Borcherds, Gowers, Kontsevich, and McMullen Receive Fields Medals

Borcherds, Gowers,

Kontsevich, and McMullen

Receive Fields Medals

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On August 18, 1998, four Fields Medals were pre-
sented at the Opening Ceremonies of the Interna-
tional Congress of Mathematicians (ICM) in Berlin.
The four medalists are: R

ICHARD

E. B

ORCHERDS

, W

ILLIAM

IMOTHY

OWERS

, M

AXIM

ONTSEVICH

, and C

URTIS

ULLEN

At the 1924 Congress in Toronto a resolution

was adopted that at each ICM two gold medals
should be awarded to recognize outstanding math-
ematical achievement. J. D. Fields, a Canadian
mathematician who was secretary of the 1924 Con-
gress, later donated funds establishing the medals,
which were named in his honor. Consistent with
Fields’s wish that the awards recognize both ex-
isting work and the promise of future achieve-
ment, the medals are awarded to young math-
ematicians, where “young” has traditionally been
interpreted to mean no more than forty years of
age in the year of the Congress. In 1966 it was
agreed that, in light of the great expansion of
mathematical research, up to four medals could be
awarded at each ICM. Today the Fields Medal is
widely recognized as the world’s highest honor in
mathematics.

Previous recipients are: Lars V. Ahlfors and

Jesse Douglas (1936); Laurent Schwartz and Atle
Selberg (1950); Kunihiko Kodaira and Jean-Pierre
Serre (1954); Klaus F. Roth and René Thom (1958);
Lars Hörmander and John W. Milnor (1962); Michael
F. Atiyah, Paul J. Cohen, Alexander Grothendieck,
and Stephen Smale (1966); Alan Baker, Heisuke
Hironaka, Sergei P. Novikov, and John G. Thomp-
son (1970); Enrico Bombieri and David B. Mumford
(1974); Pierre R. Deligne, Charles L. Fefferman,
Grigorii A. Margulis, and Daniel G. Quillen (1978);

Alain Connes, William P. Thurston, and Shing-Tung
Yau (1983); Simon K. Donaldson, Gerd Faltings, and
Michael H. Freedman (1986); Vladimir Drinfeld,
Vaughan F. R. Jones, Shigefumi Mori, and Edward
Witten (1990); Jean Bourgain, Pierre-Louis Lions,
Jean-Christophe Yoccoz, and Efim Zelmanov (1994).

The committee choosing the 1998 Fields Medal-

ists consisted of: John Ball (Oxford University),
John Coates (Cambridge University), J. J. Duister-
maat (University of Utrecht), Michael H. Freedman
(Microsoft Research), Jürg Fröhlich (Eidgenössische
Technische Hochschule, Zürich), Robert MacPher-
son (Institute for Advanced Study, Princeton), Yuri
Manin (chair, Max-Planck-Institut für Mathematik,
Bonn), Kyoji Saito (University of Kyoto), and
Stephen Smale (City University of Hong Kong).

Richard Borcherds

Richard Borcherds was born on November 29,
1959, in Cape Town, South Africa. He received his
undergraduate and doctoral degrees from the Uni-
versity of Cambridge. He has held various positions
at Cambridge and at the University of California,
Berkeley. Currently he is on leave from Berkeley and
is a Royal Society Research Professor at Cambridge.
In 1992 he received a European Mathematical So-
ciety Prize at the First European Congress of Math-
ematicians in Paris. He was an Invited Speaker at
the ICM in Zürich in 1994.

As a student of John H. Conway, Borcherds

began his research career in finite group theory.
He has distinguished himself not only by utilizing
techniques and ideas from outside of finite group
theory but also by producing results that have had
an impact in other areas. In the classification of fi-

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nite simple groups, one of the most mysterious ob-
jects found was the monster group. There are var-
ious conjectures that attempt to connect the mon-
ster to other parts of mathematics. Borcherds
invented the notion of a vertex algebra and used
it to solve the Conway-Norton conjecture, which
concerns the representation theory of the monster
group (this theory is sometimes called “monstrous
moonshine”). He used these results to generate
product formulae for certain modular and auto-
morphic forms. The first such formulae were found
in the one-dimensional case by Euler and Jacobi,
and the conventional wisdom in algebraic geome-
try was that such product formulae could not exist
in higher dimensions. Borcherds’s work is also im-
portant in physics, as it lays rigorous groundwork
for conformal field theory in two dimensions.

William Timothy Gowers

William Timothy Gowers was born on November
20, 1963, in Marlborough, England. He received his
undergraduate and doctoral degrees from the Uni-
versity of Cambridge, where he was a student of
Belá Bollobás. Gowers was a Research Fellow at
Trinity College, Cambridge, before spending four
years at University College, London. He was then
appointed as a Lecturer at Cambridge and a Fel-
low of Trinity College. Currently he holds the Rouse
Ball Chair of Mathematics at Cambridge. He was
an Invited Speaker at the ICM in Zürich in 1994. In
1996 he received a European Mathematical Soci-
ety Prize at the Second European Congress of Math-
ematics in Budapest.

Gowers works in the areas of Banach space the-

ory and combinatorics. His main achievements are
his solutions to a number of famous problems
first stated in the 1930s by Stefan Banach. Gow-
ers and B. Maurey exhibited in 1991 a Banach space
having the property that none of its infinite-di-

mensional subspaces has an
unconditional basis. An un-
conditional basis provides a
useful coordinatization of
the space, guaranteeing
many “symmetries” (auto-
morphisms).

Gowers also produced an

example of a Banach space
that is not isomorphic to
any of its hyperplanes,
thereby solving the famous
Banach hyperspace prob-
lem. He proved a “di-
chotomy theorem”, which
says that every Banach
space has either a subspace
that has an unconditional
basis, and therefore many
symmetries, or is such that
all of its subspaces have
only trivial symmetries. This
work solves in the affirma-
tive the homogeneous space
problem, one of the central problems in Banach
space theory, which asks whether a homogeneous
Banach space is a Hilbert space. A hallmark of
Gowers’s work is the way in which it combines tech-
niques of analysis with combinatorial arguments.
His work in combinatorics and combinatorial num-
ber theory includes results about Szeméredi’s
lemma and an improved proof of Szeméredi’s the-
orem on arithmetic progressions.

Maxim Kontsevich

Maxim Kontsevich was born on August 25, 1964,
in Moscow. He received his doctoral degree from
the University of Bonn, under the direction of
Don B. Zagier. After holding a professorship at

Richard Borcherds

William Timothy Gowers

Maxim Kontsevich

Curtis T. McMullen

Photographs courtesy of the ICM, Berlin, 1998.

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the University of California, Berkeley, he moved to
his present position as professor at the Institut des
Hautes Études Scientifiques in Bûres-sûr-Yvette,
France. In 1992 Kontsevich received a European
Mathematical Society Prize at the First European
Congress of Mathematics in Paris. He was a Plenary
Speaker at the ICM in 1994 in Zürich.

Kontsevich first received international atten-

tion for his doctoral thesis, in which he proved a
conjecture of Edward Witten. This conjecture says
that the generating function for intersection num-
bers on the moduli spaces of algebraic curves sat-
isfies the Korteweg-de Vries equation. Also draw-
ing on ideas of Witten, Kontsevich produced a vast
generalization of the Gauss linking number for
knots. He then used this generalization and a new
notion of “graph cohomology” to generate Vas-
siliev knot invariants as well as invariants for three-
manifolds. Kontsevich produced the first math-
ematical definition of the “number” of rational
curves on Calabi-Yau manifolds, such as three-di-
mensional quintics, and gave an explicit formula
for this number. This work was crucial for later
work in the area of mirror symmetry. Most re-
cently Kontsevich has established that any Poisson
manifold admits a formal quantization and has pro-
vided an explicit formula for the flat case.

Curtis T. McMullen

Curtis T. McMullen was born on May 21, 1958, in
Berkeley, California. He received his undergradu-
ate degree in 1980 from Williams College and his
doctoral degree in 1985 from Harvard University.
His thesis advisor was Dennis Sullivan. McMullen
has held positions at the Massachusetts Institute
of Technology, the Mathematical Sciences Research
Institute, the Institute for Advanced Study, Prince-
ton University, and the University of California,
Berkeley. At present he is a professor of math-
ematics at Harvard. In 1991 he received the Salem
Prize.

McMullen has produced important results in

several areas of mathematics, including the theory
of computation, dynamical systems, and three-
manifolds. In his doctoral thesis he used dynam-
ical systems techniques to solve completely the
question of whether there exist generally conver-
gent algorithms for finding the zeros of polyno-
mials of degree three or greater. Newton’s method
converges for almost all quadratic polynomials
and almost all initial points. McMullen exhibited
an analogous algorithm for degree three polyno-
mials and proved that no such algorithm exists for
degree four and higher. He has also made impor-
tant strides toward solving one of the central con-
jectures in one-dimensional dynamics: Are the hy-
perbolic maps of degree

dense in all maps of

degree

? McMullen proved that, given

→

, if

is in a connected component of the

Mandelbrot set that intersects the real axis, then

is hyperbolic. He also brought new ideas and in-

sights from dynamical systems to the geometriza-
tion program for three-manifolds formulated by
1982 Fields Medalist William Thurston. McMullen
has also worked with Sullivan on a “dictionary” be-
tween the theory of iterations of rational maps of
the Riemann sphere and that of Kleinian groups.

—Allyn Jackson

Wiles Receives Special Award

At the ICM Open-
ing Ceremonies,
Andrew J. Wiles of
Princeton Univer-
sity received a
one-time Special
Tribute from the
I n t e r n a t i o n a l
M a t h e m a t i c a l
Union in recogni-
tion of his work
that led to the
proof of Fermat’s

Last Theorem. Because he is over forty years
old, Wiles was not considered eligible for a
Fields Medal. Instead of a gold medal he re-
ceived the “IMU Silver Plaque”, or, as number
theorist Don B. Zagier called it, a “Quantized
Fields Medal”. Wiles’s award also differed from
the Fields Medals in that no lecture was pre-
sented about his work. Instead, the next day
Wiles himself gave a special lecture entitled
“Twenty Years of Number Theory”.

In 1993 Wiles announced that he had proved

Fermat’s Last Theorem. The ground-breaking
research he did in order to produce the proof
seemed likely to secure him a Fields Medal at
the ICM in Zürich in 1994 until a gap appeared
in the proof. The gap was not repaired until
after the Zürich Congress. With the proof com-
plete, the general consensus was that Wiles had
done work of Fields Medal quality. This view
was reinforced by the 3,000 people assembled
for the ICM Opening Ceremonies, who gave
Wiles a thundering round of applause longer
than that given to any of the other awardees.

—A. J.

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