Wednesday, August 20, 2014

The Eternal Strands of Garland (1979)

The Hofstadter butterfly is a fractal, here in color, representing the spectrum of a Schrödinger operator with quasi-periodic Mathieu.  The horizontal axis represents the energy (or chemical potential) and the vertical axis the values ​​of magnetic flux.  Warm and cool colors represent positive and negative values ​​of the so-called Hall conductance.  The butterfly was discovered by Douglas Hofstadter in his article "The energy and wave functions of Bloch electrons in rational and irrational magnetic fields levels"

The most remarkable result of Artur Avila in the field of Schrödinger operators concerning the so-called quasi-periodic Mathieu operator. He proved with the mathematician Svetlana Jitomirskaya that its spectrum was well described by a Cantor. We knew that the spectrum of this operator was especially since the results obtained in 1976 had the physicist Douglas Hofstadter. Before turning to the artificial intelligence and cognitive science, the author of the famous book Gödel, Escher, Bach:

The Eternal Strands of Garland (1979), had in fact managed to calculate the energy spectrum of a electron limited to move in a two-dimensional periodic potential under the influence of a magnetic field perpendicular. A surprising geometric representation of this spectrum was then emerged that is now known to be a fractal . He was baptized the Hofstadter butterfly . It was not until 2013 that this prediction has been verified experimentally in a structure based on graphene .

The new French winner of the Fields Medal, which became in 2008, just 29 years old, the youngest director of research at CNRS, in recent years divided his time between France, at the Institute of Mathematics of Jussieu, Paris Rive left, and Brazil, where he is still active in the international joint research unit CNRS - IMPA.

Schrödinger operators and Hofstadter butterfly

Arthur Avila, mathematician Amie Wilkinson of the University of Chicago and the mathematician Sylvain Crovisier of Paris-Sud University also looked at an old question related to the ergodicity and going back to Boltzmann. As already done Sinai, researchers considered it to what is called a billiard chaos theory. Those wishing to know more about the connections between dynamical systems and billiards are referred to the excellent article that Aurélien Alvarez and Jean-Christophe Yoccoz spent on this on the site of CNRS inescapable Images Math .

Schrödinger operators and Hofstadter butterfly

Spectral theory, in the words of the great mathematician David Hilbert , is rooted in the equations used to describe waves and oscillations in physical systems. These equations are used to calculate for example the different frequencies associated with an electromagnetic wave. These frequencies, which may be a discrete or continuous set of values ​​constitute the spectrum . The most famous case is probably that of the spectrum of light waves that can emit carbon calculated using the Schrödinger equation in quantum physics .

Mathematicians have begun to explore methodically spectra, that is to say, the different values ​​associated with matter waves in physical systems described by a Schrödinger equation energies (we generally speak of Schrödinger operator). There is for example the case of electrons in a crystal lattice . For them, the nuclei of atoms, eg forming a metal appear as a series of periodic potential wells, alternating valleys and hills so. Schrödinger equations describing matter waves in periodic potentials are naturally involve equations already considered in celestial mechanics with the stars subjected to periodic disturbances such as the Hill equation and the Mathieu.

Artur Avila contributed very significantly

It is conducted to represent the behavior to a law of evolution in continuous time (in the form of systems of equations differential) or discrete, many physical systems describing trajectories in a particular geometric space, the phase space. Basically, the geometric and topological properties of these trajectories provide information on the long-term behavior of these systems. Are they stable or do they become chaotic? One can ask this question and others related to the paths of the planets of the solar system , the ecosystem and the population growth in them, the behavior of plasmas in tokamaks, such as that of ITER , and many yet other situations as possible the behavior of cosmological models describing the beginning of our universe in general relativity .

Artur Avila contributed very significantly, with colleagues Mikhail Lyubich, Welington de Melo and Carlos Gustavo Moreira, the study of chaos in very general dynamical systems related to what is called the logistic equation, the Belgian mathematician Pierre François Verhulst was proposed in 1838 in the wake of Malthus model, which led to an exponential growth of the population. The theory of dynamical systems has natural connections with the problems of the kinetic theory of gases Boltzmann and more generally with the entropy and the notion of ergodicity as Futura-Sciences explained in the article on the 2014 Abel Prize attributed to mathematician Yakov Sinai.

Dynamical systems and chaos, it's billiard

In this video, Artur Avila briefly talks about his work on dynamical systems and their connections with the theory of billiards and Schrödinger operators (for details see text below). He divides his time between Paris and IMPA in Rio. He likes to work and think while walking on the beach. Brazil has never had a Nobel Prize and had not yet had a Fields Medal.

Avila think his example is now very motivating for Brazilian researchers. To get a rough translation in French, click the rectangle with two horizontal bars on the bottom right. Subtitles in English should then appear, if it is not already. By simply moving the mouse over the rectangle, you should see the words "Translate subtitles." Click to bring up the menu language selection, select "French" and then click "OK". © tywebbOOOOO, YouTube

Dynamical systems and chaos, it's billiard

Possible classifications of mathematicians, one distinguishes those that solve the problems of those who create theories. Artur Avila, in the opinion of Jean-Christophe Yoccoz, belongs to the first category. Technically, his work focuses on the theory of dynamical systems and spectral theory of operators.

The theory of dynamical systems is rooted in the work on celestial mechanics Poincaré and David Birkhoff, especially regarding the famous three-body problem, that is to say, the description of the movements of celestial bodies three s' attracting each other under the action of gravitation.

French Fields Medal winners

Born June 29, 1979, Avila is only 35 years old and he is a bit of an exception among the 12 French Fields Medal winners since Laurent Schwartz to Cédric Villani through Alain Connes and René Thom, to name they all come from the ENS in the Rue d'Ulm.

This is by winning the gold medal at the International Mathematical Olympiad in Toronto (1995), at 16, he draws the attention of Welington de Melo IMPA, a specialist in dynamic systems such as Jean-Christophe Yoccoz. Melo will take under his wing and the young talent while still in high school, he was already masters student at IMPA where he will begin a thesis to 19 years under the direction of the researcher. Three years later, in 2001, he finally joined the College of France as a post-doc Jean-Christophe Yoccoz.

In an article in the Journal du CNRS , Charline Zeitoun goes into more detail on some little unusual path 's Artur Avila . In particular we learn that he missed twice the input CNRS before succeeding in 2003 competition and he prefers to discuss his work and learn directly from his colleagues, including the beach, rather than reading articles. More details are yet available on the website of Quanta Magazine , the journal of the Simons Foundation. Those that math does not scare can refer to the article that wrote about Avila his colleague Etienne Ghys , CNRS research director at the Ecole Normale Supérieure de Lyon.

French and Brazilian mathematicians, a long history

With 12 winners of a Fields Medal, France is ranked second behind the United States in the global ranking of nations. The new award that just brought him Artur Avila at the International Congress of Mathematicians, which this year is being held in Seoul, South Korea, certainly confirms the excellence of the French school mathematics but also that of the Brazilian school. © Francetvinfo

French and Brazilian mathematicians, a long history

There is therefore a link between the former France and Brazil on science. This is especially true with math. In the aftermath of the Second World War, the great mathematician Jean Dieudonné, impressive founding figure of the Bourbaki group, was a professor at the University of Sao Paulo from 1946 to 1948, a few years later, it was his pupil, the famous Alexander Grothendieck who will spend some time in Brazil as a teacher but also a member of CNRS.

One of the most brilliant mathematicians of the XX th century, Grothendieck will be awarded the Fields Medal in mathematics in 1966 was closely linked to Brazil and the Instituto Nacional de Matematica Pura e Aplicada (IMPA) in Rio de Janeiro, the French Jean -Christophe Yoccoz also will pick the same medal in 1994 August 13, 2014, it was the turn of the Franco-Brazilian Artur Avila getting what many consider to be the equivalent of the Nobel Prize of mathematics. Are also winners of the Fields Medal Austrian Martin Hairer, the Canada-United States Manjul Bhargava and Iranian Maryam Mirzakhani .

Artur Avila enjoys working casually in peaceful environments

On 08/14/2014 at 17:31 - By Laurent Sacco, Futura-Sciences

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Artur Avila enjoys working casually in peaceful environments.  The new recipient of the Fields Medal in mathematics even finds inspiration on the beach in Rio de Janeiro.  His work focuses on dynamic systems found throughout the world, whether in the world of planets, stock markets or ecosystems.  © CNRS Photo / Sebastian Ruat

Many probably are unaware of the origin Brazilian flag, which has more than 100 years, and especially the national currency imposed on him. It comes in the form of a large yellow diamond containing a green background crossed by a curved white stripe navy blue disc. We read in this band Ordem e Progresso , which means "order and progress."

She was chosen deliberately in reference to the motto of positivism, the movement founded by the French philosophical Auguste Comte in the early XIX th century. Polytechnique and influenced especially by Condorcet, Comte imagined the creation of a society based entirely on science and turning his back on inherited from archaic times more designs, like astronomy and chemistry had come to emancipate beliefs and goals of astrology and alchemy.