Artur Avila

Artur Ávila Cordeiro de Melo ( born June 29, 1979 in Rio de Janeiro) is a Brazilian mathematician.

Ávila won the 1995 Gold Medal of the International Mathematical Olympiad. In 2001 he received his doctorate at the Instituto Nacional de Matematica Pura e Aplicada ( IMPA ) in Rio at Welington de Melo ( Bifurcations of unimodal maps: the topological and metric picture ). After that he was Maitre de Conference at the College de France and from 2003, researchers at the CNRS from 2008 as director of research. From 2006 he was at the Clay Mathematics Institute and at IMPA. He has been a visiting scientist at Caltech and at the State University of New York at Stony Brook. He is at the Laboratory of probability theory at the University Pierre and Marie Curie ( Paris VI University ) and at IMPA.

Ávila is concerned with dynamic systems of various kinds, as well as mathematical physics, such as spectral theory of Schrödinger operators. He expanded among other things in his dissertation and the resulting site works known from the Feigenbaum scenario investigation square pictures of the unit interval on unimodal maps ( ie continuous maps with a maximum).

With Svetlana Jitomirskaya he solved the problem of the 10 Martini Barry Simon, in which it comes to the structure of the spectrum of an electron in a magnetic field. The spectrum has fractal nature, as was Douglas Hofstadter in his dissertation recognized (nested butterflies). Avila and Jitomirskaya proved that for a large range of values ​​of the parameters.

Ávila worked with, among others, Marcelo Viana, Mikhail Lyubich and Jean -Christophe Yoccoz. In 2005 he proved a conjecture of Viana with Maxim Kontsevich and Anton Zorich on the Lyapunov exponents of the Teichmüller flow on the moduli space of abelian differentials on compact Riemann surfaces ( namely that the non-trivial Lyapunov exponents are all different ).

In 2006 he received the Salem Prize and the Bronze Medal of the CNRS. 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 price. He was Invited Lecturer at the European Congress of Mathematicians (ECM ) 2008 ( Dynamics of quasiperiodic cocycles and the spectrum of the almost Mathieu operator). In 2009 he was awarded the Herbrand Award of the French Mathematical Society.

Writings

• 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 J. Kahn, M. Lyubikh, W. Shen: Combinatorial rigidity for unicritical polynomials. Ann. of Math ( 2) 170 (2009 ), no 2, 783-797.