... for works on a broad spectrum of problems in applied analysis. His results on the Mumford-Shah conjecture in the theory of computer vision meant a breakthrough. This problem deals with a variational problem with a singular boundary set, and proposes a finite representation of the optimum solution. Bonnet obtained the first finiteness result under additional assumptions, which is a major step in understanding this difficult free boundary value problem. In a different direction, his results on partial differential equations, in particular on flame propagation and combustion, are very significant.

... whose 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 Schröder-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 Fredholm 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.

... 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 continuous étale cohomology - naturally factorise.

... 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.

... 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.

... whose achievements have combinatorial and geometric flavour; 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 derandomisation of geometric algorithms. He obtained the best results on several key problems in computational and combinatorial geometry and optimisation, 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 dimensional Banach spaces using uniform distributions of points on spheres. In mathematical logic, he found a striking example of a combinatorial unprovable statement.

... 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).

... whose 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 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.

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

... contributed in a most important way to several domains of geometry and dynamical systems, in particular to symplectic geometry. Polterovich ties together complex analytic and 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.

... has contributed greatly both to the Asymptotic Theory of Convexity and to Classical Convexity Theory. His most significant work is on valuations (additive functionals) on convex bodies and it has remodeled a central part of convex geometry. Group invariant valuations were studied since Dehn's solution of Hilbert's third problem, with later contributions by Blaschke and others, and culminating in Hadwiger's celebrated characterization theory for the intrinsic volumes. The latter theorem was considered the top result in this area for almost fifty years. Alesker has now considerably extended this theory, obtaining very complete classification results under weaker invariance assumptions. He approximated (continuous) rotation invariant valuations by polynomial valuations and characterized the latter, making use of representations of the orthogonal group. This enabled him to extend Hadwiger's theorem to tensor valued valuations. In another direction, he solved a problem of McMullen, in a much stronger form, showing that translation invariant valuations are essentially (up to linear combinations and approximation) mixed volumes. The approach is via representation theory of the general linear group and involves a surprising application of D-modules. The new approach has also opened the way to finiteness results for valuations with other group invariances.

... has became known through his results on Probability theory. Using a large deviation principle in the proper topology, Raphael Cerf has established a Wulff construction for the supercritical percolation model in three dimensions. This result is a very major advance in the subject, and provides the right formulation for the geometry of the problem. Raphael Cerf has been able to carry out this program using a correct mixture of combinatorial arguments, geometric ideas and probabilistic tools. In addition to this research, Raphael Cerf has made original contributions in genetic algorithms. He has solved a central problem in bootstrap percolation and extended to three dimensions the metastable behaviour of the stochastic Ising model in the limit of low temperatures.

... is one of the leaders in the geometric Langlands correspondence and related areas. In the modern "geometric" representation theory one replaces functions by complexes of constructible sheaves on (infinite-dimensional) algebraic varieties. In this way many deep structures appear, and classical results in the theory of automorphic forms can be seen much more clearly. In his thesis and in the subsequent work with Braverman, Gaitsgory established fundamental properties of Eisenstein series in the geometric setting. In a recent paper on nearby cycles, he proposed an extremely elegant construction of the convolution of equivariant perverse sheaves on so-called affine Grassmannians. This implies that the center of the affine Hecke algebra conincides with the whole spherical Hecke algebra. Also, it gives the best conceptual explanation of the Satake equivalence. Recent work of Gaitsgory relates finite quantum groups and chiral Hecke algebras. It is a very important step in the program of Beilinson and Drinfield in the geometric Landlands theory.

... has greatly contibuted to the asymptotic analysis of Euler and Navier-Stokes equations with large Coriolis force. The simplest case (when the equations are set on the unit cube with periodic boundary conditions) has been solved by Grenier around 1995. Later, in collaboration with Desjardins, Dormy, and Masmoudi, he gave rigorous derivations of several asymptotic models currently used in Ocean and Atmosphere modelling, or in Magnetohydrodynamics. Grenier obtained both positive and negative important results on the problem of convergence of the Navier-Stokes equations to the Euler equations in a domain with solid boundary conditions. In particular, he showed that the positive results of Caflisch and Sammartino obtained for analytic initial data, cannot be extended to Sobolev data. He also justified the hydrostatic limit of the Euler equations in a two dimensional infinitesimally thin strip. Grenier gave a very elegant proof of convergence for the semi-classical limit of the nonlinear Schrödinger equations (before appearance of shocks). He also obtained, simultaneously with E Rykow and Sinai, a hydrodynamic limit for Zelodvich adhesion particle model.

... whose work on the existence of metrics with special holomony is among the best in Riemannian geometry in the last decade. The question of the existence of Riemannian metrics with special holomony has a long history beginning with the work of Cartan. It includes some of the best work of such people as M Berger, J Simons, S T Yau and B Bryant. Using a dazzling display of geometry and analysis, Joyce constructed compact examples in the exceptional cases where the holonomy is Spin7 and G2 the only remaining possibilities, the others on Berger's list had been eliminated. Joyce also computed the dimension of the deformation spaces of such metrics and many other of their invariants. As a result, he also discovered a totally unexpected version of mirror symmetry for such spaces. Dominic Joyce is one of the leading young differential geometers.

... whose work is a major advance in the K-theory of operator algebras: the proof of the Baum-Connes conjecture for discrete co-compact subgroups of SL3 (R), SL3 (C), SL3 (Qp ) and some other locally compact group (and of more general objects). The conjecture plays a central role in non-commutative geometry and has far-reaching connections to the Novikov conjecture on higher signatures in topology, to harmonic analysis on discrete groups and the theory of C*-algebras. Lafforgue's result is the first passage of the barrier which property T of Kazhdan has posed for many years in the proof of the Baum-Connes conjecture. The proof involves several remarkable technical and conceptual developments, like a bivariant K-theory for Banach algebras (versus Kasparov's by now classical one for C*-algebras) or establishing the conjecture for various completions of the L1 algebras of the groups.

... has created the method of dynamic diophantine approximation which has led to a series of remarkable results in complex geometry of algebraic varieties. Among these results one can mention a new proof of Bloch's conjecture on holomorphic curves in closed subvarieties of abelian varieties, the proof of the conjecture of Green and Griffiths that a holomorphic curve in a surface of general type cannot be Zariski-dense, and the hyperbolicity for generic hypersurfaces in a projective space P3 of high enough degree (Kobayashi conjecture).

... has obtained several strong results on topology and complex analysis. Using modern techniques like the famous Seiberg-Witten invariants he has solved some old classical problems about sub-manifolds in complex domains. First, he generalized Thom inequality proved by P Kronheimer and T Mrowka. As a very particular case he proved that there are no nonconstant holomorphic functions in a neighbourhood of an embedded non-trivial 2-sphere in a complex projective plane. Another application of his main theorems is also very attractive. Suppose that an analytic disc is attached from the outside to a strictly pseudoconvex domain U in a complex 2-plane, then there is no smooth disc inside of U wich the same boundary. As a corollary one gets that it is impossible to attach an analytic disc from the outside to a strictly pseudoconvex domain that is diffeomorphic to a closed ball.

... became known through his work on symplectic topology. In his PhD theses he studied the fundamental question whether symplectic diffeomorphisms which are diffeotopic to the identity are also symplectically diffeotopic to the identity. He showed that the answer is negative in many cases, already in dimension 4. His counterexamples are generalized Dehn twists, his proof involves Floer homology. In further works, Seidel constructed a natural representation of the fundamental group of the group of Hamiltonian symplectomorphisms into the quantum cohomology ring. This work was basic for later work of Lalonde, McDuff, and Polterovich on the topology of the group of symplectomorphisms. There is more to say about other work. His latest work is related to mirror symmetry, showing his broad horizon.

... has obtained deep results on stochastic processes and, more precisely, he has proved a number of significant results on Brownian path properties, including the shape of Brownian islands and Brownian windings. Wendelin Werner has made remarkable contributions to the study of self-avoiding random walks and the corresponding critical exponents. More specifically, he obtained the first non-trivial upper bound of the disconnection exponent, and he developed an elegant approach for studying the limiting behaviour of the non-intersection exponents for a great number of independent Brownian motions. Among many other interesting works he constructed, with a collaborator, the so-called true self-repelling motion using an ingenious method involving infinite systems of coalescing Brownian motions.

... pioneered the use of measure-transportation techniques (due to Kantorovich, Brenier, Caffarelli, Mc Cann and others) in geometric inequalities of harmonic and functional analysis with striking applications to geometry of convex bodies. His major achievement is an inverse form of classical Brascamp-Lieb inequalities. Further contributions include discovery of a functional form of isoperimetric inequalities and a recent solution (with Artstein, Ball and Naor) of a long-standing Shannon's problem on entropy production in random systems.

... has introduced an entirely new perspective to the theory of discontinuous solutions of one-dimensional hyperbolic conservation laws, representing solutions as local superposition of travelling waves and introducing innovative Glimm functionals. His ideas have led to the solution of the long standing problem of stability and convergence of vanishing viscosity approximations. In his best individual achievement, published in 2003 in Arch. Ration. Mech. Anal., he shows convergence of semidiscrete upwind schemes for general hyperbolic systems. In the technically demanding proof the travelling waves are constructed as solutions of a functional equation, applying centre manifold theory in an infinite dimensional space.

... has made fundamental and influential contributions to symplectic topology as well as to algebraic geometry and Hamiltonian systems. His work is characterised by new depths in the interactions between complex algebraic geometry and symplectic topology. One of the earlier contributions is his surprising solution of the symplectic packing problem, completing work of Gromov, McDuff and Polterovich, showing that compact symplectic manifolds can be packed by symplectic images of equally sized Euclidean balls without wasting volume if the number of balls is not too small. Among the corollaries of his proof, Biràn obtains new estimates in the Nagata problem. A powerful tool in symplectic topology is Biràn's decomposition of symplectic manifolds into a disc bundle over a symplectic submanifold and a Lagrangian skeleton. Applications of this discovery range from the phenomenon of Lagrangian barriers to surprising novel results on topology of Lagrangian submanifolds. Paul Biràn not only proves deep results, he also discovers new phenomena and invents powerful techniques important for the future development of the field of symplectic geometry.

... has done deep and highly original work at the interface of ergodic theory and number theory. Although he has worked widely in ergodic theory, his recent proof of the quantum unique ergodicity conjecture for arithmetic hyperbolic surfaces breaks fertile new ground, with great promise for future applications to number theory. Already, in joint work with Katok and Einsiedler, he has used some of the ideas in this work to prove the celebrated conjecture of Littlewood on simultaneous diophantine approximation for all pairs of real numbers lying outside a set of Hausdorff dimension zero. This goes far beyond what was known earlier about Littlewood's conjecture, and spectacularly confirms the high promise of the methods of ergodic theory in studying previously intractable problems of diophantine approximation.

... contributed greatly to the field of asymptotic combinatorics. An extremely versatile mathematician, he found a wide array of applications of his methods. His early results include a proof of a conjecture of Olshanski on the representations theory of groups with infinite-dimensional duals. Okounkov gave the first proof of the celebrated Baik-Deift-Johansson conjecture, which states that the asymptotics of random partitions distributed according to the Plancherel measure coincides with that of the eigenvalues of large Hermitian matrices. An important and influential result of Okounkov is a formula he found in joint work with Borodin, which expresses a general Toeplitz determinant as the Fredholm determinant of the product of two associated Hankel operators. The new techniques of working with random partitions invented and successfully developed by Okounkov lead to a striking array of applications in a wide variety of fields: topology of module spaces, ergodic theory, the theory of random surfaces and algebraic geometry.

... was the first to make a systematic and impressive asymptotic analysis for the case of large parameters in Theory of Ginzburg-Landau equation. She established precisely the values of the first, second and third (with E Sandier) critical fields for nucleation of one stable vortex, vortex fluids and surface superconductivity. In micromagnetics, her work with F Alouges and T Rivière breaks new ground on singularly perturbed variational problems and provides the first explanation for the internal structure of cross-tie walls.

... whose most striking result is the proof of existence and conformal invariance of the scaling limit of crossing probabilities for critical percolation on the triangular lattice. This gives a formula for the limiting values of crossing probabilities, breakthrough in the field, which has allowed for the verification of many conjectures of physicists, concerning power laws and critical values of exponents. Stanislav Smirnov also made several essential contributions to complex dynamics, around the geometry of Julia sets and the thermodynamic formalism.

... has made fundamental contributions to Harmonic and Complex Analysis. His most outstanding work solves Vitushkin's problem about semiadditivity of analytic capacity. The problem was raised in 1967 by Vitushkin in his famous paper on rational approximation in the plane. Tolsa's result has important consequences for a classical (100 years old) problem of Painlevé about a geometric characterization of planar compact sets are removable in the class of bounded analytic functions. Answering affirmatively Melnikov's conjecture, Tolsa provides a solution of the Painlevé problem in terms of the Menger curvature. Xavier Tolsa has also published many important and influential results related to Calderón-Zygmund theory and rational approximation in the plane.

... has given a rigorous proof that the Lorenz attractor exists for the parameter values provided by Lorenz. This was a long standing challenge to the dynamical system community, and was included by Smale in his list of problems for the new millennium. The proof uses computer estimates with rigorous bounds based on higher dimensional interval arithmetics. In later work, Warwick Tucker has made further significant contributions to the development and application of this area.

... has made a number of important discoveries in both the algebraic and arithmetic aspects of non-commutative Iwasawa theory, especially on problems which appeared intractable from the point of view of the classical commutative theory. In arithmetic geometry, Iwasawa theory is the only general technique known for studying the mysterious relations between exact arithmetic formulae and special values of L-functions, as typified by the conjecture of Birch and Swinnerton-Dyer. Venjakob's work applies quite generally to towers of number fields whose Galois group is an arbitrary compact p-adic Lie group (which is not, in general, commutative), and has done much to show that a rich theory is waiting to be developed. His most important results include the proof of a good dimension theory for modules over Iwasawa algebras, and the proof of the first case of a structure theory for modules over these algebras. With Hachimori he discovered the first examples of arithmetic Iwasawa modules which are completely faithful, as well as proving a remarkable asymptotic upper bound for the rank of the Mordell Weil group of elliptic curves in certain towers of number fields over Q whose Galois group is a p-adic Lie group of dimension 2. Very recently, he found they key to the problem of defining, in non-commutative Iwasawa theory, the analogue of the characteristic series of modules over Iwasawa algebras.

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