The International Congress of Mathematicians (Hyderabad, India) has been awarding four Fields Medals, the Nevanlinna Prize, the Gauss Prize, and the new Chern Medal.

**FIELDS MEDALS**

**Elon Lindenstrauss** (Israel), “*for his results on measure rigidity in ergodic theory, and their applications to number theory*“. Ergodic theory is the study of measure preserving transformations (areas, volumes, …) for various spaces. With these ergodic theoretic and arithmetical ideas, he and his collaborators have found many applications to problems in classical number theory, such as the Littlewood Conjecture.

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**Ngô Bảo Châu** (France-Vietnam), “*for his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebro-geometric methods*“. He established a brillant proof of the Fundamental Lemma, a conjecture suggested in 1987 that is a key aspect of the Langland Programme. It is worth to note that the Shimura-Taniyama-Weil conjecture demonstrated by Andrew Wiles in his proof of the Fermat’s theorem, as well as the work of the Fields medalist Laurent Lafforgue, did also concern the Langland Programme.

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**Stanislav Smirnov** (Russia), “*for the proof of conformal invariance of percolation and the planar Ising model in statistical physics*“. Conformal invariance are transformations which locally preserve angles but not necessarily distances. In particular, Smirnov demonstrated the conformal invariance of percolation on triangular lattices. He also demonstrated the conformal invariance of the bidimensional Ising model, which is extremely useful in statistical physics.

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**Cédric Villani** (France), “*for his proofs of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation*“. At the end of the XIXth century, Ludwig Boltzmann (1844-1906) proposed to study the statistical behavior of gases instead of tracking each particle individually. In particular, he established the first equations of evolution to statistical equilibrium. However, the convergence rate to equilibrium when the gas is initially not close to equilibrium is a very difficult problem. Villani obtained first results to this problem in collaboration with Desvillettes in the 1990’s. Furthermore, in 1946, Lev Laudau (1908-1968) studied damping effects in plasma physics, and proposed a simplified linear equation of plamas similar to the Boltzmann equation. Recently, Villani has established the complete nonlinear equation, in collaboration with Mouhot, which definitely confirmed the theory of Laudau.

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**NEVANLINNA PRIZE**

**Daniel Spielman** (USA), “*for smoothed analysis of Linear Programming, algorithms for graph-based codes and applications of graph theory to Numerical Computing*“. This prize is awarded for remarkable contributions to the mathematical aspects of information science. Spielman, in collaboration with Teng, has established the efficiency of the “simplex method”, a widely used algorithm in linear programming. In addition, he provided very efficient techniques in coding theory.

**GAUSS PRIZE**

**Yves Meyer** (France), “*for fundamental contributions to number theory, operator theory and harmonic analysis, and his pivotal role in the development of wavelets and multiresolution analysis*“. In the 1970’s, Meyer developed mathematical tools in number theory, especially to the study of quasi-crystals (regular structures generally aperiodic). Later, he made important contributions to the development of wavelet theory, which is similar but more powerful than Fourier analysis to decompose non stationary signals such as sounds or images. Today, wavelets are widely used in computers such as the image compression standard JPEG-2000. Finally, Meyer found a connection between his early work on quasy-crystals and the technique of compressed sensing used for acquiring and reconstructing a signal from sparse or compressed informations. This kind of algorithm has been used by the European Space Agency in the mission Herschel.

**CHERN MEDAL**

**Louis Nirenberg** (USA), “*for his role in the formulation of the modern theory of non-liner elliptic partial differential equations and for mentoring numerous students and post-docs in this area*“. The first Chern Medal was awarded to Louis Nirenberg. His work is focused on partial differential equations, ubiquitous in mathematics and their applications (an example among many others is Navier-Stokes). He also made contributions to differential geometry, complex analysis and topology, which are domains often in conjunction with the study of partial differential equations.

Sources : icm2010, cnrs, pour la science