Artur Avila

Artur Avila
Breithaupt, Katrin · CC BY-SA 2.0 de · source
Name	Artur Avila
Birth date	29 June 1979
Birth place	Rio de Janeiro, Brazil
Nationality	Brazilian, French
Fields	Mathematics
Workplaces	IMPA, CNRS, Université Paris-Saclay
Alma mater	IMPA
Doctoral advisor	Welington de Melo
Known for	Dynamical systems, spectral theory, one-dimensional maps, Schrödinger operators

Artur Avila is a Brazilian and French mathematician known for work in dynamical systems, spectral theory, and one-dimensional real and complex dynamics. He received broad recognition for combining techniques from ergodic theory, harmonic analysis, and operator theory to resolve long-standing problems in the theory of one-dimensional maps and quasi-periodic Schrödinger operator spectral properties. Avila helped bridge research communities across Brazil, France, United States, and Europe through collaborations and institutional leadership.

Early life and education

Avila was born in Rio de Janeiro and attended local schools before enrolling at the IMPA. At IMPA he studied under Welington de Melo and worked alongside mathematicians associated with the Brazilian Mathematical Society and visiting scholars from institutions like Princeton University, École Normale Supérieure, and University of Chicago. He completed his doctorate at IMPA, contributing to topics related to real dynamics, complex dynamics, and renormalization problems influenced by the work of Michael Feigenbaum, Dennis Sullivan, and Jean-Christophe Yoccoz.

Research and mathematical contributions

Avila's research spans rigorous results in dynamical systems, spectral theory, and ergodic theory, often addressing problems posed by predecessors such as Ya. Sinai, Andrew Katok, and Jacques Hadamard. He made foundational contributions to the global theory of one-dimensional maps, proving results on the measure and structure of attractors and the prevalence of stochastic and regular behavior in families of maps originally studied by Mitchell Feigenbaum and P. Coullet. Avila developed techniques combining renormalization, complex analytic methods from holomorphic dynamics inspired by Curt McMullen, and thermodynamic formalism related to works of David Ruelle and Yakov Sinai.

In spectral theory, Avila solved outstanding conjectures concerning the spectral type and localization for quasi-periodic Schrödinger operators, building on earlier research by Barry Simon, Svetlana Jitomirskaya, and Eli Lieb. His "global theory" of one-frequency Schrödinger operators clarified transitions between absolutely continuous spectrum and Anderson localization, employing tools from subharmonic function theory, Lyapunov exponent analysis linked to Oseledec's theorem, and reducibility techniques related to KAM theory developed by Kolmogorov, Arnold, and Moser. Avila also worked on continuity of the Lyapunov exponent, the Ten Martini Problem lineage related to work by Mark Kac and Avi Izrailev, and criteria for purely singular continuous spectrum echoing earlier problems studied by Boris Simon and Eugene Wigner.

He collaborated with a wide range of mathematicians including Jean-Christophe Yoccoz, Sylvain Crovisier, Raphaël Krikorian, and Stéphane Jitomirskaya to address rigidity, universality, and measurable dynamics questions. His work influenced research programs at IMPA, CNRS, Collège de France, and major international conferences like the International Congress of Mathematicians.

Career and positions

Avila held positions at IMPA and later joined the French CNRS and Université Paris-Saclay research environments. He served as a researcher affiliated with the Institute of Mathematics of the Federal University of Rio de Janeiro and maintained collaborations with groups at Princeton University, Massachusetts Institute of Technology, ETH Zurich, and Imperial College London. Avila was an invited speaker at international venues including the International Congress of Mathematicians, the European Mathematical Society meetings, and seminars at IHÉS and Collège de France.

Awards and honors

Avila received the Fields Medal at the International Congress of Mathematicians in 2014, a landmark as the first Latin American-born recipient, recognized for contributions to dynamical systems and spectral theory. He won other distinctions such as the Salem Prize, the Berwick Prize, and fellowships from institutions like CNRS and IMPA. His work earned invitations to prestigious lectureships at IHÉS, Clay Mathematics Institute, and membership in academies including the Brazilian Academy of Sciences and participation in committees of the European Research Council and international awarding bodies.

Personal life

Avila held dual Brazilian and French citizenship and maintained professional ties in Rio de Janeiro and Paris. He balanced research with mentorship of students and collaborations with colleagues from institutions such as University of California, Berkeley, Universidade de São Paulo, and ENS Lyon. Known for a reserved demeanor, he engaged in international mathematical outreach through lectures, summer schools, and collaborations linking research groups across South America and Europe.

Category:Brazilian mathematicians Category:Fields Medalists