http://www.snopes.com/science/nobel.asp
http://nobelprizes.com/nobel/why_no_math.html
http://www.maa.org/devlin/devlin_04_04.html
Mathematics and the Nobel Prize
Although Nobel did not will a prize for mathematics, over the years many mathematicians have won a Nobel Prize. Taking a fairly generous interpretation for what constitutes being a mathematician, the mathematical Laureates are:
• 1902 Lorentz (Physics)
• 1904 Rayleigh (Physics)
• 1911 Wien (Physics)
• 1918 Planck (Physics)
• 1921 Einstein (Physics)
• 1922 Bohr (Physics)
• 1929 de Broglie (Physics)
• 1932 Heisenberg (Physics)
• 1933 Schroedinger (Physics)
• 1933 Dirac (Physics)
• 1945 Pauli (Physics)
• 1950 Russell (Literature)
• 1954 Born (Physics)
• 1962 Landau (Physics)
• 1963 Wigner (Physics)
• 1965 Schwinger (Physics)
• 1965 Feynman (Physics)
• 1969 Tinbergen (Economics)
• 1975 Kantorovich (Economics)
• 1983 Chandrasekhar (Physics)
• 1994 Selten (Economics)
• 1994 Nash (Economics)
Overall, a fairly good showing for mathematics. Still, this isn't the same as having a prize for mathematics itself.
The Abel Prize
There is no Nobel Prize for mathematics, but many mathematicians have won the prize, most commonly for physics but occasionally for economics, and in one case for literature. For instance, when mathematician John Nash won a Nobel Prize in 1994, it was for a result that had a major impact in economics. (Nash's achievement was celebrated in director Ron Howard's 2002 movie A Beautiful Mind, starring Russell Crowe.)
The Abel Prize is intended to give the mathematicians their own equivalent of a Nobel Prize. Such an award was first proposed in 1902 by King Oscar II of Sweden and Norway, just a year after the award of the first Nobel Prizes. However, plans were dropped as the union between the two countries was dissolved in 1905. As a result, mathematics has never had an international prize of the same dimensions and importance as the Nobel Prize.
Plans for an Abel Prize were revived in 2000, and in 2001 the Norwegian Government granted NOK 200 million (about $22 million) to create the new award. Niels Henrik Abel (1802-1829), after whom the prize is named, was a leading 19th-century Norwegian mathematician whose work in algebra has had lasting impact despite Abel's early death aged just 26. Today, every mathematics undergraduate encounters Abel's name in connection with commutative groups, which are more commonly known as "abelian groups" (the lack of capitalization being a tacit acknowledgement of the degree to which his name has been institutionalized).
As it happens, Abel's own field of group theory plays a role in the Atiyah-Singer Index Theorem, but this is not a condition for the award of the Abel Prize.
The Abel Prize is awarded annually, and is intended to present the field of mathematics with a prize at the highest level. Laureates are appointed by an independent committee of international mathematicians.
As a result of Norway's action, made in part to celebrate the 200th anniversary of Abel's birth in 2002, mathematicians now too have an award equivalent to the Nobel Prize. The question is, will the new prize achieve the international luster of a real Nobel? The Nobel Prize in Economics (as it is popularly, but incorrectly, called) achieved that status after it was introduced in 1968, but in that case the Bank of Sweden, which created the award, attached the magic name Nobel to it. (See later.) One could hardly expect Norway to name their prize after a famous Swede, especially when they have Abel to recognize.
The Fields Medal
The Fields Medal is often cited as being "the mathematical equivalent of the Nobel Prize."
The Field Medals were first proposed at the 1924 International Congress of Mathematicians in Toronto by Professor J.C. Fields, a Canadian mathematician who was the secretary of the Congress that year. He later donated funds to establish the medals. Fields wanted the awards to recognize both existing work and the promise of future achievement, as a result of which it was agreed to restrict the medals to mathematicians not over forty at the year of the Congress. Medals are awarded every four years, at the Congress, by its organizing body, the International Mathematical Union. Initially, up to two medals were awarded every four years; in 1966 it was agreed that, in light of the great expansion of mathematical research, up to four medals could be awarded at each Congress.
"Fields Medals" are more properly known by their official name, "International medals for outstanding discoveries in mathematics." The medal is accompanied by a cash prize of CND$15,000.
Atiyah himself received a Fields Medal in 1966.
There are some unique characteristics of the Fields Medal that make it different from a Nobel Prize. First it is awarded only every fourth year. Second, it is given for mathematical work done before the recipient is 40 years of age. Third, the monetary prize that goes with the Fields Medal is considerably less than the Nobel Prize. Fourth, the Fields Medal does not come out of Scandinavia.
When the Norwegian Academy of Science decided to create a prize for mathematics in honor of Abel, they did so with the intention of rectifying what they saw as an omission on the part of Nobel.