**William Timothy Gowers'** work has made the geometry of
Banach spaces look completely different. To mention some of his spectacular
results: he solved the notorious Banach hyperplane problem,
to find a Banach space which is not isomorphic to any of its hyperplanes.
He gave a counterexample to the Schroeder-Bernstein theorem
for Banach spaces. He proved a deep dichotomy principle for Banach
spaces which if combined with a result of Komorowski and Tomczak-Jaegermann
shows that if all closed infinite-dimensional subspaces of a
Banach space are isomorphic to the space, then it is a Hilbert space.
He gave (jointly with Maurey) an example of a Banach space such that
every bounded operator from the space to itself is a Eredholm operator.
His mathematics is both very original and technically very strong. The
techniques he uses are highly individual; in particular, he makes very
clever use of infinite Ramsey theory.

**Annette Huber** developed a difficult and important theory,
the theory of the derived category of mixed motivic realizations. The
theory of motives was discovered by Alexander Grothendieck in the
60's. This important topic is still largely conjectural. The definition of
mixed motives is one of the central problems of this theory. Annette
Huber defines a derived category of the category of mixed realisations
defined by Jannsen. She constructs a functor from the category of
simplicial varieties to this derived category, whose cohomology objects
are precisely the mixed realisations of the variety. She then defines an
absolute cohomology theory, over which the usual absolute theories -
absolute Hodge-Deligne and continous &aecute;tale cohomology - naturally
factorise.

**Aise Johan de Jong** has produced in a large variety of deep
results on various aspects of arithmetic algebraic geometry. His personal
influence on the work in the field is impressive. His work is
characterized by a truly geometric approach and a abundance of new
ideas. Among others, his results include the resolution of a conjecture
of Veys and the answer to a long-standing question of Mumford on moduli
spaces. Resolution of singularities by modification is difficult and
unknown in most cases; in a recent outstanding work, de Jong found an
elegant method for the resolution of singularities by alterations, which
is a slightly weaker question but sufficient for most applications. This
basic method combines geometric insight and technical knowledge.

**Dmitri Kramkov** has important results in statistics and the
mathematics of finance. He did fundamental work in filtered statistical
experiments. In particular, he obtained a deep result on the structure
of Le Cam's distance between two filtered statistical experiments,
and proved very general theorems about the structure of the limit experiments
which cover many results in the asymptotic mathematical
statistics of stochastic processes. Recently he proved a remarkable
"Optional decomposition of supermartingales" which is an extension
of the fundamental Doob-Meyer decomposition for the case of many
probability measures. This unexpected result is rather difficult and
refined technically, and, from the conceptual point of view, very important.
In the direction of mathematical finance, Kramkov obtained
impressive results on pricing formulas for certain classes of "exotic"
options based on geometric Brownian motion. He succeeded in computing
explicit solutions for "Asian options" where the pay-off is given
by a time-average of geometric Brownian motion.

**Jiri Matousek's** achievements have combinatorial and geometric
flavor; his research is characterized by its breadth, by its algorithmic
motivation, as well as the difficulty of the problems he attacks.
He gave constructions of epsilon-nets in computational geometry, which
provide tools for derandomization of geometric algorithms. He obtained
the best results on several key problems in computational and
combinatorial geometry and optimization, such as linear programming
algorithms and range searching. He solved several long-standing problems
(going back to the work of K.F.Roth) in geometric discrepancy
theory, in particular on the discrepancy of halfplanes and of arithmetical
progressions. He solved a problem by Johnson and Lindenstrauss
on embeddings of finite metric spaces into Banach spaces. He also
obtained sharp results on almost isometric embeddings of finite dimenisional
Banach spaces using uniform distributions of points on spheres.
In mathematical logic, he found a striking example of a combinatorial
unprovable statement.

**Loic Merel** proved an absolute bound for the torsion of elliptic
curves. Thereby he gave a solution to a long-standing problem,
open for more than 30 years, that has resisted the efforts of the greatest
specialists of elliptic curves. The group of torsion points of an elliptic
curve over a number field is finite. Merel found a bound of the order
of this group in terms of the degree of the number field; such a bound
was known in a very few cases only (the case of the rational numbers
(Mazur 1976), number fields of degree less than 8 (Kamieny-Mazur
1992), and number fields of degree less than 14 (Abramovitch 1993).

**Grigory Perelman's** work played a major role in the development
of the theory of Alexandrov spaces of curvature bounded from
below, giving new insight into to what extent the results of Riemannian
geometry rely on the smoothness of the structure. Now, mainly
due to Perelman, the theory is rather complete. His results include a
structure theory of these spaces, a stability theorem (new even for Riemannian
manifolds), and a synthetic geometry a'la Aleksandrov. He
proved a conjecture of Gromov concerning an estimation of the product
of weights, and the the Cheeger-Gromov conjecture. This last problem
attracted the attention and efforts of many geometers for more than 20
years, and the method developed by Perelman yielded an astonishingly
short solution.

**Ricardo Perez-Marco** solved several outstanding problems,
and obtained basic results, in the theory of dynamics of non-linearizable
germs and non-linearizable analytic diffeomorphisms of the circle, and
in the theory of centralizers, a natural complement of non-linearizability.
He discovered a new arithmetic condition under which a germ
without periodic orbits is linearizable. He gave a negative answer to
a question of Arnold on the linearizability of analytic diffeomorphisms
of the circle without accumulating periodic orbits. Perez-Marco developed
a theory of analytic non-linearizable germs based on an important
and useful compact invariant.

**Leonid Polterovich** contributed in a most important way to
several domains of geometry and dynamical systems, in particualr to
symplectic geometry. Polterovich ties together complex analytic amd
dynamical ideas in a unique way, leading to significant progress in both
directions. In particular, he brings complex analysis into the realm of
Hamiltonian mechanics, which marks a principally new step in a this
classical field. Among others, he established (with Bialy) an anti-KAM
estimate in terms of the Hofer displacement of a Hamiltonian flow.
Polterovich found the first non-trivial restriction on the Maslov class
of an embedded Lagrangian torus, and (with Eliashberg) completely
solved the knot problem in the real 4-space.