lower than the roughly US$1.5 million given with each Nobel prize. Other major awards in mathematics, such as the Abel Prize and the Chern Medal, have a large monetary prize like a Nobel.

Fields medalists[link]

Year	ICM location	Medalists[7]	Institution	Nationality
1936	Oslo	Lars Ahlfors Jesse Douglas	University of Helsinki Massachusetts Institute of Technology
1950	Cambridge	Laurent Schwartz Atle Selberg	University of Nancy Institute for Advanced Study
1954	Amsterdam	Kunihiko Kodaira Jean-Pierre Serre	Institute for Advanced Study Centre National de la Recherche Scientifique
1958	Edinburgh	Klaus Roth René Thom	Imperial College London University of Strasbourg
1962	Stockholm	Lars Hörmander John Milnor	University of Stockholm Princeton University
1966	Moscow	Michael Atiyah Paul Joseph Cohen Alexander Grothendieck Stephen Smale	University of Oxford Stanford University Institut des Hautes Études Scientifiques University of California, Berkeley	None
1970	Nice	Alan Baker Heisuke Hironaka Sergei Novikov John G. Thompson	University of Cambridge Harvard University Moscow State University University of Cambridge
1974	Vancouver	Enrico Bombieri David Mumford	University of Pisa Harvard University
1978	Helsinki	Pierre Deligne Charles Fefferman Grigory Margulis Daniel Quillen	Institut des Hautes Études Scientifiques Princeton University Moscow State University Massachusetts Institute of Technology
1983	Warsaw	Alain Connes William Thurston Shing-Tung Yau	Institut des Hautes Études Scientifiques Princeton University Institute for Advanced Study
1986	Berkeley	Simon Donaldson Gerd Faltings Michael Freedman	University of Oxford Princeton University University of California, San Diego
1990	Kyoto	Vladimir Drinfeld Vaughan F. R. Jones Shigefumi Mori Edward Witten	Kharkov Institute for Low Temperature Physics University of California, Berkeley University of Kyoto Institute for Advanced Study
1994	Zürich	Jean Bourgain Pierre-Louis Lions Jean-Christophe Yoccoz Efim Zelmanov	Institute for Advanced Study Paris Dauphine University Paris-Sud 11 University University of California, San Diego
1998	Berlin	Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis T. McMullen	University of California, Berkeley and University of Cambridge University of Cambridge Institut des Hautes Études Scientifiques and Rutgers University Harvard University
2002	Beijing	Laurent Lafforgue Vladimir Voevodsky	Institut des Hautes Études Scientifiques Institute for Advanced Study
2006	Madrid	Andrei Okounkov Grigori Perelman Terence Tao Wendelin Werner	Princeton University None (Saint Petersburg) University of California, Los Angeles Paris-Sud 11 University
2010	Hyderabad	Elon Lindenstrauss Ngo Bao Chau Stanislav Smirnov Cédric Villani	Hebrew University of Jerusalem and Princeton University Paris-Sud 11 University and Institute for Advanced Study University of Geneva École Normale Supérieure de Lyon and Institut Henri Poincaré	-
2014	Seoul	TBD

Landmarks[link]

In 1954, Jean-Pierre Serre became the youngest winner of the Fields Medal, at 27. He still retains this distinction.

In 1966, Alexander Grothendieck boycotted the ICM, held in Moscow, to protest Soviet military actions taking place in Eastern Europe.^[8]

In 1970, Sergei Novikov, due to restrictions placed on him by the Soviet government, was unable to travel to the congress in Nice to receive his medal.

In 1978, Grigory Margulis, due to restrictions placed on him by the Soviet government, was unable to travel to the congress in Helsinki to receive his medal. The award was accepted on his behalf by Jacques Tits, who said in his address: "I cannot but express my deep disappointment — no doubt shared by many people here — in the absence of Margulis from this ceremony. In view of the symbolic meaning of this city of Helsinki, I had indeed grounds to hope that I would have a chance at last to meet a mathematician whom I know only through his work and for whom I have the greatest respect and admiration."^[9]

In 1982, the congress was due to be held in Warsaw but had to be rescheduled to the next year, due to martial law introduced in Poland Dec. 13, 1981. The awards were announced at the ninth General Assembly of the IMU earlier in the year and awarded at the 1983 Warsaw congress.

In 1990, Edward Witten became the first and so far only physicist to win this award.

In 1998, at the ICM, Andrew Wiles was presented by the chair of the Fields Medal Committee, Yuri I. Manin, with the first-ever IMU silver plaque in recognition of his proof of Fermat's Last Theorem. Don Zagier referred to the plaque as a "quantized Fields Medal". Accounts of this award frequently make reference that at the time of the award Wiles was over the age limit for the Fields medal.^[10] Although Wiles was slightly over the age limit in 1994, he was thought to be a favorite to win the medal; however, a gap (later resolved by Taylor and Wiles) in the proof was found in 1993.^[11][12]

In 2006, Grigori Perelman, who proved the Poincaré conjecture, refused his Fields Medal^[4] and did not attend the congress.^[13]

The medal[link]

The medal was designed by Canadian sculptor R. Tait McKenzie.^[14]

• On the obverse is Archimedes and a quote attributed to Marcus Manilius which reads in Latin: "Transire suum pectus mundoque potiri" (Rise above oneself and grasp the world).

The reverse of the Fields Medal

• On the reverse is the inscription (in Latin):

CONGREGATI

EX TOTO ORBE

MATHEMATICI

OB SCRIPTA INSIGNIA

TRIBUERE

Translation: "Mathematicians gathered from the entire world awarded [understood "the winners"] for their outstanding writings."

In the background, there is the representation of Archimedes' tomb, with the carving illustrating his theorem on the sphere and the cylinder, behind a branch. (This is the mathematical result of which Archimedes was reportedly most proud: Given a sphere and a circumscribed cylinder of the same height and diameter, the ratio between their volumes is equal to 2/3.)

The rim bears the name of the prizewinner.

See also[link]

	Book: Fields Medal
Wikipedia books are collections of articles that can be downloaded or ordered in print.

• List of prizes, medals, and awards in mathematics

References[link]

Further reading[link]

• Riehm, Elaine (2011). Turbulent Times in Mathematics: The Life of J.C. Fields and the History of the Fields Medal. Providence, RI: American Mathematical Society. ISBN 0-8218-6914-0 
• Monastyrsky, Michael (1998). Modern Mathematics in the Light of the Fields Medal. Wellesley, MA: A. K. Peters. ISBN 1-56881-083-0 
• Tropp, Henry S. (1976). "The Origins and History of the Fields Medal". Historia Mathematica 3 (2): 167–181. DOI:10.1016/0315-0860(76)90033-1 .

External links[link]

Wikimedia Commons has media related to: Fields medal

• Official website

Stephen Smale
Born	(1930-07-15) July 15, 1930 (age 81) Flint, Michigan
Nationality	American
Fields	Mathematics
Institutions	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 Jacob Palis Themistocles M. Rassias
Notable awards	Wolf Prize (2006/07) National Medal of Science (1996) Fields Medal (1966)

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).

Education and career[link]

He entered the University of Michigan 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^{[citation needed]}. 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".^[1] 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.^[2]

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.^[3]

In 2007, Smale was awarded the Wolf Prize in mathematics.^[4]

Important publications[link]

• 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.

References[link]

External links[link]

• 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, http://www-history.mcs.st-andrews.ac.uk/Biographies/Smale.html .
• Weisstein, Eric W., "Smale's Problems" from 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 Website at Universities[link]

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

Terence Chi-Shen Tao - Хуяо
Tao at the March 2006 at the Erdős Memorial Conference in Memphis, Tennessee
Born	(1975-07-17) 17 July 1975 (age 36) Adelaide, Australia
Residence	Los Angeles, California
Nationality	Australian[1] United States[1]
Fields	Mathematics
Institutions	University of California, Los Angeles
Alma mater	Flinders University Princeton University
Doctoral advisor	Elias M. Stein
Known for	Green–Tao theorem Tao's inequality Kakeya conjecture Horn conjecture
Notable awards	Salem Prize (2000) Bôcher Memorial Prize (2002) Clay Research Award (2003) Australian Mathematical Society Medal (2005) Ostrowski Prize (2005) SASTRA Ramanujan Prize (2006) Levi L.Conant Prize (2005) Fields Medal (2006) MacArthur Award (2007) Fellow of the Royal Society (2007) Alan T. Waterman Award (2008) Onsager Medal (2008) King Faisal International Prize (2010) Nemmers Prize in Mathematics (2010) Polya Prize (2010) Crafoord Prize (2012)

Terence "Terry" Chi-Shen Tao FRS (simplified Chinese: 陶哲轩; traditional Chinese: 陶哲軒) (born 17 July 1975, Adelaide), is an Australian born American mathematician working in harmonic analysis, partial differential equations, combinatorics, and analytic number theory. He currently holds the James and Carol Collins chair in mathematics at the University of California, Los Angeles. He was one of the recipients of the 2006 Fields Medal.

Personal life[link]

Tao was a child prodigy,^[2] one of the subjects in the longitudinal research on exceptionally gifted children by education researcher Miraca Gross.^[3] His father told the press that at the age of two, during a family gathering, Tao attempted to teach a 5-year-old child Arithmetic and English. According to Smithsonian Online Magazine, Tao could carry out basic arithmetic by the age of two. When asked by his father how he knew numbers and letters, he said he learned them from Sesame Street.^[4] Aside from English, Tao speaks Cantonese, but does not write Chinese.

Tao exhibited extraordinary mathematical abilities from an early age, attending university level mathematics courses at the age of nine. He is one of only two children (besides Lenhard Ng) in the history of the Johns Hopkins' Study of Exceptional Talent program to have achieved a score of 700 or greater on the SAT math section while just 8 years old (he scored a 760).^[5] In 1986, 1987, and 1988, Tao was the youngest participant to date in the International Mathematical Olympiad, first competing at the age of ten, winning a bronze, silver, and gold medal respectively. He remains the youngest winner of each of the three medals in the olympiad's history winning the gold medal when he barely turned thirteen. At age 14, Tao attended the Research Science Institute. When he was 15 he published his first assistant paper. He received his bachelor's and master's degrees (at the age of 16) from Flinders University under Garth Gaudry. In 1992 he won a Fulbright Scholarship to undertake postgraduate study in the United States. From 1992 to 1996, Tao was a graduate student at Princeton University under the direction of Elias Stein, receiving his Ph.D. at the age of 20.^[6] He joined the faculty of the University of California, Los Angeles in 1996.

When he was 24, he was promoted to full professor at UCLA and remains the youngest person ever appointed to that rank by the institution. Tao's father was born and grew up in Shanghai, and Tao's mother is Cantonese by ethnicity.^[7] His parents are first generation immigrants from Hong Kong to Australia.^[8] His father, Billy Tao (Chinese: 陶象國; pinyin: Táo Xiàngguó; Cantonese Yale: tòuh jeuhng gwok) is a pediatrician, and his mother is a physics and mathematics graduate from the University of Hong Kong, formerly a secondary school teacher of mathematics in Hong Kong.^[9]

Tao has two brothers living in Australia, both of whom represented Australia at the International Mathematical Olympiad.

• Nigel Tao is part of the team at Google Australia that created Google Wave.^[10]
• Trevor Tao has a double degree in maths and music and will soon be featured in a book on autistic savants.^[10]

Tao, his wife Laura (an engineer at NASA's Jet Propulsion Laboratory),^[11] their son and daughter live in Los Angeles, California.

Research and awards[link]

Tao has won numerous honors and awards.^[12] He received the Salem Prize in 2000, the Bôcher Memorial Prize in 2002, and the Clay Research Award in 2003, for his contributions to analysis including work on the Kakeya conjecture and wave maps. In 2005 he received the American Mathematical Society's Levi L. Conant Prize with Allen Knutson, and in 2006 he was awarded the SASTRA Ramanujan Prize.

In 2004, Ben Green and Tao released a preprint proving what is now known as the Green–Tao theorem. This theorem states that there are arbitrarily long arithmetic progressions of prime numbers. The New York Times described it this way:^[13][14]

For this and other work Tao was awarded the Australian Mathematical Society Medal of 2004.

In August 2006, at the 25th International Congress of Mathematicians in Madrid, he became one of the youngest persons, the first Australian, and the first UCLA faculty member ever to be awarded a Fields Medal.^[15][16] An article by New Scientist^[17] writes of his ability: