Artur Avila

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Artur Ávila, Oberwolfach 2012

Artur Ávila Cordeiro de Melo (born June 29, 1979 in Rio de Janeiro ) is a Brazilian and French mathematician and Fields Medal Prize winner.

Life

Ávila won a gold medal at the International Mathematical Olympiad in 1995 . In 2001 he received his doctorate in his hometown at the Instituto de Matemática Pura e Aplicada (IMPA) with Welington de Melo ; the topic of his dissertation was Bifurcations of unimodal maps: the topological and metric picture . He was then Maître de Conférences at the Collège de France and from 2003 researcher at the CNRS , from 2008 as Research Director. From 2006 he was at the Clay Mathematics Institute and at IMPA. Among other things, he was visiting scholar at Caltech and at the State University of New York at Stony Brook . Since 2009 he has been at the Institute for Probability Theory at the University of Pierre and Marie Curie (University of Paris VI) and at the IMPA . From September 2018 Ávila will be a full professor at the University of Zurich.

plant

Ávila deals with dynamic systems of various kinds as well as mathematical physics such as the spectral theory of Schrödinger operators. Among other things, in his dissertation and the work resulting from it, he expanded the investigation of quadratic mappings of the unit interval, known from the Feigenbaum scenario, to unimodal mappings ( i.e. continuous maps with a maximum).

With Svetlana Jitomirskaya he solved the 10 Martini problem of Barry Simon , which concerns an electron in a magnetic field to the structure of the spectrum. The spectrum has a fractal nature, as Douglas Hofstadter recognized in his dissertation (“nested butterflies”). Ávila and Jitomirskaya proved this for a wide range of parameters.

Ávila worked with Marcelo Viana , Mikhail Lyubich and Jean-Christophe Yoccoz , among others . In 2005 he and Viana proved a conjecture by Maxim Kontsevich and Anton Zorich about the Lyapunov exponents of the Teichmüller river on the modular space of Abelian differentials on compact Riemann surfaces (namely that the non-trivial Lyapunov exponents are all different). Recently he and Sylvain Crovisier and Amie Wilkinson proved that generic volume-preserving diffeomorphisms with positive metric entropy form an ergodic -dynamic system.

Awards

In 2006 he received the Salem Prize and the CNRS Bronze Medal . For the International Congress of Mathematicians (ICM) 2010 he was invited to a plenary lecture (Dynamics of renormalization operators). In 2008 he received the EMS Prize . He was an Invited Lecturer at the European Congress of Mathieu (ECM) 2008 ( Dynamics of quasiperiodic cocycles and the spectrum of the almost Mathieu Operator ). In 2009 he received the Herbrand Prize from the French Mathematical Society. In 2014 he received the Fields Medal . In 2019, Ávila was elected to the National Academy of Sciences as a foreign member .

Fonts

  • with M. Viana: Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture. Acta Math. 198 (2007), no. 1, 1-56.
  • with S. Jitomirskaya: The Ten Martini Problem. Ann. of Math. (2) 170 (2009), no. 1, 303-342.
  • with Raphaël Krikorian : Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles , Annals of Mathematics, Volume 164, 2006, pp. 911-940, Arxiv
  • with J. Kahn , M. Lyubikh , W. Shen: Combinatorial rigidity for unicritical polynomials. Ann. of Math. (2) 170 (2009), no. 2, 783-797.
  • Dynamics of renormalization operators. Proceedings of the International Congress of Mathematicians. Volume I, 154-175, Hindustan Book Agency, New Delhi, 2010. pdf
  • Global theory of one-frequency Schrödinger operators. Acta Math. 215 (2015), no. 1, 1-54.
  • with Amie Wilkinson, Sylvain Crovisier: Diffeomorphisms with positive metric entropy, Arxiv 2014

literature

  • Forni, Giovanni: On the Brin Prize work of Artur Avila in Teichmüller dynamics and interval-exchange transformations. J. Mod. Dyn. 6 (2012), no. 2, 139-182.
  • Lyubich, Mikhail: Forty years of unimodal dynamics: on the occasion of Artur Avila winning the Brin Prize. J. Mod. Dyn. 6 (2012), no. 2, 183-203.
  • Étienne Ghys : The work of Artur Avila . In: Sun Young Jang, Young Rock Kim, Dae-Woong Lee, Ikkwon Yie (Eds.): Proceedings of the International Congress of Mathematicians, Seoul 2014: Volume I, Plenary Lectures and Ceremonies . Kyung Moon Sa, Seoul 2014, ISBN 978-89-6105-804-9 , pp. 47–54 ( Online [PDF; 18.5 MB ; accessed on August 31, 2017]).

Web links

Individual evidence

  1. ↑ http://www.media.uzh.ch/de/medienmitteilungen/2018/Artur-Avila.html "Fields-Medaille" winner Artur Avila becomes full professor at UZH , press release of the University of Zurich on July 24, 2018
  2. ↑ Avila, Viana Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture , Acta Mathematica, Volume 198, 2007, pp. 1–56 (PDF; 441 kB)
personal data
SURNAME Ávila, Artur
ALTERNATIVE NAMES Ávila Cordeiro de Melo, Artur (full name)
BRIEF DESCRIPTION Brazilian mathematician
DATE OF BIRTH June 29, 1979
PLACE OF BIRTH Rio de Janeiro , Brazil