During the Congress, which is held every four years, four Fields Medals are awarded to outstanding mathematicians under the age of forty. In view of their significance the Fields medals are often dubbed the "Nobel Prize of Mathematics." Four years ago, Andrew J. Wiles was a hot favorite for an award, since in 1993 he had presented a proof of Fermat's Last Theorem -- one of the most famous mathematical puzzles, which had remained unsolved for more than 350 years. Shortly afterwards, however, colleagues found a gap in the proof, which Wiles was only able to close up a year later. But this was too late for the Fields Medal, because Wiles was then over the age limit of forty. With its special tribute, the International Mathematical Union acknowledged Andrew J. Wiles's outstanding achievement.
Andrew J. Wiles(born 11 April 1953) is Professor of Mathematics
at Princeton University. Since 1995 he has also been a member of the Institute
for Advanced Study (IAS). Wiles studied in England at Cambridge University
before going to America as assistant professor at Harvard in 1974. In 1982,
he became professor in Princeton. His fields of research are number theory
and arithmetic geometry.
The Moonshine conjecture provides an interrelationship between the so-called "monster group" and elliptic 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. The monster group, in contrast, only seemed to be of importance in pure mathematics. 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 1053). Hence the name "monster." In his proof, Borcherds uses many ideas of string theory - a surprisingly fruitful way of 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.
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.
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 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.
A further result of McMullen relates to the Mandelbrot set. This set describes dynamic systems which can be used to model complicated natural phenomena such as weather or fluid flow. The point of interest is where a system drifts apart and which points move towards centers of equilibrium. The border between these two extremes is the so-called Julia set, named after the French mathematician Gaston Julia, who laid the foundations for the theory of dynamic systems early in the twentieth century. The Mandelbrot set shows the parameters for which the Julia set is connected, i.e. is mathematically attractive. This description is very crude, but a better characteristic of the boundary set was not available. Curtis T. McMullen made a major advance, however, when he showed that it is possible to decide in part on the basis of the Mandelbrot set which associated dynamic system is "hyperbolic" and can therefore be described in more detail. For these systems a well-developed theory is available. McMullen's results were suspected already in the sixties, but nobody had previously been able to prove this exact characterization of the Julia set.
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.
Peter
Shor has carried out pioneering work in combinatorial analysis and the
theory of quantum computing. He received worldwide recognition in 1994
when he presented a computational method for "factoring large numbers"
which, theoretically, could be used to break many of the coding systems
currently employed. The drawback is that Shor's algorithm works on so-called
quantum computers, of which only prototypes currently exist. Quantum computers
do not operate like conventional ones, but make use of the quantum states
of atoms, which offers a computing capacity far in excess of current parallel
supercomputers. Shor's result unleased a boom in research amongst physicists
and computer scientists. Experts predict the quantum computers could already
become a reality within the next decade, but this rapid development is
also a cause of concern for some observers. Shor has been able to prove
mathematically that the new computers would mean that current standard
encrypting methods such as "RSA," which are used for electronic cash and
on-line signatures, would no longer be secure. "RSA" was developed in 1977
by the mathematicians Ronald Rivest, Adi Shamir and Leonard Adelmann (hence
the acronym). It makes use of the fact that factoring a number is so-called
one-way function. This means that while it is very easy to make a large
number from smaller ones, it takes much longer to find all the factors
of a large number. This time factor is the basis for the security offered
by many encryption methods. Using Shor's algorithms, factoring large numbers
on a quantum 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).
Peter Shor's web site
(http://www.research.att.com/~shor) contains, among other things, his paper
Polynomial-time
algorithms for prime factorization and discrete logarithms on a quantum
computer (in PostScript
format).