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The Work of the 2014 Fields Medalists The Work of the 2014 Fields Medalists The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation De Melo, Welington, Bjorn Poonen, Jeremy Quastel, and Anton Zorich. “The Work of the 2014 Fields Medalists.” Notices of the American Mathematical Society 62.11 (December 2015): 1334–1349. As Published http://dx.doi.org/10.1090/noti1317 Publisher American Mathematical Society (AMS) Version Final published version Citable link http://hdl.handle.net/1721.1/107740 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The Work of the 2014 Fields Medalists Welington de Melo, Bjorn Poonen, Jeremy Quastel, and Anton Zorich Mathematicians. ©2010–2015 International Congress of L–R: Martin Hairer, Manjul Bhargava, South Korean president Park Geun- hye, Maryam Mirzakhani, IMU president Ingrid Daubechies, and Artur Avila. The Notices solicited the following articles about the works of the four individuals to whom Fields Medals were awarded at the International Congress of Mathematicians in Seoul, South Korea, in August 2014. This was a historic occasion, as it marked the first time since the medal was established in 1936 that a woman was among the recipients. The International Mathematical Union also issued news releases de- scribing the medalists' work, and these appeared in the October 2014 issue of the Notices. —Allyn Jackson 1334 Notices of the AMS Volume 62, Number 11 Welington de Melo is that, for typical one-dimensional families of unimodal maps, one does not see the third region; The Work of Artur Avila i.e., the set of parameter values corresponding to Artur Avila was awarded the Fields Medal in maps in the third region has Lebesgue measure zero. 2014 for his deep contributions to dynamical For a typical one-parameter family of unimodal systems and to the spectral theory of one-frequency maps we have a decomposition of the parameter Schrödinger operators. Many of his profound space with the same properties as the quadratic achievements—in one-dimensional dynamics, both family. real and complex, as well as in flat billiards and in Another important result in his early career is the spectral theory of Schrödinger operators—are the unexpected rigidity in the space of unimodal characterized by an intense use of the powerful maps obtained in [10]. In this article it is shown that ideas of renormalization. He has also made deep there is a large set R of unimodal maps such that advances in the theory of conservative dynamical any typical one-parameter family intersects R in a systems in any dimension and in the stable set of parameters of full Lebesgue measure in the ergodicity of partially hyperbolic systems. complement of the regular parameters and any two Artur Avila was born in Rio de Janeiro, Brazil, maps in R that are topologically conjugate are in where he lived until he finished his PhD studies at fact smoothly conjugate. In particular, the authors the Instituto de Matemática Pura e Aplicada (IMPA) provide a combinatorial formula to calculate the in 2001. Before starting his postdoc position multipliers of all periodic points for maps in R. in the Collège de France in Paris, Artur, with More recently, in complex one-dimension dy- collaborators, obtained a complete description namics, Avila and Lyubich described several of typical dynamics of unimodal interval maps, fundamental properties on the geometry of the i.e., smooth interval maps with a unique nonflat Julia set of Feigenbaum-type quadratic polynomials critical point. It was already known that the space [6], [7], [2], [3], [4]. In particular, the existence of of unimodal maps contains two disjoint regions examples of such quadratic polynomials with Julia where the dynamics is well understood. For maps in sets of positive Lebesgue measure is proven. the regular region, there exists a unique attracting In most of these results, Avila uses as a fixed point, and the trajectories of almost all fundamental tool the renormalization theory. This initial conditions, in the Lebesgue measure sense, theory started with experimental discovery by the are asymptotic to this periodic point. The typical physicists Feigenbaum and Coullet-Tresser in the dynamical behavior is therefore periodic. For maps 1970s on the transition from simple to chaotic in the stochastic region the dynamics is chaotic dynamics in families of unimodal maps. They but with a good statistical description: there exists formulated a conjecture that involves a nonlinear an absolutely continuous invariant measure that operator in the space of unimodal maps. The controls the dynamics of a typical orbit in the sense proof of the conjecture and some generalizations that the time average of any physical observable involve the work of several mathematicians such is equal to the space average of the observable as Sullivan, McMullen, and Lyubich. Finally, Avila with respect to this measure. In particular, the and Lyubich in [16] gave a very conceptual and frequency of visits of a typical orbit to any interval much simpler proof of the general conjecture that is equal to the measure of this interval. In the holds also in the space of unimodal maps with complement of these two regions there are maps higher criticality. where the dynamics can be completely described A second area of dynamical systems where but there are other maps whose dynamics is Avila made fundamental contributions is the pathological. These different dynamical behaviors dynamics of interval exchange transformations, were already present in the famous quadratic family regular polygonal billiards, and ergodic properties x 2 [0; 1] , qµ = µx(1−x), where the parameter µ of the geodesic Teichmüller flow in the moduli belongs to the interval (0; 4]. The set of parameter space of Abelian differentials on Riemann surfaces. values corresponding to regular maps is open and Given a partition I1;:::;In of the interval [0; 1] in dense by a difficult result proved in [18] and [20]. d ≥ 2 intervals and a permutation σ of f1; : : : ; dg, On the other hand, the set of parameter values we can define the mapping T : [0; 1] ! [0; 1] by corresponding to stochastic maps has positive X X T (x) = x − λ + λ Lebesgue measure, as proved in [20], and the j j j<i σ (j)<σ (i) complement has measure zero [22] but positive Hausdorff dimension. The main result in [5], [9] if x 2 Ii, where λj is the length of the interval Ij . This is what is called an interval exchange W. de Melo is a professor at the Instituto de Matemática Pura transformation, and it is completely characterized e Aplicada. His email address is [email protected]. by the permutation σ and by the vector (λ1; : : : ; λd) d For permission to reprint this article, please contact: that belongs to a simplex in R+. The space of [email protected]. such maps is therefore finite dimensional. We say DOI: http://dx.doi.org/10.1090/noti1317 that a subset is typical if the complement has December 2015 Notices of the AMS 1335 zero Lebesgue measure. Veech proved that for particular, it was known that if one multiplies an any irreducible permutation the typical interval analytic potential by a coupling constant generating exchange transformation is uniquely ergodic. In a one-parameter family of operators, then for very 1980 Katok proved that no interval exchange small values of the coupling constant the spectrum transformation is mixing. Recall that a measure- of the operator is absolutely continuous, and preserving transformation f : (X; µ) ! (X; µ) is for very large values the spectrum is pure point mixing if for typical values of the frequency. Not much −n was known for values of the coupling constant limn!1[µ(f (A) \ B − µ(A)µ(B)] = 0 in between. However, it was known that to an for any measurable sets A and B, and it is weak operator corresponds a one-parameter family of mixing if SL(2; R) cocycles, parametrized by the energy, and N it was known that the spectrum coincides with the 1 X −n limN!1 [µ(f (A) \ B − µ(A)µ(B)] = 0: bifurcation set of the dynamical object. These are N n=0 the values of the parameter such that the cocycle The joint work of Avila and Forni [11] solved the is not uniformly hyperbolic. Probably inspired by main problem in the ergodic theory of interval ex- the results in one-dimensional dynamics, Avila change transformation: a typical interval exchange described three regions in this space of cocycles: UR transformation is weakly mixing for any irreducible (uniform hyperbolic), SpC (supercritical), and SbC permutation that is not a rotation. This work is (subcritical) and called the complement of these connected to a recent result of Avila and Delecroix C (critical). The cocycles in SpC are not uniformly proving that almost all regular polygonal billiards hyperbolic but have positive Lyapunov exponents are weakly mixing. Again, one of the main tools and were already well understood. The part of the for these results is a renormalization operator spectrum of the operator that lies in this region that consists of considering the first return map corresponds typically to the pure point spectrum. to a smaller interval and rescaling to the original Avila proved that the part of the spectrum that lies size. It is called the Rauzy-Veech renormalization in SbC corresponds to the absolutely continuous operator. Veech and Masur proved that it has an part of the spectral measure. Finally, he proved absolutely continuous invariant measure, and the the existence of a stratification of the space into ergodic properties of the operator give important submanifolds of positive codimension and that information about the ergodic properties of a typi- the set of critical cocycles is a small subset, in the cal interval exchange transformation.
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