Artur Avila

Artur Avila
Born	Artur Avila Cordeiro de Melo 29 6, 1979
Birthplace	Rio de Janeiro, Brazil
Nationality	Brazilian, French
Occupation	Mathematician, professor
Employer	University of Zurich, IMPA, CNRS
Known for	Dynamical systems, spectral theory, Fields Medal (2014)
Education	PhD in Mathematics (2001), IMPA
Awards	Fields Medal (2014), TWAS-Lenovo Science Prize (2015), Salem Prize (2006)

Artur Avila Cordeiro de Melo (born 29 June 1979) is a Brazilian-French mathematician whose work in dynamical systems and spectral theory has reshaped understanding of chaotic behavior and mathematical physics. In 2014, he became the first Latin American and the first lusophone (Portuguese-speaking) mathematician to receive the Fields Medal, the most prestigious award in mathematics, often described as the discipline's equivalent of the Nobel Prize. Born and raised in Rio de Janeiro, Avila showed extraordinary mathematical talent from an early age, competing in the International Mathematical Olympiad at the age of sixteen and completing his doctorate at IMPA (Instituto de Matemática Pura e Aplicada) by the age of twenty-one. Throughout his career, he has maintained research positions at both IMPA in Brazil and the Centre national de la recherche scientifique (CNRS) in France, splitting his time between Rio de Janeiro and Paris. Since September 2018, he has also served as a professor at the University of Zurich. Avila's contributions span a remarkably wide range of problems, from one-dimensional dynamics and renormalization theory to the spectral properties of Schrödinger operators and the behavior of interval exchange transformations and translation flows.

Early Life

Artur Avila Cordeiro de Melo was born on 29 June 1979 in Rio de Janeiro, Brazil. He grew up in the city and displayed an aptitude for mathematics at an early age. As a teenager, he became involved in mathematical competitions, a common pathway for gifted young mathematicians in Brazil. In 1995, at the age of sixteen, Avila represented Brazil at the International Mathematical Olympiad (IMO), where he earned a gold medal. His performance at the olympiad signaled his exceptional promise and brought him to the attention of the mathematical community in Brazil, particularly the researchers at IMPA, one of the foremost mathematical research institutes in Latin America.

IMPA, located in Rio de Janeiro, has long served as a center for advanced mathematical training in Brazil and has produced several internationally recognized mathematicians. Avila's early contact with the institute shaped the trajectory of his career. He began studying there as a teenager and was quickly immersed in an environment of high-level mathematical research. According to a profile in the Brazilian publication Revista piauí, Avila's precociousness was evident to those around him, but he was also known for a relaxed and informal demeanor that belied the intensity of his mathematical focus.

Growing up in Rio de Janeiro, Avila was shaped by the culture of the city—he has been described as a carioca (a native of Rio) who enjoys the social life, music, and beaches of his hometown. Despite his early success in competitions, Avila later indicated that his interest shifted relatively quickly from olympiad-style problem solving to the deeper, more open-ended challenges of mathematical research.

Education

Avila pursued his graduate education at IMPA, where he worked under the supervision of Welington de Melo, a prominent Brazilian mathematician specializing in dynamical systems. His doctoral dissertation, titled Bifurcações de transformações unimodais sob os pontos de vistas topológico e métrico (Bifurcations of unimodal transformations from the topological and metric points of view), was completed in 2001, when Avila was just twenty-one years old. The thesis addressed fundamental questions in the theory of one-dimensional dynamical systems, an area that would remain central to his research for years to come.

By earning his doctorate at such a young age, Avila joined a small cohort of mathematicians who have completed their doctoral training before the age of twenty-five. His time at IMPA provided him with a deep grounding in dynamical systems theory and connected him with a network of Brazilian and international collaborators who would prove instrumental in his subsequent work.

Career

Early Research and CNRS Appointment

Following the completion of his doctorate, Avila held a position as a Clay Mathematics Institute research fellow, which allowed him to pursue independent research at an early stage of his career. During this period, he began to develop the body of work that would establish his international reputation.

Avila joined the CNRS in France, where he was affiliated with the Laboratoire de Probabilités et Modèles Aléatoires at the Université Pierre-et-Marie-Curie (now part of Sorbonne University) in Paris. He adopted a distinctive working arrangement, spending roughly half of each year at the CNRS in Paris and the other half at IMPA in Rio de Janeiro. This dual appointment allowed him to remain rooted in the Brazilian mathematical community while also engaging with the rich mathematical culture of France, which has historically been one of the strongest countries in the world for mathematics.

Avila's early research focused on one-dimensional dynamics, extending the work begun in his doctoral thesis. He made significant contributions to the understanding of the dynamics of unimodal maps—simple-looking maps of an interval to itself that can exhibit extraordinarily complex behavior. Working with collaborators including Mikhail Lyubich, he achieved breakthrough results on the density of hyperbolicity for real quadratic maps, settling a long-standing conjecture that had been a central open problem in the field.

Dynamical Systems and Renormalization

A substantial portion of Avila's work concerns the theory of renormalization as applied to dynamical systems. Renormalization is a technique, originally developed in physics, that involves studying a system at different scales to understand its behavior. In the context of dynamical systems, renormalization methods have been used to analyze the transition from regular to chaotic behavior. Avila, together with collaborators such as Lyubich and Welington de Melo, made important advances in understanding the renormalization of one-dimensional maps and its connection to universality phenomena—the observation that very different dynamical systems can exhibit the same qualitative behavior at the onset of chaos.

Avila's approach to problems in dynamical systems has been characterized by a willingness to work across traditional boundaries within mathematics. He has combined techniques from analysis, probability theory, and ergodic theory to tackle problems that had resisted more narrowly focused approaches. This versatility has been noted by colleagues and commentators as a distinguishing feature of his mathematical style.

Spectral Theory and Schrödinger Operators

Another major thread of Avila's research concerns the spectral theory of Schrödinger operators, which are fundamental objects in quantum mechanics. In mathematical physics, the Schrödinger operator describes how a quantum particle behaves in a given potential, and its spectral properties determine the possible energy levels of the system. When the potential is generated by a dynamical system—for instance, a quasiperiodic function—the spectral theory becomes deeply intertwined with dynamics.

Avila made major contributions to the understanding of the almost Mathieu operator, a specific Schrödinger operator with a quasiperiodic potential that serves as a model for an electron in a two-dimensional crystal subjected to a magnetic field. This operator is connected to the famous Hofstadter's butterfly, a fractal pattern that arises in its spectral diagram. Avila, working with Svetlana Jitomirskaya, proved the Ten Martini Problem—a conjecture named by Barry Simon after he offered ten martinis to anyone who could prove it—which asserts that the spectrum of the almost Mathieu operator is a Cantor set for all irrational frequencies and all non-zero coupling constants.

Avila also developed what he called a "global theory" of one-frequency analytic Schrödinger operators, which provided a comprehensive framework for understanding the spectral behavior of these operators. His work showed that, in a precise mathematical sense, the behavior of such operators is typically well-organized rather than pathological, resolving several open questions in the field.

A column by mathematician Marcelo Viana in the Brazilian newspaper Folha de S. Paulo highlighted the significance of Avila's work on the Schrödinger equation, noting that by 2013, the mathematical community had widely recognized Avila's contributions as among the most important of his generation.

Interval Exchange Transformations and Teichmüller Theory

Avila has also made significant contributions to the study of interval exchange transformations and Teichmüller theory. An interval exchange transformation is a map that cuts an interval into several pieces and rearranges them. Despite their simple description, these transformations are connected to deep questions in geometry and topology, particularly through their relationship to the geodesic flow on moduli spaces of Riemann surfaces.

Working with Giovanni Forni, Avila proved that almost every interval exchange transformation (in the measure-theoretic sense) is weakly mixing, a result that resolved a question that had been open for decades. This work combined ideas from Teichmüller theory, ergodic theory, and combinatorics in a novel way. The result was praised in the Fields Medal citation issued by the International Mathematical Union, which noted that Avila's work on interval exchange transformations and translation flows had "transformed the subject."

Invited Speaker at the ICM

Avila was an invited speaker at the International Congress of Mathematicians (ICM), a recognition of the significance of his research within the global mathematical community.

Appointment at the University of Zurich

In September 2018, Avila took up a professorship at the University of Zurich in Switzerland, while continuing his affiliations with IMPA and CNRS. This appointment added a third institutional base to his already unusual bi-continental career, reflecting the international demand for his expertise and the increasingly global nature of mathematical research at the highest levels.

Personal Life

Avila has been described as having a relaxed, sociable personality that contrasts with stereotypical images of mathematicians. Profiles in Brazilian media have emphasized his connection to the culture of Rio de Janeiro, including its music and nightlife. A feature in VICE described him standing outside a bar in the Leblon neighborhood of Rio de Janeiro, far from any stereotypical academic setting. In an interview with Chalkdust Magazine, Avila discussed his approach to mathematics and his life, offering insight into how he balances the demands of research with everyday life.

Avila holds both Brazilian and French nationality, a reflection of his long-standing dual career between the two countries. His decision to maintain active research bases in both Rio de Janeiro and Paris has been noted by colleagues and commentators as unusual and as a testament to his commitment to fostering mathematical research in Latin America while remaining at the center of European mathematics.

Recognition

Fields Medal (2014)

In August 2014, the International Mathematical Union awarded Avila the Fields Medal at the International Congress of Mathematicians held in Seoul, South Korea. The citation recognized his "profound contributions to dynamical systems theory," noting that he had "made outstanding contributions to a broad array of topics, including the theory of quasiperiodic Schrödinger operators, the theory of interval exchange maps and translation flows, and the theory of one-dimensional maps." The award received significant media attention in Brazil, where it was celebrated as a milestone for Latin American science.

The Guardian featured Avila among the 2014 Fields Medal winners, highlighting his work alongside the other recipients: Manjul Bhargava, Martin Hairer, and Maryam Mirzakhani.

Other Awards

Prior to the Fields Medal, Avila received the Salem Prize in 2006, awarded for outstanding contributions to analysis. He was also recognized by the TWAS (The World Academy of Sciences) as a winner of its 2013 prize. In 2015, Avila was awarded the TWAS-Lenovo Science Prize for his research in mathematics, which was described as having solved "daunting mathematical" problems.

In 2016, Americas Quarterly named Avila among its top five Latin American academics, noting his Fields Medal and his contributions to the visibility of Latin American science on the world stage.

In 2017, Avila received the Łojasiewicz Prize from the Jagiellonian University in Kraków, Poland, further recognizing his contributions to mathematical analysis and dynamical systems.

Avila has also been recognized by the International Association of Mathematical Physics (IAMP) with a prize for his contributions at the interface of mathematics and physics.

Legacy

Artur Avila's career has been significant not only for his individual mathematical achievements but also for what it represents in the broader context of mathematics in the developing world. As the first Latin American and the first Portuguese-speaking mathematician to receive the Fields Medal, his recognition brought international attention to the strength of mathematical research in Brazil and, more specifically, to the institutional role of IMPA in nurturing mathematical talent.

Avila's work has had a lasting impact across several branches of mathematics. His contributions to the theory of one-dimensional dynamical systems, including the resolution of long-standing conjectures on the density of hyperbolicity, helped close chapters that had been open for decades. His global theory of one-frequency Schrödinger operators provided tools and perspectives that continue to influence research in mathematical physics and spectral theory. His results on interval exchange transformations and translation flows connected disparate areas of mathematics—dynamics, geometry, and topology—in ways that have stimulated further research.

The breadth of Avila's research output, spanning from pure dynamical systems to mathematical physics, has been cited as an example of the kind of cross-disciplinary thinking that drives progress in modern mathematics. His dual career between Brazil and France, and later his appointment in Switzerland, has also served as a model for how mathematicians from countries outside the traditional European and North American centers can maintain active roles at the forefront of research while contributing to the development of their home institutions.