Artur Avila

Native spelling: Artur Ávila

fl. 1979, Rio de Janeiro, Brazil

Differential equations and dynamical systemsAnalysisFields Medal 2014

Brazilian-French mathematician who unified one-dimensional dynamics and Teichmüller dynamics through renormalization, and solved the Ten Martini problem; Fields Medal 2014.

Artur Avila was born in Rio de Janeiro in 1979. A prodigy in Brazil's mathematical olympiad community, he began research at IMPA as a teenager and completed his doctorate there in 2001 under Welington de Melo, while still in his early twenties; his thesis already contained important results on unimodal maps. He moved to France, became a CNRS researcher, and later a research director at the Institut de Mathématiques de Jussieu–Paris Rive Gauche, while keeping ties with IMPA. In 2018 he took a professorship at the University of Zurich.

Avila's signature method is renormalization: zooming into a dynamical system at smaller and smaller scales to reveal a self-similar operator, then analyzing the dynamics of that operator in a space of systems. In one-dimensional dynamics this gave the regular-or-stochastic dichotomy for real quadratic maps with Mikhail Lyubich — almost every non-regular map admits an absolutely continuous invariant measure. In quasiperiodic Schrödinger operators, the same dynamical view let him settle long-standing conjectures, including the Ten Martini problem and, more broadly, a global theory of the Lyapunov exponent for analytic cocycles. In Teichmüller dynamics, joint work with Yoccoz, Forni and others yielded precise control of Lyapunov spectra of translation surfaces.

He received the Fields Medal in 2014 in Seoul — the first awarded to a mathematician whose entire training took place in Brazil — alongside Manjul Bhargava, Martin Hairer and Maryam Mirzakhani. Avila continues to work between Paris, Zurich and Rio, on one-dimensional and billiard dynamics and on spectral theory of quasiperiodic operators.

Workplaces: University of Zurich, CNRS and Institut de Mathématiques de Jussieu, Instituto Nacional de Matemática Pura e Aplicada (IMPA)

Brazil, France

Award-winning work

Fields Medal 2014The medal cites his profound contributions to dynamical systems theory, which changed the face of the field by uniting renormalization as a unifying idea. Avila solved the Ten Martini problem with Svetlana Jitomirskaya: the spectrum of the almost-Mathieu operator is a Cantor set for all irrational frequencies and nonzero couplings, settling a conjecture driven by physicists' observations of the Hofstadter butterfly. With Jean-Christophe Yoccoz, Giovanni Forni and others he revolutionized Teichmüller dynamics, proving results on the Lyapunov exponents and deviations of interval exchange transformations and billiards in rational polygons. He also established the dichotomy that almost every real quadratic map is either regular or stochastic (with Lyubich and collaborators), advancing the program on the density of hyperbolicity in unimodal families.

Contributions, linked to the library

Library topicChaos theoryProved that almost every real quadratic map is either regular or stochastic, a keystone of chaotic one-dimensional dynamics.Library topicErgodic theoryUsed renormalization and ergodic methods to control Lyapunov exponents of cocycles, interval exchanges and translation surfaces.Great problemLocal connectivity of the Mandelbrot set (MLC)His regular-or-stochastic results on real quadratic maps advanced the broader rigidity and density-of-hyperbolicity program behind local connectivity questions.