Jean-Christophe Yoccoz

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Jean-Christophe Yoccoz (in Oberwolfach 2005)

Jean-Christophe Yoccoz (born May 29, 1957 in Paris ; † September 3, 2016 ) was a French mathematician . He was awarded the Fields Medal in 1994 for his work in the field of dynamic systems .

Life

Yoccoz won a gold medal at the International Mathematical Olympiad in 1974 , studied from 1975 at the École normal supérieure (where he was the best in the entrance exams as well as in those of the École polytechnique ) with the completion of the Agrégation in mathematics in 1977. From 1979 he was he Attaché de Recherches and later Chargé des Recherches of the CNRS . 1981 to 1983 he did his military service in Brazil during a research stay at the IMPA . He received his doctorate in 1985 with Michael Herman at the École polytechnique in Paris, with a thesis on dynamic systems (Centralisateurs et conjugaison différentiable des difféomorphismes du cercle). In 1988 he became a professor at the University of Paris-South and also a member of the Institut Universitaire de France . Since 1996 he has been a professor at the Collège de France . He died in 2016 after a long illness.

Yoccoz was a member of the Académie des Sciences (1994) and belonged to the Nicolas Bourbaki collective of authors .

Ricardo Pérez-Marco is one of his PhD students .

Services

His field of work on the theory of dynamic systems has a long tradition in France, which goes back to the investigation of the three-body problem by Henri Poincaré , but also to the iteration of complex functions by Gaston Julia or in the theory of fractals by Benoît Mandelbrot . In the 1950s, the famous KAM theory was developed there by Kolmogorov , Wladimir Arnold and Jürgen Moser , which describes the existence of quasi-periodic orbits around tori in phase spaces ( tori , as defined by closed orbits corresponding to rational periods) - depending on the degree of irrationality of the Periods. Another pioneer was Carl Ludwig Siegel and Stephen Smale for the application of topology . From the 1970s the field experienced a new boom thanks to chaos theory (existence of “strange attractors” in phase space around which the orbits are asymptotically close).

Yoccoz not only examined the qualitative behavior of such dynamic systems (“hyperbolic” chaotic systems or quasi-regular etc.), but also the Julia sets and Mandelbrot sets in complex dynamics. He proved that the Mandelbrot set is locally connected for finitely renormalizable parameter values. That the Mandelbrot set connected is already proven in the 1980s Adrien Douady and John H. Hubbard and she suspected that this is locally connected for all parameter values, which is one of the great unsolved problems of complex dynamics, since it after Douady and Hubbard Far-reaching conclusions can be drawn about the topological structure of the Mandelbrot set.

Honors

In 1988 Yoccoz received the Salem Prize . In 1994 he was awarded the Fields Medal (plenary lecture at the ICM in Zurich: Recent Developments in Dynamics ). In 1990 he was invited speaker at the International Congress of Mathematicians in Kyōto (Optimal arithmetic conditions in some small divisor theorems). In 1991 he became a member of the Institut Universitaire de France. He was a member of the Brazilian Academy of Sciences and the Third World Academy of Sciences and was Knight of the Legion of Honor and Officer of the Ordre du Mérite in 1995 (2000).

Further prizes are the bronze medal of the CNRS (1984), the Mathematics Prize from IBM (1985), the Prix Jaffe of the Académie des Sciences in 1991 and the Grand Cross of the Scientific Order of Merit of Brazil in 1998.

Works (selection)

  • Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation verifie une condition diophantienne. Ann. Sci. École Norm. Sup. (4) 17 (1984) no. 3, 333-359.
  • With Lennart Carleson , PW Jones: Julia and John. Bol. Soc. Brazil. Mat. (NS) 25 (1994) no. 1, 1-30.
  • An introduction to small divisor problems. In: Michel Waldschmidt , Claude Itzykson , Jean-Marc Luck, Pierre Moussa (editor): Number Theory and Physics. Les Houches 1989, Springer 1992.
  • Continued fraction algorithms for interval exchange maps: an introduction. In: Pierre Cartier et al .: Frontiers in Number Theory, Physics and Geometry. Volume 1, Springer Verlag, 2006.
  • With Stefano Marmi, Pierre Moussa: Some properties of real and complex Brjuno functions. In: Pierre Cartier et al .: Frontiers in Number Theory, Physics and Geometry. Volume 1, Springer Verlag, 2006.
  • Petits diviseurs en dimension 1. Astérisque, 231 (1995).
  • Il n'y a pas de contre-example de Denjoy analytique. CR Acad. Sci. Paris, Series I, t. 1984, 298: 141-144.
  • With J. Palis: Rigidity of centralizers of diffeomorphisms. Ann. Sci.ENS, 4e série, t. 22: 81-98 (1989).
  • Conjugaison analytique des difféomorphismes du cercle. 1989 (unpublished).
  • With Jacob Palis : Homoclinic tangencies for hyperbolic sets of large Haussdorff dimension. Acta Mathematica, 172: 92-136 (1994).
  • With Ricardo Pérez-Marco : Germes de feuilletages holomorphes à holonomie prescrite. Astérisque, 222 (1994) 345-371.
  • With P. Le Calvez: Un théorème d'indice pour les homéomorphismes du plan au voisinage d'un point fixe. Annals of Mathematics, 146: 241-293 (1997).
  • With C. Moreira: Stable intersections of regular Cantor sets with large Hausdorff dimension. Annals of Math., 154: 45-96 (2001).
  • With S. Marmi, P. Moussa: Complex Brjuno functions. Journal of the American Math. Soc., 14 (2001), 783-841.

literature

Web links

Individual evidence

  1. ^ Le mathématicien français Jean-Christophe Yoccoz, médaille Fields, est mort. At: Liberation.fr . September 5, 2016 (French).
  2. ^ Décès du mathématicien français Yoccoz, médaille Fields 1994. At: Ouest-France.fr. September 5, 2016 (French).
  3. ↑ Maurice Mashaal: Bourbaki. AMS 2006, p. 17; members are quoted at a meeting in 1995.
personal data
SURNAME Yoccoz, Jean-Christophe
BRIEF DESCRIPTION French mathematician
DATE OF BIRTH May 29, 1957
PLACE OF BIRTH Paris , Seine department , France
DATE OF DEATH 3rd September 2016