## Devlin's Angle |

The prize is being given for the work that led to the names Atiyah and Singer being forever linked in the field of mathematics: the "Atiyah-Singer Index Theorem", which they formulated and proved in a series of papers they published in the early 1960s. The Index Theorem provides a bridge between pure mathematics (differential geometry, topology, and analysis) and theoretical physics (quantum field theory) that has led to advances in both fields.

The Norwegian Academy of Science, which oversees and manages the new prize, referred to the Index Theorem as "one of the great landmarks of 20th century mathematics". In fact, it is no exaggeration to say that the result changed the landscape of mathematics. Atiyah, who trained as an algebraic geometer and topologist, and Singer, who came from analysis, worked on ramifications of the theorem for twenty years.

Sir Michael, quoted in an article in
Britain's *Daily Telegraph* newspaper
(March 26) commented: "the Index Theorem
provides a Trojan horse that mathematicians
have used to get into physics and vice
versa. When we first did it, we had no
inkling that this would follow."

Singer, speaking to BBC News (March 26), said: "I am delighted to win this prize with Sir Michael. The work we did broke barriers between different branches of mathematics and that's probably its most important aspect. ... It has also had serious applications in theoretical physics." Referring to the creation of the Abel Prize, he added "I appreciate the attention mathematics will be getting. It's well-deserved because mathematics is so basic to science and engineering."

Atiyah and Singer will receive their award from Norway's King Harald at a ceremony in Oslo on May 25.

The prize amount is 6 million Norwegian Kroner, currently about $875,000. The award of the first Abel Prize, made in 2003 with relatively little fanfair outside of Scandinavia, went to the French mathematician Jean-Pierre Serre for his work in algebraic geometry and number theory.

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.

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)

A number of theories have been put forward to explain the omission of mathematics from Nobel's original list. The most colorful suggestion is that Nobel was miffed at mathematicians after discovering that his wife had had an affair with the Swedish mathematician Magnus Mittag-Leffler. Of all the theories, this is the easiest to dismiss, for the simple reason that Nobel never had a wife. Another oft-repeated suggestion is that Nobel hated mathematics after doing poorly in it at school. It may or may not be true that Nobel wasn't good at math, but there is no evidence to suggest that a negative high school experience in the math class led to a desire to get back at the mathematicians later in life by not giving them one of his prizes.

By far the most likely explanation, I think, is that he viewed mathematics as merely a tool used in the sciences and in engineering, not as a body of human intellectual achievement in its own right. He also did not single out biology, possibly likewise regarding it as just a tool for medicine, a not unreasonable view to have in the late 19th century.

"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.

Michael Francis Atiyah was born in 1929 in London. He got his B.A. and his doctorate from Trinity College, Cambridge. He spent the greatest part of his academic career at the Universities of Cambridge and Oxford. He was the driving force behind the creation of the Isaac Newton Institute for Mathematical Sciences in Cambridge and became its first director. He was elected a Fellow of the Royal Society in 1962 at the age of 32, and was the Society's president from 1990 to 1995. He was knighted in 1983 (hence "Sir Michael") and made a member of the Order of Merit in 1992.

Because of its technical and highly abstract nature of the Index Theorem, it isn't possible to give a precise statement in a column such as this. The exact formulation requires a heady mix of K-theory, functional analysis, and global analysis. Not long after I got my Ph.D., I met Atiyah in Oxford, and rushed off to read up about his famous theorem. I soon gave up, having decided that life was too short and I had my own mathematical research career to work on.

The press announcement of the award released by the Norwegian Academy of Sciences tried to convey the essence of the result with these words:

"We describe the world by measuring quantities and forces that vary over time and space. The rules of nature are often expressed by formulas involving their rates of change, so-called differential equations. Such formulas may have an "index", the number of solutions of the formulas minus the number of restrictions which they impose on the values of the quantities being computed. The index theorem calculated this number in terms of the geometry of the surrounding space. ... The Atiyah-Singer Index Theorem was the culmination and crowning achievement of a more than 100-year old development of ideas, from Stokes's theorem, which students learn in calculus classes, to sophisticated modern theories like Hodge's theory of harmonic integrals and Hirzebruch's signature theorem."Let's try to get a bit closer to the real thing.

Start with a compact smooth manifold
(without boundary) and an elliptic operator
*E* on it. (*E* is a differential
operator acting on smooth sections of a
given vector bundle. Laplacians are
examples of elliptic operators.) The
property of being *elliptic* is
expressed by a *symbol s* that can be
seen as coming from the coefficients of the
highest order part of *E*; *s* is
a bundle section and required to be non-zero.
(In the case of a Laplacian, *s*
is a positive-definite quadratic form.) The
*index* of *E* is defined as the
difference between the dimension of the
kernel of *E* and the dimension of the
cokernel of *E*. The "index problem"
is to compute the index of *E* using
only the symbol *s* and topological
information about the manifold and the
vector bundle. This problem seems to have
emerged in the late 1950s. The Atiyah-Singer
Index Theorem is its solution.

Devlin's Angle is updated at the beginning of each month.

Devlin's most recent book is The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, published by Basic Books (hardback 2002, paperback 2003).