Jean-Christophe Yoccoz †
Born 29 May 1957 in Paris (France)
Died 3 September 2016
Fields Medal (1994) for his contributions to the study of dynamical systems
Even as a teenager, Jean-Christophe Yoccoz’s mathematical accomplishments were considerable. In 1973 he won a silver medal at the International Mathematics Olympiad and the following year he took a gold medal. In 1975 he graduated from the École Normale Supérieure and two years later he received the Agrégation de Mathématiques (a teaching qualification). After he completed military service in Brazil – a country with which he still maintains a close relationship, among other things because his late wife was Brazilian – he received his doctorate (“Thèse d’Etat”) in 1985 advised by Michael Robert Herman, a leading expert in the study of dynamical systems, with a dissertation on circle diffeomorphisms. He taught at the Université de Paris-Sud (Orsay) from 1987 to 1997. He has taught at the Collège de France since 1997. He has also been a member of the Institut Universitaire de France (1991) and of the Working Group ‘Topologie et Dynamique’ at the Centre National de la Recherche Scientifique (CNRS) in Orsay.
In addition to the Fields Medal he has received many other awards, including the IBM Award in Mathematics (1985) and the Salem Prize (1988). He is Chevalier of the French Légion d’ Honneur (1995), winner of the Grand Cross of the Brazilian Scientific Order of Merit (1998) and Officer of the French Ordre du Mérite (2000). Yoccoz has been a member of the French and Brazilian Academies of Sciences since 1994, and since 2004 an associate member of The World Academy of Sciences (TWAS).
Jean-Christophe Yoccoz’s research focus is the theory of dynamical systems (often referred to by the public as chaos theory). The core principle of the theory of dynamical systems is iteration: a function is evaluated and the result is reused as a new input for the next iteration, and the process is repeated again and again. Through this repetition, the system can react extremely sensitively to changes in the initial values at certain points, which can lead to unpredictable (‘chaotic’) behavior.
Iteration is associated with a discrete model for time. When time is viewed as a continuous variable, a dynamical system is typically defined by a differential equation (or a partial differential equation). In fact, the theory of dynamical systems was first developed through the study of the differential equations governing the three-body-problem by the French mathematician Henri Poincaré at the end of the 19th century, describing the orbits of three celestial bodies whose gravity affects each other.
Yoccoz’s research is at the cutting edge of this theory, where he pursues a very wide range of interests. He has contributed to KAM theory (the persistence of quasiperiodic behavior), has studied the various properties of different hyperbolic sets of high fractal dimension, coming for instance from cuts or combinations of Cantor sets, and he has developed a combinatorial method – known as the Yoccoz puzzle – with which Julia sets can be examined. They can be described as maps, which show how chaotically a dynamical system behaves in any point.