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The 2014 Fields Medals

Jean-Christophe Yoccoz
p. 42-43
Cet article est une traduction de :
Les médailles Fields 2014

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Source: La lettre, no. 39, March 2015

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1The 27th International Congress of Mathematicians was held in Seoul from
13 to 21 August 2014. This event, which has taken place every four years since the inaugural congress in Zurich in 1897 (with interruptions due to the two World Wars), brought together over 5,000 participants and is the most important one in the field of mathematical sciences.

Artur Avila and Prof Jean-Christophe Yoccoz

2At the Paris Congress in 1900, David Hilbert proposed a list of 23 problems that were to have a profound influence on the development of mathematics in the twentieth century. The 28th International Congress of Mathematicians will be held in Rio de Janeiro in August 2018. Since the Oslo Congress in 1936, the International Congress of Mathematicians has been the occasion on which the International Mathematical Union awards the Fields Medals: between two and four at each Congress. This distinction, which rewards the work of mathematicians under the age of 40, is considered as the most prestigious one in mathematics. The laureates this year were Manjul Bhargava, Martin Hairer, Maryam Mirzakhani and Artur Avila.

3Manjul Bhargava, born in Canada into a family from Rajasthan, is a professor at Princeton University, where he obtained his PhD under Andrew Wiles’s supervision. In his work, which focuses on number theory, he has obtained remarkable results on the average rank of elliptic curves. Martin Hairer is an Austrian national. He studies mathemat-
ical physics problems in which randomness plays a major role, modelled by stochastic partial differential equations. After obtaining a PhD from the University of Geneva under the supervision of Jean-Pierre Eckmann, and a brief stay at the Courant Institute in New York, he is now professor at the University of Warwick. Maryam Mirzakhani, from Iran, was the first woman to receive the Fields Medal. She obtained a PhD from Harvard University under Curt McMullen’s supervision, and is now a professor at Stanford University. Her research is mainly on the geometry and dynamics of the moduli spaces of complex algebraic curves and of quadratic or Abelian differentials.

Artur Avila

4Given his ties with the Collège de France, I will devote more attention to Artur Avila. He was born in Rio de Janeiro in 1979. In 1995 (alongside Maryam Mirzakhani) he won a gold medal at the International Mathematical Olympiad.

5He obtained his PhD in 2001, at the Instituto Nacional de Matemática Pura e Aplicada (IMPA) in Rio de Janeiro, under Welington de Melo’s supervision. From 2001 to 2003, he did a post-doc at the Collège de France as an associate lecturer in my Chair of Differential Equations and Dynamic Systems. He entered the CNRS as a researcher in 2003, and became a senior researcher there in 2008. In 2005 he delivered the Peccot lecture series at the Collège de France and won numerous international distinctions before receiving the Fields Medal, including the Salem Prize in 2005, a prize from the European Mathematical Society in 2008, the Herbrand Prize of the French Academy of Sciences in 2009, and the Michael Brin Prize in 2011.

6Artur Avila splits his time between France, having recently acquired French citizenship, and Brazil, as the IMPA has been a CNRS international joint research unit since 2006.

7Most of Artur Avila’s work fits within the framework of the theory of dynamical systems. This discipline, founded by Henri Poincaré at the end of the nineteenth century, focuses on understanding the long-term behaviour of systems with a known short-term evolution. The applied conceptual tools depend on the degree of predictability of the dynamical systems considered. For highly predictable systems, so-called renormalization methods have proven highly powerful. They consist in carrying out a series of spatial and temporal scale changes on the systems considered. The succession of these changes defines a new evolution, whose properties reveal those of the initial system. Artur Avila and his colleagues have contributed significantly to our understanding of the dynamics of these renormalization transformations.

8For systems with a more chaotic nature, Lyapunov exponents provide a quantitative measurement of the rate of information loss caused by the evolution. Much of Artur Avila’s work is devoted to the crucial question of whether or not these exponents are null.

A Collège de France Visiting Professor Chair in Rio de Janeiro
Relations between the Collège de France and Brazil are longstanding and are a continuation of the contact established by Claude Lévi-Strauss in the 1930s, mainly at the University of São Paulo, where a first Visiting Professor Chair was created in 1998, which is now known as the “Lévi-Strauss Chair”. The Rio de Janeiro Chair is a more recent initiative dating back to 2009, the Year of France in Brazil, during which ten Collège de France professors took part in scientific events. The two institutions chose to place it under the auspices of Claude Bernard, Experimental Medicine (1855-1878).
On the French side, Profs Jean-Christophe Yoccoz and Roger Chartier coordinate the exchange. On the Brazilian side, several people are involved, particularly Prof. Nelson Maculan, an informatician, Prof. Luiz Davidovich, a physicist and member of the Collège de France International Scientific and Strategic Committee (Comité International d’Orientation Scientifique et Stratégique, COSS), and Academician Jacob Palis, current president of the Brazilian Academy of Sciences and former member of the COSS. On 17 November 2014 the convention was renewed until 2019, at a ceremony where Prof. Pierre Rosanvallon represented the Collège de France.

Fields Medals at the Collège de France
Four professors of the Collège de France have won the Fields Medal
Jean-Pierre Serre
Algebra and Geometry
Fields Medal 1954
Alain Connes
Analysis and Geometry
Fields Medal 1982
Pierre-Louis Lions
Partial Differential Equations and Applications
Fields Medal 1994
Jean-Christophe Yoccoz
Fields Medal 1994

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Légende Artur Avila and Prof Jean-Christophe Yoccoz
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Jean-Christophe Yoccoz, « The 2014 Fields Medals », La lettre du Collège de France [En ligne], 9 | 2015, mis en ligne le 25 septembre 2015, consulté le 18 juin 2018. URL :

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Jean-Christophe Yoccoz

Differential Equations and Dynamical Systems

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