Lieu : salle 3L8

Résumé : We shall consider the two-parameter family f_a,b of self-maps of R^2 given by (x,y) → (ax+1/y, by-b/y-abx) where 0<a≤b≤1. When a=b=1 this map is known as the « Hénon-Devaney map ». We will give some dynamical and ergodic properties of these maps.
For all the parameters, we exhibit two transversal invariant C^1 foliations. For a<b, there is a global unbounded transitive attractor exhibiting a type of SRB measure. If b=1 the measure is infinite, and if b<1 the measure is finite. Moreover, in the last case the attractor is robustly transitive and stable in the sense that for nearby systems there is a conjugated attractor. For the Hénon-Devaney map (a=b=1), we get a conjugation to a subshift, providing a global understanding of the map’s behavior.

Lieu : Salle 3L8

Résumé : The Duflo Theorem states the isomorphism of the center of the universal enveloping algebra with the ring of invariant polynomials. The Kashiwara-Vergne (KV) problem on the properties of the Baker-Campbell-Hausdorff series is one of the strategies of proving the Duflo Theorem. Surprisingly, it turns out that the KV problem is related to many other fields of mathematics including braid groups, Drinfeld associators and multiple zeta-values.
In this talk, we will start by reviewing the Duflo Theorem and the KV problem, and then we will explain the connection between the KV problem and the Goldman bracket and Turaev cobracket defined by intersections and self-intersections of curves on 2-manifolds. Our toolbox will include the Knizhnik-Zamolodchikov connection and the non-commutative version of the divergence.
The talk is based on joint works with B. Enriquez, N. Kawazumi, Y. Kuno, E. Meinrenken, F. Naef and C. Torossian.

Lieu : Salle 3L15, IMO (Bâtiment 307)

Résumé : Using an operator-theoretic framework in a Hilbert-space setting, we perform a detailed spectral analysis of the one-dimensional Laplacian in a bounded interval, subject to specific non-self-adjoint connected boundary conditions modelling a random jump from the boundary to a point inside the interval. In accordance with previous works, we find that all the eigenvalues are real. As the new results, we derive and analyse the adjoint operator, determine the geometric and algebraic multiplicities of the eigenvalues, write down formulae for the eigenfunctions together with the generalised eigenfunctions and study their basis properties. It turns out that the latter heavily depend on whether the distance of the interior point to the centre of the interval divided by the length of the interval is rational or irrational. Finally, we find a closed formula for the metric operator that provides a similarity transform of the problem to a self-adjoint operator. This is joint work with Martin Kolb.

Lieu : salle 3L8

Résumé : In the last two decades several works have been done on stable ergodicity
of volume-preserving diffeomorphisms. In the partially hyperbolic world,
all the works on stable ergodicity use a notion called accessibility. In
this talk, I will present the main points in the proof of the stable
ergodicity of an example of a volume-preserving, partially hyperbolic
diffeomorphism introduced by Pierre Berger and Pablo Carrasco. The main
novelty of this proof is that it does not use accessibility.

Lieu : Salle 3L15, IMO (Bâtiment 307)

Résumé : In the first part of my talk, I will show that a quantum system with translational invariance may host dispersionless wave packets and that a sufficient condition for their existence is presence of a linear energy band in the spectrum of the corresponding Hamiltonian. This condition is, of course, very restrictive, so we will also briefly study what happens if it is slightly violated. Still, there are some rare examples of systems with exactly linear bands – those I am aware of are always governed by 2D Dirac Hamiltonian. In the second part of my talk, I will show how to construct a new example of such a system employing some basic self-adjoint extension techniques.

Lieu : salle 3L8

Résumé : Un système dynamique stochastique (SDS) est un processus aléatoire défini récursivement par X_n(x) = Psi_n(X_n-1(x)) et X_0(x)=x, où les Psi_n sont des transformations continues aléatoires iid. Nous considérons la classe SDS de la droite réelle définie par des transformations asymptotiquement linéaires en +∞ et -∞. Cette classe inclut des processus intéressants tels que la récursion affine, la marche aléatoire réfléchie et différents modèles venant de la biologie et de la finance. Nous étudions les conditions d’existence d’une mesure de probabilité stationnaire et décrivons le comportement à l’infini d’une telle mesure.

Lieu : Salle 3L8

Résumé : Nous nous intéressons à la question de l’unicité pour des solutions (expansives ou auto-similaires) au flot d’applications harmoniques lissant des applications 0-homogènes à valeurs dans une variété riemannienne fermée. Pour ce faire, nous introduisons une entropie relative qui permet d’établir deux résultats. D’une part, nous montrons l’existence de deux solutions expansives associées à toute solution lissant une application 0-homogène par un procédé d’éclatement parabolique : cela démontre une conjecture d’Ilmanen dans ce contexte. D’autre part, un phénomène d’unicité générique pour de telles solutions ayant une entropie relative nulle est montré.

Lieu : Salle 3L8

Résumé : L’asymptotique du noyau de Bergman associé à
la suite des puissances tensorielles
d’un fibré en droites positif a reçu beaucoup d’attention
dans les dernières
années. Dans cet exposé, nous considérons plus
généralement une suite de fibrés positifs
dont les courbures satisfont à une condition faible de
croissance, et nous étudions l’asymptotique de
de leurs noyaux de Bergman sur la diagonale et en
dehors de la diagonale.

Lieu : Salle 3L8

Résumé : \’Elie Cartan described arbitrary systems of partial differential equations in terms of differential forms and submanifolds rather than equations and functions.
Many examples of such systems arise in differential geometry, sometimes in a manner that does not even require coordinates to state.
Cartan used this point of view to prove a theorem of existence and generality of solutions, further developed by Kaehler.
Their theorem applies only in the real analytic category, and only describes local solutions, so from a modern PDE perspective it seems a very poor theorem.
Nonetheless, because the theorem is coordinate independent, it can sometimes apply easily to problems which are easier to describe in differential forms than in coordinates.
For some problems, there is no other analytical tool to prove local existence of solutions, so the Cartan—Kaehler theorem is surprisingly rich.
The theorem has an algebraic flavour, as it demands no estimates, so it appeals to algebraic geometers, while analysts tend to find it unattractive for the same reason.
However, the theorem demands strong nondegeneracy and smoothness hypotheses, which make it frustrating to everyone.
I want to explain how to state and employ the Cartan Kaehler theorem, with simple examples, and give an overview of some points of the proof.
This talk will explain very old ideas, with no new ideas.

Lieu : salle 3L8

Résumé : Il s’agit d’un travail en cours avec Dawei Yang (Suzhou). Nous démontrons qu’un attracteur partiellement hyperbolique de dimension 3 avec singularités admet pour tout potentiel Hölder au plus une mesure d’équilibre régulière (c.à.d. qui ne charge pas les singularités). Comme corollaire, nous en déduisons l’existence et l’unicité de la mesure d’entropie maximale.
La preuve consiste à se ramener à un système induit bidimensionnel appelé « mille-feuille » qui permet de construire localement l’équilibre.
J’expliquerai la construction de ce mille-feuille en me basant sur l’exemple de l’attracteur de Lorenz dont la géométrie est assez simple du fait d’une section de Poincaré globale.

Lieu : Salle 3L8

Résumé : We describe an application of equivariant integration to the topology of the moduli spaces of rank-2 Higgs bundles on Riemann surfaces.
Our integral formulas naturally lead us to certain transcendental equations, the Bethe equations, and to a proof of a conjecture concerning the cohomology of this moduli space : the P=W conjecture.
This work is joint with Simone Chiarello and Tamas Hausel.

Lieu : Salle 3L8

Résumé : We describe an application of equivariant integration to the topology of the moduli spaces of rank-2 Higgs bundles on Riemann surfaces.
Our integral formulas naturally lead us to certain transcendental equations, the Bethe equations, and to a proof of a conjecture concerning the cohomology of this moduli space : the P=W conjecture.
This work is joint with Simone Chiarello and Tamas Hausel.