TUTORIALS:

Fields Medal Prize Winners
Fields Medal Prize Winners (1998)
Richard Ewen Borcherds
(born 29 November 1959)
He 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.
Richard E. Borcherds will receive a medal for his work in the fields of
algebra and geometry, in particular for his proof of the socalled Moonshine
conjecture. This conjecture was formulated at the end of '70s by the British
mathematicians John Conway and Simon Norton and presents two mathematical
structures in such an unexpected relationship that the experts gave it the
name "Moonshine". In 1989, Borcherds was able to cast some more light on the
mathematical background of this topic and to produce a proof for the
conjecture. The Moonshine conjecture provides an interrelationship between
the socalled "monstergroups" and elliptical functions. These functions are
used in the construction of wireframe structures in twodimensions, 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.
William Timothy Gowers
(born 20 November 1963)
He 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.
William Timothy Gowers has provided important contributions to functional
analysis, making extensive use of methods from combination theory. These two
fields apparently have little to do with each other, and a significant
achievement of Gowers has been to combine these fruitfully. Functional
analysis and combination analysis have in common that many of their problems
are relatively easy to formulate, but extremely difficult to solve. Gowers
has been able to utilise complicated mathematical constructions to prove
some of the conjectures of the Polish mathematician Stefan Banach
(18921945), including the problem of "unconditional bases". Banach was an
eccentric, preferring to spend his time in the café rather than in his
office in the University of Lvov. In the twenties and thirties he filled a
notebook with problems of functional analysis while sitting in the "Scottish
Café", so that this later became known as the Scottish Book. Gower has made
significant contribution above all to the theory of Banach spaces. Banach
spaces are sets whose members are not numbers but complicated mathematical
objects such as functions or operators. However, in a Banach space it is
possible to manipulate these objects like numbers. This finds applications,
for example, in quantum physics. A key question for mathematicians and
physicists concerns the inner structure of these spaces, and what symmetry
they show. Gowers has been able to construct a Banach space which has almost
no symmetry. This construction has since served as a suitable
counterexample for many conjectures in functional analysis, including the
hyperplane problem and the Schr?derBernstein problem for Banach spaces.
Gowers' contribution also opened the way to the solution of one of the most
famous problems in functional analysis, the socalled "homogeneous space
problem". A year ago, Gowers attracted attention in the field of combination
analysis when he delivered a new proof for a theorem of the mathematician
Emre Szemeredi which is shorter and more elegant than the original line of
argument. Such a feat requires extremely deep mathematical understanding.
Maxim Kontsevich
(born 25 August 1964)
He is a 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.
Maxim Kontsevich has established a reputation in pure mathematics and
theoretical physics, with influential ideas and deep insights. He has been
influenced by the work of Richard Feynmann and Edward Witten. Kontsevich is
an expert in the socalled "string theory" and quantum field theory. He made
his name with contributions to four problems of geometry. He was able to
prove a conjecture of Witten and demonstrate the mathematical equivalence of
two models of socalled quantum gravitation. The quantum theory of gravity
is an intermediate step towards a complete unified theory. It harmonises
physical theories of the macrocosm (mass attraction) and the 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)
He 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.
Curtis T. McMullen is being awarded a medal primarily in recognition of his
work in the fields of geometry and "complex dynamics", a branch of the
theory of dynamic systems, better known perhaps as chaos theory. McMullen
has made contributions in numerous fields of mathematics and fringe areas.
He already provided one important result in his doctoral thesis. The
question was how to calculate all the solutions of an arbitrary equation.
For simple equations it is possible to obtain the solutions by simple
rearrangement. For most equations, however it is necessary to use
approximation. One wellknown form is the "Newton method"  already known in
a rudimentary form in ancient times. For seconddegree polynomials this
provides very good results without exception. A key question therefore was
whether a comparable method  which happened not to have been discovered 
also existed for equations of higher degrees. Curtis T. McCullen's
conclusion was that there is definitely no such universal algorithm for
equations above degree three; only a partially applicable method is
possible. For degreethree equations he developed a "new" Newtonian method
and could thus completely solve the question of approximation solutions. 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 centres of equilibrium. The
border between these two extremes is the socalled 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
characterisation 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 welldeveloped theory is available. McMullen's results were
suspected already in the sixties, but nobody had previously been able to
prove this exact characterisation of the Julia set.
Andrew J. Wiles awarded the "IMU silver plaque"
Andrew J. Wiles
(born 11 April 1953)
He 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 British mathematician Andrew J. Wiles has been honoured with the "IMU
silver plaque" at 18 August. The chairman of the Fields Medals Committee,
Yuri Manin, presented him with this award during the opening ceremony of the
International Congress of Mathematicians in the Berlin International
Congress Centre. This world congress of mathematicians, the largest and most
important one worldwide, is taking place until 25th August. Some 3500
mathematicians from all over the world will be visiting Berlin to
participate. 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
favourite 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
