Richard E. Borcherds

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.

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.

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.

The Nevanlinna Prize is awarded to:

Peter Shor

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).