We consider the classification problem for second order (nonsingular) ordinary differential equations. This study is motivated by the fact that, unlike the case of first order equations, two randomly chosen second order ODEs appear to be inequivalent under local diffeomorphisms of the plane. We aim to construct a normal form for each such equation - that is, we search for distinguished coordinates, where the initial ODE looks maximally simple. Despite its seeming simplicity, this fundamental problem exhibits a very interesting geometry and has not been understood well till very recently. Some normal form construction in this setting was previously suggested by Arnold. However, Arnold's normal form is incomplete: a second order ODE has infinite-dimensional space of Arnold's normal forms.

In our joint work with Zaitsev, we use a very different approach to construct a complete convergent normal form for all second order ODEs. This normal form allows us to produce various geometric invariants of an ODE. In particular, we obtain chains - certain distinguished curves, canonically associated with every ODE. Our approach is inspired by an interesting parallel between Differential Equations and Several Complex Variables (even though we formally do not use Several Complex Variables in this work). We are going to shortly outline this parallel, which previously allowed for solving a number of difficult problems in Several Complex Variables, and now found its applications in Differential Equations as well.

Dans cet exposé, nous nous intéresserons aux groupes de tresses de surfaces, qui généralisent les groupes de tresses classiques d'Artin. Ces groupes sont fortement reliés aux groupes modulaires ("mapping class groups") des surfaces. Les groupes de tresses de la sphère et du plan projectif ont un intérêt particulier en raison de l'existence de torsion. Nous montrerons que leurs sous-groupes finis sont très proches des groupes de symétrie des polyèdres réguliers. Ensuite, en utilisant le lien entre les groupes de tresses d'une surface M et les espaces de configuration de celle-ci, nous donnerons des résultats récents sur l'inclusion naturelle de ces espaces dans le produit cartésien \(M^n\) (travail en commun avec D.Gonçalves, São Paulo).

In this work, we show that, the periodic orbit obstruction vanishing condition still implies the existence of solution of the cohomological equation for this general cocycles. This result is a generalization of classical Livsic theorem in the 70's. The key idea is approximation of Lyapunov exponents of the system, using singular values of the linearized cocycle along orbit segments. This is based on some kind of pseudo-orbit shadowing technique, and this is a joint work with Artur Avila and Alejandro Kocsard.

Given an interval \(I\), an interval exchange transformation, or i.e.t., is a bijection \(f\colon I \to I\) such that for some splitting \((I_i)_i\) of \(I\) into \(d>1\) subintervals, \(\restriction{f}{I_i}\) is just a translation. It can be parametrized by two data: a length vector \(\lambda \in \mathbb{R}_+^d\) and a permutation \(\pi\in \mathfrak{S}_d\). Such a map can be seen as a discrete version of the geodesic flow on some translation surface. The introduction of these objects was motivated by the study of the billiard flow on rational polygons.



In this talk, we will focus on the ergodic properties of i.e.t.'s. By works of Masur and Veech, we know that for a typical i.e.t., Lebesgue measure is the unique invariant measure, while Katok has proved that i.e.t.'s are never mixing.

A criterion due to Veech shows that the weak mixing property is related to the dynamics of a cocycle over a renormalization operator on the space of i.e.t.'s. Thanks to this fact, Avila and Forni developed a probabilistic argument which allowed them to show that when \(\pi\) is irreducible and is not a rotation, \(f(\lambda,\pi)\) is weak mixing for almost every \(\lambda\). In a joint work with A. Avila, we prove that in fact, the set of \(\lambda\) such that \(f(\lambda,\pi)\) is not weak mixing does not have full Hausdorff dimension. We show that the dynamics in parameter space has a property called ``fast-decay''. We will explain how to improve the estimates of Avila-Forni to get a large deviation result; combining this with techniques introduced by Avila-Delecroix, it is then possible to estimate the Hausdorff dimension of non weak-mixing parameters.

There are considered germs of two-dimensional maps and three-dimensional vector fields with resonances of Siegel type. For some partial cases, functional modules of analytic classification of such germs will be constructed.

La conjecture de Sard fait partie, avec le problème de régularité des singulières minimisantes, des grands problèmes ouverts de géométrie sous-riemannienne. Cet exposé aura pour but de présenter la conjecture et d'expliquer quelques résultats obtenus récemment dans le cas de distributions de rang 2 en dimension 3. Nous expliquerons également comment ce problème est lié à des questions de résolutions des singularités. Il s'agit d'un travail en collaboration avec André Belotto. Cet exposé devrait être accessible au plus grand nombre.

Dans cet exposé, nous nous intéresserons au type de champs de vecteurs singuliers qui apparait à l'infini dans les équations de Painlevé (P1),..., (P5) (pour des valeurs génériques des paramètres), après compactification à poids de l'espace complexe tridimensionnel. Nous commencerons par rappeler quelques généralités sur l'étude des champs de vecteurs en dimension 2 (en particuliers le cas du noeud-col étudié par Martinet et Ramis) pour voir comment généraliser cela en dimension 3 (on s'appuiera également sur les résultats de classification analytique de Stolovitch pour les champs 1-résonants). On commencera ensuite par donner une classification formelle de la famille de champs de vecteurs considérée sous l'action de certaines transformations formelles fixant la singularité, en exhibant pour cela des formes normales (formelles) uniques. Ensuite, on énoncera un théorème de normalisation sectorielle, généralisant ainsi un théorème de Hukuhara-Kimura-Matuda pour les noeud-cols en dimension 2. Enfin, on verra comment l'étude des applications de recollements entre les différentes normalisations sectorielles permet d'obtenir un résultat de classification analytique à la Martinet-Ramis, en exhibant des invariants analytiques appelés difféomorphismes de Stokes. Si le temps le permet, on terminera par mettre ces résultats en lien avec la première équation de Painlevé (P1).

In this talk we will be interested in the following basic question:

given a (measurable) conjugacy between dynamical systems, how regular is it?

A well-known theorem of Margulis classifies measurable factors of flag manifolds, and an analogous result in the continuous category has been established by Dani. In this talk, we discuss

a classification of smooth factors under some hyperbolicity assumptions. This is a joint work with R. Spatzier.

We will be interested in the problem whether using information about correlations order two, one can deduce information about higher-order correlations.

Although this fails in general, we develop a method that allows establishing quantitative estimates for higher-order correlations assuming only estimates

on correlations of order two. We apply our approach to prove exponential multiple mixing for semisimple Lie groups, and as an application, we deduce a Ramsey-type result regarding the structure of lattices in semisimple Lie groups. This is a joint work with Bjourklund and Einsiedler.

I will briefly introduce the quantitative almost reducibility of analytic quasi-periodic linear systems and cocycles. The applications include: Dry Ten Martini problem in the non-critical case and Andre-Aubry-Jitomirskaya conjecture (with A. Avila and Q. Zhou); Gap estimates (joint with M. Leguil, Z. Zhao and Q. zhou); the regularity of Liapunov exponents of quasi-periodic cocycles (joint with L. Ge).

Irrational pseudo-rotations form a special class of Hamiltonian disk which are interesting because they include ergodic systems with zero topological entropy (indeed are the only such systems on genus zero surfaces). In this talk I will explain how pseudo-holomorphic curve methods from symplectic geometry always yields a sequence of approximating disk maps with a surprising amount of order, namely they are conjugates of rigid rotations.

We study smooth group actions and random walks on surface under a mild assumption called "weakly expanding". In particular, we prove that some

statistical properties (large deviation, equidistribution, etc.) for non-abelian semigroup of linear action on the torus persist under C^1 conservative perturbation of the generators. In addition we give a sufficient condition and an example for robust minimality of the action of semigroup generated by conservative diffeomorphisms on the surfaces. This is a joint work with X. Liu

avec Luc Hillairet & Emmanuel Trélat