|Department of Mathematics|
Address: Mathematics 253-37 | Caltech | Pasadena, CA 91125
Telephone: (626) 395-4335 | Fax: (626) 585-1728
Project MATHEMATICS! | Caltech Home
7th Annual Thomas Wolff Memorial Lecture
Renormalization and quasiperiodicity in some low-dimensional dynamical systems
We will first go through a few ways in which the key concept of renormalization may appear in the analysis of low-dimensional dynamics. In the understanding of a very non-linear system, like a quadratic map, renormalization may give rise to an attractor consisting of non-linear systems, the mere existence of which is far from obvious. Renormalization of a less non-linear system, like diffeomorphisms of the circle or quasiperiodic cocycles (with not too big parameters) may be regularizing and take us to an attractor of linear models, near which more analytical tools are available (such as KAM schemes). One may also be interested directly in the renormalization of linear systems, where one can get into very precise questions, as the geodesic flow on the modular surface testifies (but I will concentrate on its higher dimensional friend, the Teichmüller flow).
We will then go more deeply on the Teichmüller flow on moduli spaces of Abelian differentials, which can be seen as a renormalization operator of translation flows, closely related to interval exchange transformations. The dynamics of the Teichmüller flow itself is very chaotic, and we will see how this can be used to understand the underlying (non-chaotic) low-dimensional dynamics with probabilistic techniques.
The discussion then moves on to quasiperiodic cocycles, especially in the way dynamics can be used in the solution of concrete questions about the almost Mathieu operator (and similar objects). The dynamical picture here is very enriched with Aubry-André duality and the presence of a phase transition at the critical (self-dual) point. Starting with an application of renormalization to the understanding of the critical point, we will move towards estimates on Fourier series and holomorphic functions to achieve a more detailed account of the dynamics and the spectrum.
Artur Avila was born in 1979 in Rio de Janeiro and got his PhD in 2001 at IMPA, as a student of Welington de Melo. He was a post-doc at the Collège de France for two years before becoming a CNRS researcher at the Laboratoire de Probabilités et Modèles Aléatoires in Paris. He holds a Clay Research Fellowship from July 2006 to June 2009 and is currently at IMPA. Avila gave a “Cours Peccot" at the Collège de France in 2005 and has been awarded the Bronze Medal of the CNRS and the Salem Prize ”for his work on Lyapounov exponents and quasiperiodic behaviour in unimodal maps, Schrödinger-like cocycles, interval exchange maps and Teichmüller flows".
Avila's work typically combines dynamical intuition with techniques from probability, combinatorics and complex analysis. His sequence of results on quasiperiodic cocycles with Raphaël Krikorian, Svetlana Jitomirskaya and David Damanik has led to the complete solution of three outstanding problems on the almost Mathieu operator in Barry Simon's list. Among his other main contributions, one finds results with Moreira and Lyubich concerning the statistical properties of unimodal maps and the Hausdorff dimension of Julia sets, and works with Forni and Viana on the mixing properties of interval exchange transformations and the Lyapunov exponents of the the Teichmüller flow which aswered conjectures of Veech and Kontsevich-Zorich.
You are invited to attend a dinner following theThomas Wolff Memorial Lectures in Mathematics at The Athenaeum on Tuesday, February 19, 2008
Host bar 5:45 p.m.
MENUSalad of Roasted Red and Golden Beets
Roasted Breast of Mediterranean Chicken
Brownie Cappuccino Parfait
Please indicate if you require a vegetarian or kosher meal
For reservations, please contact Stacey V. Croomes at 626-395-4336 or send payment by Friday,
Stacey V. Croomes
The Thomas Wolff lectures, sponsored by donations from his widow and his parents, memorialize Caltechs great analyst who was tragically killed at age 46 in an automobile accident in July 2000. Wolff was a specialist in analysis, particularly harmonic analysis. Professor Wolff made numerous highly original contributions to the mathematical fields of Fourier analysis, partial differential equations, and complex analysis. A recurrent theme of his work was the application of finite combinatorial ideas to infinite, continuous problems.
His early work on the Corona theorem, done as a Berkeley graduate student, stunned the mathematical community with its simplicity. Tom never wrote it up himself since several book writers asked for permission to include the proof in their books where it appeared not long after he discovered it. After producing a number of very significant papers between 1980 and 1995, he turned to the Kakeya problem and its significance in harmonic analysis, works whose impact is still being explored.
Peter Jones, professor of mathematics at Yale, described Tom’s contributions as follows: “The hallmark of his approach to research was to select a problem where the present tools of harmonic analysis were wholly inadequate for the task. After a period of extreme concentration, he would come up with a new technique, usually of astonishing originality. With this new technique and his well-known ability to handle great technical complications, the problem would be solved. After a few more problems in the area were resolved, the field would be changed forever. Tom would move on to an entirely new domain of research, and the rest of the analysis community would spend years trying to catch up. In the mathematical community, the common and rapid response to these breakthroughs was that they were seen not just as watershed events, but as lightning strikes that permanently altered the landscape.”
Tom Wolff was noted for his analytic prowess, the depth of his insights, and the passion with which he nurtured the talents of young mathematicians. We miss him.