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In his Will, John Charles \htmladdnormallink{fields}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} proposed to establish the Fields Medal which has played since 1936 the role that the Nobel Prize might have placed if it were awarded to mathematicians (which it is not). His proposal was accepted at The International Congress of Mathematicians at Z\"urich in 1932. However, it was not until the next congress, held at Oslo in 1936, that the Fields Medal was first awarded. Fields Medals were then not awarded during World War II so that the second Fields Medals were not awarded until 1950.
In his Will Mr.Fields wished that the awards should recognize both existing mathematical \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} and also the promise of future achievement, and to fit these criteria the Fields Medals can only be awarded to eminent mathematicans that are under the age of 40 at the time when the award decision is being made.
Unlike the Nobel, the Fields Medal can be shared by four, not three, researchers.
(A similar proposal was discussed without success between Sweeden and Norway
in 1905 for the establishment of an Abel Prize in Mathematics and \htmladdnormallink{physical mathematics/}{http://planetphysics.us/encyclopedia/PhysicalMathematics2.html} Physics. In 2001, Norway alone established the substantial Abel prize for eminent mathematicians and also mathematical physicists on a par with the Sweedish prize for sciences other than mathematics. Considering the existing Crafoord prize which is also in Mathematics, it would seem that mathematicians may easily become either over-prized or `over-priced'(?), whichever comes first.)
The list of the Fields Medal winners is as follows:
\begin{itemize}
\item 1936 L V Ahlfors; J. Douglas
\item 1950 L. Schwartz; A. Selberg
\item 1954 K. Kodaira; J-P. Serre
\item 1958 K. F. Roth; Ren\'ee Thom in France, for results in Topology
(not awarded for mathematical biology or catastrophy theory)
\item 1962 L. V. H\"ormander; J. W. Milnor
\item 1966 Sir M. F. Atiyah, in UK; P. J. Cohen, in UK; \htmladdnormallink{Alexander Grothendieck}{http://planetphysics.us/encyclopedia/AlexanderGrothendieck.html} (w. PhD advisor L. Schwartz) at \htmladdnormallink{IHES}{http://planetphysics.us/encyclopedia/IHES.html}; S. Smale.
\item 1970 A. Baker; H. Hironaka; S. P. Novikov; J. G. Thompson
\item 1974 E. Bombieri; D. B. Mumford
\item 1978 P. R. Deligne (whose PhD advisor was A. Grothendieck) at IHES; C. L. Fefferman; G. A. Margulis; D. G. Quillen.
\item 1982 A. Connes at IHES, Paris, France; W. P. Thurston; S-T. Yau.
\item 1986 S Donaldson; G Faltings; M Freedman
\item 1990 V. Drinfel'd; V. Jones (some of his work is also fundamental in \htmladdnormallink{quantum operator algebras}{http://planetphysics.us/encyclopedia/Groupoid.html})
\item 1990 S. Mori; E. Witten
\item 1994 P-L. Lions; J-C. Yoccoz; J. Bourgain; E .Zelmanov
\item 1998 R. Borcherds;
William T. Gowers, FRS, at Trinity College, Cambridge University, in UK, for his work in functional analysis and combinatorics; (he is now asking a `tricky', but potentially important question: \htmladdnormallink{Is massively-collaborative mathematics possible ?!}{http://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible/}. His answer. like his question, is non-trivial: ``The idea would be that anybody who had anything whatsoever to say about the problem could chip in''... a virtual forum (N.A.))
\item 1998 Maxim Kontsevich, for his work in mathematical physics;
Curtis T. McMullen of Harvard University, for his work on
holomorphic \htmladdnormallink{dynamics}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html} and geometry of 3-dimensional \htmladdnormallink{manifolds}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html}.
\item 2002 L. Lafforgue; V. Voevodsky
\item 2006 Andrei Okounkov;Grigori Perelman (declined to accept the Fields Medal); Terence Tao; Wendelin Werner
\end{itemize}
A special award of the ``IMU silver plaque'' was given to Andrew J. Wiles at Princeton University Institute for Advanced Study for his proof of Ferma's Last \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html}.
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