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In physics or literature they have the Nobel Prize, and in mathematics there is the "Fields Medal". This highest scientific award for mathematicians was presented today at the opening ceremony of the "International Congress of Mathematicians" to Richard E. Borcherds, Maxim Kontsevich, William Timothy Gowers and Curtis T. McMullen. The International Mathematical Union also awarded the "Nevanlinna Prize" for outstanding work in the field of theoretical computer science to the mathematician Peter Shor.

The **Fields Medal** is the highest scientific award for mathematicians.
The awards are presented every four years at the International
Congress of Mathematicians (ICM) together with a prize of 15 000
Canadian dollars (approx. DM 17 500). Four medals are presented
at each ceremony to mathematicians who are not more than forty
years old. The age limit is intended to guarantee that not only
past work is rewarded. The Fields Medal is also intended to encourage
the winners to make further contributions.

"Fields Medal" is in fact only the unofficial name for the "International medal for outstanding discoveries in mathematics". John C. Fields (1863 - 1932), a Canadian mathematician, was the organiser of the International Congress of Mathematicians in 1924 in Toronto. Fields was able to attract so many sponsors that money was left over at the end of the congress, and this was used to fund the medals. The first Fields Medal was awarded in 1936 at the world congress in Oslo. Due to the great expansion in mathematical research, four medals have been presented at each congress since 1966. The awards are often referred to as the "Nobel Prize for Mathematics", since the Swedish Academy of Sciences itself can only honour mathematicians indirectly through the natural sciences or social sciences. There is no Nobel Prize for mathematics.

The Fields Medal is made of gold, and shows the head of Archimedes (287 - 212 BC) together with a quotation attributed to him: "Transire suum pectus mundoque potiri" (Rise above oneself and grasp the world"). The reverse side bears the inscription: "Congregati ex toto orbe mathematici ob scripta insignia tribuere." (The mathematicians assembled here from all over the world pay tribute for outstanding work).

The only German to have received the Fields Medal to date is Gerd Faltings, professor at the Max-Planck Institute of Mathematics in Bonn. He was honoured in 1986 for his proof of the Mordell Conjecture and his work in algebraic geometry.

The **Nevanlinna Prize** has been awarded since 1983 for outstanding
work in the fields of theoretical computer science. The prize
is also in the form of a gold medal and a cash award of 15 000
Canadian dollars (DM 17 500). It is donated by the University
of Helsinki in memory of the Finnish mathematician Rolf Nevanlinna,
who was president of the International Mathematical Union 1959
- 1962 and organiser of the World Congress in Stockholm in 1962.
One side of the medal shows the bust of Nevanlinna, and the other
bears the seal of Helsinki University and a rectangle of noughts
and ones. The word "Helsinki" in coded form.

In order to select the winners of the Fields Medals and the Nevanlinna Prize, the Executive Committee of the International Mathematical Union appoints two bodies, the "Fields Medal Committee" (in this form since 1962) consisting of eight mathematicians, and the "Nevanlinna Prize Committee" with three mathematicians.

The Moonshine conjecture provides an interrelationship between the so-called "monster-groups" and elliptical functions. These functions are used in the construction of wire-frame structures in two-dimensions, and can be helpful, for example, in chemistry for the description of molecular structures. Monster groups, in contrast, only seemed to be of importance in pure mathematicians. Groups are mathematical objects which can be used to describe the symmetry of structures. Expressed technically, they are a set of objects for which certain arithmetic rules apply (for example all whole numbers and their sums form a group). An important theorem of algebra says that all groups, however large and complicated they may seem, all consist of the same components - in the same way as the material world is made up of atomic particles. The "monster group" is the largest "sporadic, finite, simple" group - and one of the most bizarre objects in algebra. It has more elements than there are elementary particles in the universe (approx. 8 x 10^53). Hence the name "monster". In his proof, Borcherds uses many ideas of string theory - a surprisingly fruitful way a making theoretical physics useful for mathematical theory. Although still the subject of dispute among physicists, strings offer a way of explaining many of the puzzles surrounding the origins of the universe. They were proposed in the search for a single consistent theory which brings together various partial theories of cosmology. Strings have a length but no other dimension and may be open strings or closed loops.

**Richard Ewen Borcherds** (born 29 November 1959) has been
"Royal Society Research Professor" at the Department
of Pure Mathematics and Mathematical Statistics at Cambridge University
since 1996. Borcherds began his academic career at Trinity College,
Cambridge before going as assistant professor to the University
of California in Berkeley. He has been made a Fellow of the Royal
Society, and has also held a professorship at Berkeley since 1993.

microcosm (forces between elementary particles). Another result of Kontsevich relates to knot theory. Knots mean exactly the same thing for mathematicians as for everyone else, except that the two ends of the rope are always jointed together. A key question in knot theory is, which of the various knots are equivalent? Or in other words, which knots can be twisted and turned to produce another knot without the use of scissors? This question was raised at the beginning of the 20th century, but it is still unanswered. It is not even clear which knots can be undone, that is converted to a simple loop. Mathematicians are looking for ways of classifying all knots. They would be assigned a number or function, with equivalent knots having the same number. Knots which are not equivalent must have different numbers. However, such a characterisation of knots has not yet been achieved. Kontsevich has found the best "knot invariant" so far. Although knot theory is part of pure mathematics, there seem to be scientific applications. Knot structures occur in cosmology, statistical mechanics and genetics.

**Maxim Kontsevich** (born 25 August 1964) is professor at
the Institute des Hautes Etudes Scientific (I.H.E.S) in France
and visiting professor at the Rutgers University in New Brunswick
(USA). After studying at the Moscow University and beginning research
at the "Institute for Problems of Information Processing",
he gained a doctorate at the University of Bonn, Germany in 1992.
He then received invitations to Harvard, Princeton, Berkeley and
Bonn.

**William Timothy Gowers** (born 20 November 1963) is lecturer
at the Department of Pure Mathematics and Mathematical Statistics
at Cambridge University and Fellow of Trinity College. From October
1998 he will be Rouse Professor of Mathematics. After studying
through to doctorate level at Cambridge, Gowers went to University
College London in 1991, staying until the end of 1995. In 1996
he received the Prize of the European Mathematical Society.

**Curtis T. McMullen** (born 21 May 1958) is visiting professor
at Harvard University. He studied in Williamstown, Cambridge University
and Paris before gaining a doctorate in 1985 at Harvard. He lectured
at various universities before becoming professor at the University
of California in Berkeley. Since 1998 he has taught at Harvard.
The Fields Medal is his tenth major award. In 1998 he has been
elected to the American Academy of Arts and Sciences.

computer would be just as fast as multiplication. "RSA" and other procedures would no longer be safe. Experts have been making reassuring noises, since a lot of work remains to be done before such computers can even be constructed, but cryptographers are already working on the next generation of encryption techniques.

**Peter Shor** (born 14 August 1959) is mathematician at the
AT&T Labs in Florham Park, New Jersey (USA). His research
interests include quantum computing, algorithmic geometry, and
combinatorial analysis. After studying at California Institute
of Technology (Caltech) he gained a doctorate at Massachusetts
Institute of Technology (MIT). Before going to AT&T in 1986,
he was postdoc for a year at the Mathematical Research Center
in Berkeley, California (USA).

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