Séminaire Géométrie, analyse et dynamique

Séminaire Géométrie, analyse et dynamique

(2021-2022)

Laboratoire Dieudonné-CNRS-UNS UMR 7351

Le Séminaire a lieu le Mardi à 14h00 en salle I du LJAD

Accès au laboratoire J.A. Dieudonné




Prochain exposé


Mardi 26 Octobre            
     Ao Cai (Universidade de Lisboa (Portugal))

Lyapunov exponent and spectrum of 1-d lattice Schrödinger operators
Résumé

In this talk, we will focus on two specific examples of dynamically defined 1-d lattice Schrödinger operators. Namely, the quasi-periodic model and the mixed random-quasiperiodic model. We will compare the influence of quasi-periodic potentials and random potentials on the Lyapunov exponent and the spectrum from the perspective of Metal-Insulator Transition. We show that randomness somehow dominates quasi-periodicity.



This includes a joint work with Pedro Duarte ( from University of Lisbon) and Silvius Klein (from PUC-Rio).


Au programme


Novembre

Mardi 9 Novembre            
     Emmanuel Militon (LJAD)
TBA


Janvier

Mardi 10 Janvier             Exceptionnellement à 15h00
     Valentino Tosatti (McGill University (Canada))
(en distanciel)


Mardi 25 Janvier            
     Viet Anh Nguyen (Université de Lille)
TBA


Exposés passés


Septembre

Mardi 7 Septembre      Alexis Drouot (University of Washington (US))
Dirac operators and topological insulators
Résumé

In this talk, I will study Dirac operators that emerge in the macroscopic analysis of topological insulators. I will analytically construct canonical edge states: coherent states that propagate along interfaces, but do not admit natural counter-propagating companions. I will illustrate the results with various numerical simulations.



Mardi 28 Septembre      Marc Chaperon (IMJ)
Faits élémentaires et questions naïves
Résumé

On propose ici un point de vue un peu dissident sur les singularités, où elles apparaissent comme limites de situations plus régulières et plus faciles à expliquer. Dans cet exposé, les cas singuliers sont quasi-périodiques, les plus réguliers étant périodiques.



Octobre

Mardi 5 Octobre      Xianghong Gong (University of Wisconsin - Madison (US))
Global Newlander-Nirenberg theorem for domains with \(C^2\) boundary
Résumé

The Newlander-Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the standard complex structure in the complex Euclidean space. We consider two natural generalizations of the Newlander-Nirenberg theorem under the presence of a \(C^2\) strictly pseudoconvex boundary. When a given formally integrable complex structure \(X\) is defined on the closure of a bounded strictly pseudoconvex domain with \(C^2\) boundary \(D \subset \mathbb C^n\), we show the existence of global holomorphic coordinate systems defined on \(\overline{D}\) that transform \(X\) into the standard complex structure provided that \(X\) is sufficiently close to the standard complex structure. Using the global Newlander-Nirenberg theorem, we prove the existence of local one-sided holomorphic coordinate systems provided that \(M\) is strictly pseudoconvex with respect to the given complex structure. This is joint work with Chun Gan.



Mardi 12 Octobre      Zhiyan Zhao (LJAD)
Geometry of hyperbolic Cauchy-Riemann (CR) singularities and KAM-like theory for holomorphic involutions
Résumé

We consider the real analytic perturbed Bishop quadric surfaces in \((\mathbb{C}^2,0)\), having an isolated CR singularity at the origin. There are two kinds of stable CR singularities: elliptic and hyperbolic. The elliptic case was studied by Moser-Webster, where they showed that such a surface is locally, near the CR singularity, holomorphically equivalent to normal form from which lots of geometric features can be read off.



We focus on the hyperbolic case. As shown by Moser-Webster, such a surface can be transformed to a formal normal form by a formal change of coordinates (but usually not holomorphic in any neighborhood of origin). For a non-degenerate real analytic surface \(M\) having a hyperbolic CR singularity at the origin, we prove the existence of Whitney smooth family of curves intersecting \(M\) along holomorphic hyperbolas. This is shown by a KAM-like theorem for a pair of holomorphic involutions at the origin, a common fixed point. Joint work with L. Stolovitch.



Mardi 19 Octobre      Jonathan DeWitt (University of Chicago (US))
(en distanciel) Simultaneous Linearization of Diffeomorphisms of Isotropic Manifolds
Résumé

Suppose that M is a closed isotropic Riemannian manifold and that R_1,...,R_m generate the isometry group of M. Let f_1,...,f_m be smooth perturbations of these isometries. We show that the f_i are simultaneously conjugate to isometries if and only if their associated uniform Bernoulli random walk has all Lyapunov exponents zero. This extends a linearization result of Dolgopyat and Krikorian from S^n to real, complex, and quaternionic projective spaces.




Archives du séminaire de géométrie, analyse et dynamique: 2016/2017, 2017/2018, 2018/2019
Archives du séminaire de géométrie et analyse: 2007/2008, 2008/2009, 2009/2010, 2010/2011, 2011/2012, 2012/2013, 2013/2014, 2014/2015, 2015/2016
Archives du séminaire de géométrie et dynamique: ici

Organisation: Zhiyan Zhao (écrire) et Emmanuel Militon (écrire)