Person: Ávila, Artur
Artur Ávila is a Brazilian mathematician who has made deep contributions to dynamical systems theory which has transformed the subject. His work has been honoured with numerous prizes including a Fields Medal in 2014.
Mathematical Profile (Excerpt):
- It was Raimundo who taught Artur to read and write when he was three years old and, as part of this, first introduced him to mathematics.
- When he was six years old Artur began his schooling at Colégio de São Bento in Rio de Janeiro, a Benedictine school for boys.
- Artur was rather bored by the lessons and looked to learn more by reading advanced books on his own.
- It was his mathematics teacher at São Bento, Luiz Fabiano Pinheiro, who gave Artur his firm commitment to mathematics when he told him about the local junior Mathematical Olympiad competition and suggested he take part.
- It was the religious educations classes that caused Ávila problems.
- Ávila said that whether one believed in the existence of God or not, these could only be treated as philosophical arguments, not as logical proofs.
- It provided good teaching but Ávila found the mathematics classes far too easy and began missing lessons.
- Despite this Ávila was awarded a gold medal.
- Ávila attended the class 'Introduction to Real Analysis' given by Nicolau Corção Saldanha.
- This gave Ávila a totally new view of what mathematics was.
- Now he was learning complex ideas about which he had to think deeply; he loved it! In the test at the end of the course students were allowed books and notes to help them solve the problems but Ávila only took a pen to write his solutions.
- Up till that time Ávila came to lectures but said nothing and was never noticed by the lecturer.
- After this, Ávila was not interested in any more Mathematical Olympiad competitions and, despite lots of pressure to be part of the 1996 Brazilian team, he refused.
- For the Master's degree Ávila was advised by Elon Lages Lima but soon came into contact with Welington de Melo.
- Ávila asked de Melo if he would be his Ph.D. advisor and he began his studies in 1998.
- At this conference it was arranged that early in 1999 de Melo and Ávila would travel to Stony Brook and discuss a Ph.D. topic for Ávila with Lyubich.
- It was his proposal that Artur explore this line of research." ...
- "That was in March", remembers Artur.
- The joint work of Ávila, de Melo and Lyubich, completed by Ávila's "weird idea", was published in an impressive 100-page paper Regular or stochastic dynamics in real analytic families of unimodal maps in which they proved a result that had been attacked unsuccessfully by many mathematicians.
- This was only one of several outstanding papers produced by Ávila around this time.
- Ávila gave the Cours Peccot at the Collège de France in 2004-05, delivering the course Dynamique des cocycles quasi périodiques et spectres de l'opérateur presque-Mathieu Ⓣ(Dynamics of quasiperiodic cocycles and spectra of the almost-Mathieu operator).
- Ávila continued to receive awards and honours: the International Association of Mathematical Physics Early Career Award (2012); the Prize of the Brazilian Mathematical Society (2013); and the Bellow Lectures by the Northwestern University (2014).
- Ávila's work on real one-dimensional dynamics brought completion to the subject, with full understanding of the probabilistic point of view, accompanied by a complete renormalization theory.
- In the spectral theory of one-frequency difference Schrödinger operators, Ávila came up with a global description of the phase transitions between discrete and absolutely continuous spectra, establishing surprising stratified analyticity of the Lyapunov exponent.
- In the theory of flat billiards, Ávila proved several long-standing conjectures on the ergodic behaviour of interval-exchange maps.
- Ávila's collaborative approach is an inspiration for a new generation of mathematicians.
- Ávila was awarded the TWAS-Lenovo Science Prize in 2015.
- In 2017 Ávila gave the Lojasiewicz Lecture at the Jagiellonian University.
- He was elected to the US National Academy of Sciences in 2019 and in the same year the Fields Medal Symposium in Toronto honoured Ávila and explored the current and potential impact of his work.
- In this public lecture, Ávila will reflect on how his field has changed over time, and examine current questions at the forefront of dynamical systems theory.
- Ávila was appointed as a full professor at the University of Zurich in 2018.
- Let us now give some quotes by Ávila taken from various interviews.
Born 29 June 1979, Rio de Janeiro, Brazil.
View full biography at [MacTutor](https://mathshistory.st-andrews.ac.uk/Biographies/Avila/
Adapted from other CC BY-SA 4.0 Sources:
- O’Connor, John J; Robertson, Edmund F: MacTutor History of Mathematics Archive