Jeudi 21 juin 14:00-15:00 Viviana del Barco (Orsay)
Almost hermitian structures on real flag manifolds

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Lieu : salle 2L8 (IMO, bâtiment 307)

Résumé : Real flag manifolds are submanifolds of complex flags ; the latter ones have been thoroughly studied and they admit complex, symplectic, Hermitian and even Kahler structures. It is natural then to ask about the possibility of having these structures on real flags. We show that, contrary to the complex case, these are never symplectic and therefore not Kahler. Nevertheless integrable complex structures can be found in type C : some specific manifolds of flags of isotropic subspaces of R^2n with respect to a symplectic structure carry complex structures. On these
particular cases we see where the pairs of invariant Riemannian metric-almost complex structures fit in the classification of Gray-Hervella of almost Hermitian structures.
The talk is based on (on-going) works in collaboration with Ana Paula Cruz de Freitas and Luiz San Martin, from UNICAMP, Brazil.

Notes de dernières minutes : Café culturel assuré à 13h par Andrei Moroianu.

Almost hermitian structures on real flag manifolds  Version PDF


Jeudi 14 juin 14:00-15:00 Matthew Foreman (UCI (University of California, Irvine))
A global structure theorem for measure preserving systems

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Lieu : IMO, salle 2L8

Résumé : We define two classes of symbolic systems, the odometer based and the circular systems. The odometer based systems are presentations of ergodic measure preserving transformations that have odometer factors. The circular systems are symbolic presentations of Anosov-Katok diffeomorphisms.
The main result is that these two classes are isomorphic by a functor that preserves the factor structure, including compact and weakly mixing factors. We derive two consequences :

  • for every Choquet simplex K there is an ergodic measure preserving diffeomorphism T of the 2-torus with K affinely homeomorphic to the T-invariant measures.
  • there are ergodic measure-distal (generalized discrete spectrum) diffeomorphims of the 2-torus with arbitrarily large countable ordinal height, in particular with height 3.

A global structure theorem for measure preserving systems  Version PDF
Jeudi 7 juin 14:00-15:00 Oscar Bandtlow (Queen Mary University)
Ruelle transfer operators with explicit spectra

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Lieu : IMO, salle 2L8

Résumé : In a seminal paper Ruelle showed that the long time asymptotic behaviour of analytic hyperbolic systems can be understood in terms of the eigenvalues, also known as Pollicott-Ruelle resonances, of the so-called Ruelle transfer operator, a compact operator acting on a suitable Banach space of holomorphic functions.
Until recently, there were no examples of Ruelle transfer operators arising from analytic hyperbolic circle or toral maps, with non-trivial spectra, that is, spectra different from 0,1.
In this talk I will survey recent work with Wolfram Just and Julia Slipantschuk on how to construct analytic expanding circle maps or analytic Anosov diffeomorphisms on the torus with explicitly computable non-trivial Pollicott-Ruelle resonances. I will also discuss applications of these results.

Notes de dernières minutes : Café culturel à 13h par Hans-Henrik Rugh

Ruelle transfer operators with explicit spectra  Version PDF
Jeudi 31 mai 14:00-15:00 Erlend Grong  (Orsay)
Curvature and comparison theorems for the sub-Laplacians

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Lieu : salle 2L8 (IMO, bat. 307)

Résumé : We study hypoelliptic second order differential operators that are not elliptic, but satisfy the strong Hörmander condition. Just as elliptic operators correspond to a Riemannian geometric structure, such hypoelliptic operators correspond to a sub-Riemannian geometric structure. One can consider sub-Riemannian manifolds as the limit of Riemannian manifolds where the length of vectors in a certain subbundle go to infinity. Unfortunately, this limit will make the Ricci curvature become unbounded. Hence, we loose important results in the process, such as the Laplace comparison theorem.
We will show how to recover a comparison theorem for certain cases. We will consider sub-Riemannian manifolds that appear as limits of totally geodesic foliations. In particular, we will focus on Sasakian manifolds. The results we obtain allows us to determine global properties of our manifold, such as compactness and diameter bound, from just partial properties of its curvature.
This result is a joint work with Fabrice Baudoin, Kazumasa Kuwada and Anton Thalmaier.

Notes de dernières minutes : Café culturel assuré à 13h par Patrick Massot.

Curvature and comparison theorems for the sub-Laplacians  Version PDF
Jeudi 24 mai 13:45-16:45  
Exposés de doctorants

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Lieu : IMO, salle 2L8

Résumé :

13h45 - 14h15 : Davi Obata - On the stable ergodicity problem in conservative dynamics

Résumé : Ergodicity is an important feature that a conservative dynamical system may have. It states that from the probabilistic point of view the system cannot be decomposed. In this talk we will study the question : When is a conservative dynamical system ergodic and every other conservative system close to it is also ergodic ? Such systems are called stably ergodic. This type of problem dates back to Kolmogorov in 1954, but the firsts examples were given in the 60’s by Anosov, the so called hyperbolic (or Anosov) systems. In this talk I will state some recent results on the stable ergodicity problem.

14h30 - 15h : Yuntao Zang - Bounding the measure-theoretical entropy by uniform entropy on Submanifolds

Résumé : In this talk, I will introduce an inequality which gives an upper bound for the measure-theoretical entropy of a C^{1+\alpha} diffeomorphism. The inequality can be viewed as a mixture between the sum of the positive Lyapunov exponents and uniform dimensional entropy on submanifolds. I will also introduce some consequences of this inequality about hyperbolic measures.

15h30 - 16h : Antoine Fermé - Cobordismes irréversibles de fronts d’onde

Résumé : La propagation d’une perturbation (onde de choc, lumière) dans
un milieu se matérialise par une hypersurface dépendant du temps : son
front d’onde. Le front peut développer des singularités ce qui rend a
priori difficile la détermination de sa forme future. Cependant, en
assemblant les images successives du film du front en une variété de
dimension supérieure, on obtient un cobordisme reliant les fronts
initial et futur. Et comme le film est réversible, ceci définit une
relation d’équivalence, pour laquelle la classification des fronts est
(mal)heureusement trop simple. On verra qu’en introduisant de
l’irréversibilité dans nos films, on arrive à une diversité bien plus

16h15 - 16h45 : Gabriel Pallier - Géométrie asymptotique sous-linéairement lipschitzienne : hyperbolicité, autosimilarité. Invariants.

Résumé : On s’intéresse à la catégorie des applications sous-linéairement bilipschitziennes à grande échelle (quasiisométries généralisées) introduite par Yves de Cornulier et provenant de l’étude des cônes asymptotiques des groupes de Lie. En particulier, on décrira dans cet exposé le prolongement de ces applications aux sphères à l’infini des espaces hyperboliques au sens de Gromov, avant d’en déduire des invariants. On donnera finalement quelques pistes de travail futur, visant à affiner ces invariants et élargir la classe d’espaces en jeu.

Exposés de doctorants  Version PDF
Jeudi 17 mai 14:00-15:00 Milena Pabiniak (Cologne)
Cohomological rigidity via toric degenerations

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Lieu : IMO, salle 2L8

Résumé : In order to study the homeomorphism type of manifolds, algebraic topology provides quite powerful invariants, as for example the integral cohomology ring. While this invariant does not distinguish (the homeomorphism type of) smooth manifolds, its restriction to certain natural classes of manifolds is known to be complete (for example to the class of simply connected closed 4-manifolds).
In this talk we will restrict our attention to a natural class of symplectic manifolds, called toric, which admit an action of a torus of large dimension, and we will pose a symplectic cohomological rigidity problem : is any ring isomorphism from the integral cohomology of M to that of N, which maps the class of a symplectic form of M to the class of a symplectic form of N, induced by a symplectomorphism ? Due to the symmetries coming from the torus action, toric symplectic manifolds are quite rigid, giving a hope for a positive answer to the above question.
To approach such a question one needs a tool for creating symplectomorphisms and I will explain how to use the construction of toric degenerations from algebraic geometry to that purpose. In particular, I will show that the cohomological rigidity problem holds for the family of Bott manifolds with rational cohomology ring isomorphic to that of a product of copies of CP^1. This is based on joint work with Sue Tolman.

Cohomological rigidity via toric degenerations  Version PDF
Vendredi 18 mai 15:15-16:15 Milena Pabiniak (Universität zu Köln)
Cohomological rigidity for symplectic toric manifolds via toric degenerations
Vendredi 18 mai 14:00-15:00 Yang Huang (Uppsala University)
A prelude to convex hypersurface theory in contact topology