Séminaire international et très sérieux de Géométrie et dynamique

Organisateurs : Jérémy Toulisse et Selim Ghazouani

Le séminaire a lieu tous les lundis et jeudis à 16h (heure française) sur l'internet (un lien Zoom sera partagé sur cette page avant le début des exposés).

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Jeudi 9 avril 2020 Vincent Pecastaing Actions de réseaux de rang supérieur sur des structures conformes et projectives
Lien Zoom

Résumé : L'idée phare du programme de Zimmer est qu'en rang supérieur ou égal à 2, la rigidité des réseaux des groupes de Lie semi-simples est telle qu'on peut comprendre leurs actions sur des variétés compactes. Après un bref survol donnant une idée plus précise des conjectures de Zimmer et de leur contexte, je présenterai des résultats récents portant sur les actions conformes ou projectives de réseaux cocompacts. L'absence de forme volume naturelle invariante sur ces structures est l'une des motivations principales. On verra que le rang réel est borné comme lorsque le groupe de Lie ambiant agit, et qu'à la valeur critique, la variété est globalement équivalente à un espace homogène modèle. Les preuves s'appuient en outre sur un 'principe d'invariance' introduit récemment par Brown, Rodriguez-Hertz et Wang, assurant l'existence de mesures finies invariantes dans certains contextes dynamiques.

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Lundi 13 avril 2020 Andrea Seppi Examples of four-dimensional geometric transition
Lien Zoom

Résumé : Roughly speaking, a geometric transition is a deformation of geometric structures on a manifold, by “transitioning” between different geometries. Danciger introduced a new such transition, which enables to deform from hyperbolic structures to Anti-de Sitter structure, going through another type of real projective structures called “half-pipe”, and provided conditions for a compact 3-manifold to admit a geometric transition of this type. By extending a construction of Kerckhoff and Storm, I will describe examples of finite-volume geometric transition in dimension 4. This is joint work with Stefano Riolo.

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Jeudi 16 avril 2020 Bertrand Deroin Non orderability of higher rank lattices
Lien Zoom

Résumé : I will report on some joint work with Sebastian Hurtado establishing that an irreducible lattice of a connected semi-simple Lie group of rank at least two and with finite center is not left-orderable.

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Lundi 20 avril 2020 Françoise Pène Functional limit theorems for billiards in domains with cusps
Lien Zoom

Résumé : We are interested in the study of the stochastic properties of billiards in domains with convex scatterers and cusps. The most famous example is Machta's model which is delimitated by three isometric tangent circles. For this model, a rate of decay of correlations has been established by Chernov and Zhang and a functional limit theorem with a non standard normalization has been proved by Bálint, Chernov and Dolgopyat. For billiards with cusps of higher flatness, Zhang established a rate of decay of correlations depending on the flatness of the cusps. In the case of a single symmetric cusp of higher flatness, Jung and Zhang proved a non standard limit theorem (convergence to a stable random variable). We extend this result by proving a non standard functional limit theorem (convergence to a Lévy process) for more general billiards with cusps (allowing several cusps, with more general shape, possibly assymetric, with possibly different flatness order).

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Jeudi 23 avril 2020 Léo Benard Asymptotic of twisted Alexander polynomials and hyperbolic volume Slides pour l'exposé
Lien Zoom

Résumé : Given a hyperbolic 3-manifold of finite volume M, we compute the asymptotic of the family of twisted Alexander polynomials on the unit circle. We prove that this family growth asymptotically as the volume times the square of the dimension of the representation. The proof goes through the study of the analytic torsion of some compact hyperbolic manifolds obtained by Dehn surgery on M. Joint work with J. Dubois, M. Heusener and J. Porti.

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Lundi 27 avril 2020 Ramanou Santharoubane Quantum representations of surface groups
Lien Zoom

Résumé : I will show how from Witten-Reshetikhin-Turaev TQFT, we can produce interesting representations of surface groups. The key fact is the following : these "quantum representations" of surface groups have infinite images but every simple loop acts with finite order. Using this key fact and integral TQFT, we will see how to build regular finite covers of surfaces where the integral homology is not generated by pullbacks of simple closed curves on the base. This talk represents joint work with T.Koberda.

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Jeudi 30 avril 2020 Martin Leguil Entropy rigidity for three-dimensional conservative Anosov flows
Lien Zoom

Résumé : Among smooth Anosov flows on 3-manifolds preserving a smooth volume, algebraic models are believed to distinguish themselves in many ways. The question of entropy rigidity asks whether algebraic models can be characterized (up to smooth conjugacy) by the property that the volume measure has maximal entropy. I will present some results in this direction obtained in a joint project with J. De Simoi, K. Vinhage and Y. Yang. We shall see that a key step is to obtain precise estimates on the Lyapunov exponents of periodic orbits with a prescribed combinatorics. We will also present some results towards the question of entropy rigidity for dispersing billiards.

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Lundi 4 mai 2020 Bruno Duchesne Maximal representations of complex hyperbolic lattices in isometry groups of infinite dimensional hermitian symmetric spaces. Slides pour l'exposé
Lien Zoom

Résumé : Unlike higher rank Lie groups, real and complex hyperbolic lattices are much more flexible and their representations can be wilder. Nonetheless for the complex ones, their representations to Hermitian Lie groups have a numerical invariant: the Toledo number. When this number is maximal, one can hope to recover rigidity (in the spirit of Margulis super-rigidity). In this talk, we will be interested in infinite dimensional Hermitian Lie groups and their associated symmetric spaces. The complex hyperbolic space of infinite dimension being the simplest one among them. After introducing them and seeing why they can be different and interesting, we will extend Toledo number in this setting and prove a super-rigidity statement for maximal representations to the group $\mathrm{U}(p,\infty)$. This is a joint work with J. Lécureux and B. Pozzetti.

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Jeudi 7 mai 2020 Olivier Glorieux Random geometric process on Riemannian manifolds.
Lien Zoom

Résumé : We will start with the following question : « what is the probability for a triangle on a manifold to be homotopically trivial ? » and give an answer for all flat tori (j.w A. Yarmola). In a second part we will explain a related geometric process : after taking a random sequence of points $X_i$ on a manifolds we will be interested in the path given by the sequence of minimising segments between $X_i$ $X_{I+1}$. We will give its asymptotic behaviour and a central limit theorem for non amenable covers of compact negatively curved manifolds (j.w. A. Boulanger). After the break, we will give details on the proof of the second theorem, more precisely we will exhibit a spectral gap property for the Markov operator associated to the process.

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Lundi 11 mai 2020 Jérémy Alsmooth Plateau’s problem in pseudo-hyperbolic spaces
Lien Zoom

Résumé : The pseudo-hyperbolic space H^{2,n} is the pseudo-Riemannian analogue of the classical hyperbolic space. In this talk, I will explain how to solve an asymptotic Plateau problem in this space: given a topological circle in the boundary at infinity of H^{2,n}, we construct a unique complete maximal surface bounded by this circle. This construction relies on Gromov's theory of pseudo-holomorphic curves. This is a joint work with François Labourie and Mike Wolf.

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Jeudi 14 mai 2020 Joan Porti Projective structures on a certain 3-orbifold
Lien Zoom

Résumé : The moduli space of convex projective structures on surfaces is well understood, thanks to Choi and Goldman, but on 3-dimensional manifolds it is not. The aim of the talk is to overview the situation in dimension three and to present an explicit example, the moduli space of a certain 3-orbifold, jointly worked out with Stephan Tillmann.

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Lundi 18 mai 2020 François Labourie Actions affines propres de groupes de surfaces
Lien Zoom

Résumé : The goal of this talk is to give a thermodynamical proof to the following recent result of Danciger and Zhang : a surface group acting on the affine space with Hitchin linear part does not act properly. In the introduction, i will give an overview of entropy and why it is relevant to this question, as well as an introduction to the Margulis invariant and its relation to entropy and properness.

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Jeudi 21 mai 2020 Maxime Fortier-Bourque Kissing numbers of manifolds and graphs
Lien Zoom

Résumé : The kissing number of a metric space X is defined as the number of distinct homotopy classes of shortest closed geodesics in X. We are interested in how large the kissing number can get within certain families of metric spaces. This problem has been studied for flat tori and hyperbolic surfaces before. I will discuss joint work with Bram Petri where we obtain upper bounds for the kissing number of closed hyperbolic manifolds of any dimension and of regular graphs.

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Lundi 25 mai 2020 Romain Dujardin Itération aléatoire sur les surfaces complexe
Lien Zoom

Résumé : La donnée d'une mesure de probabilité \nu sur le groupe des automorphismes d’une surface complexe compacte X définit un système dynamique aléatoire sur X, obtenu en composant des difféomorphismes aléatoires de loi \nu. Une mesure de proba sur X est \nu-stationnaire si \int f_* \mu \, d\nu(f) = \mu: ce sont les objets de base de la théorie ergodique des transformations aléatoires. L’objet de cet exposé est de décrire certains résultats que nous avons récemment obtenus avec Serge Cantat sur la classification de ces mesures stationnaires.

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Jeudi 28 mai 2020 Carlos Fougeron Dynamiques des systèmes simpliciaux et des algorithmes de fraction continue multidimensionnels
Lien Zoom

Résumé : Motivés par la richesse de l'algorithme de Gauss qui permet de calculer efficacement les meilleurs approximations d'un nombre réel par des rationnels, beaucoup de mathématiciens ont proposé des généralisations de ces algorithmes pour approcher des vecteurs de dimension supérieure à 1. Citons pour exemple l'algorithme de Poincaré introduit à la fin du 19e siècle ou ceux de Brun et Selmer à la moitié du 20e siècle. Depuis le début des années 90 à aujourd'hui il y a eu un certain nombre de travaux pour comprendre la convergence de ces algorithmes. Schweiger et Broise ont notamment démontré que les algorithmes de Selmer et Brun sont convergent et ergodiques. Mais, plus surprenant peut-être, Nogueira a démontré que l'algorithme proposé par Poincaré ne convergeait presque jamais. Dans mon exposé j'aborderai une nouvelle présentation combinatoire de ces algorithmes qui permet le passage d'un point de vu déterministe à une approche probabiliste pour ceux-ci. Dans ce modèle, prendre un vecteur aléatoire pour la mesure de Lebesgue correspond à suivre une marche aléatoire "avec mémoire" dans un graphe étiqueté nommé système simplicial. Les lois pour cette marche aléatoire sont élémentaires et nous pouvons développer des techniques probabilistes pour étudier leur comportement dynamique générique. Cela nous mènera à décrire un critère purement de théorie des graphes pour montrer la convergence ou non d'un algorithme de fraction continue.

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Lundi 1 juin 2020 Lucas Kaufmann Random walks on SL_2(C) via holomorphic dynamics
Lien Zoom

Résumé : The study of random walks on matrix groups, also known as products of random matrices, is a classical subject dating back to the fundamental work of Furstenberg and Kesten. In this talk I will focus on the case of 2 by 2 matrices and show how such a problem can be viewed as a deterministic (generalized) dynamical system on the Riemann Sphere. Such point of view allows us to use recent tools from holomorphic dynamics and prove new results about the random walk. In particular, we are able to obtain a spectral gap result for the Markov operator under optimal moment conditions. As a corollary, we obtain new equidistribution results and a simplified proof of Benoist-Quint’s Central Limit Theorem. This is joint work with T.-C. Dinh and H. Wu.

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Jeudi 4 juin 2020 Nicolas Tholozan Quotients compacts d’espaces pseudo-hyperboliques
Lien Zoom

Résumé : L’espace pseudo-hyperbolique H^{p,q} est le modèle des variétés pseudo-riemanniennes de signature (p,q) et de courbure sectionnelle constante négative. On aimerait décrire ses quotients compacts (par un sous-groupe proprement discontinu d’isométries). Nous commencerons par présenter certaines obstructions classiques'' à l’existence de tels quotients (pour p et q impairs, pour p inférieur à q), et les quelques exemples connus (pour q=1,3, ou 7). Nous formulerons ensuite une conjecture sur la géométrie de ces quotients, et expliquerons comment cette conjecture inspire de nouvelles obstructions, obtenues avec Fanny Kassel et Yosuke Morita.

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Lundi 8 juin 2020 Bac Dang On the spectrum of self-similar groups and holomorphic dynamics
Lien Zoom

Résumé : We will discuss the spectrum of the Laplacian on some self-similar groups, like the Grigorchuk group, the lamplighter group and the Hanoi group. Classical Schur renormalization transformations act on spectral parameters as a rational map in two variables. We will show that the spectrum in question can be interpreted as the asymptotic distribution of sliced pullbacks of certain algebraic curves under this rational map. For groups under consideration, this rational map happens to be fibered over a quadratic polynomial. We will discuss the algebra-geometric nature of this integrability phenomenon. Based upon a joint work with R. Grigorchuk and M. Lyubich.

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Jeudi 11 juin 2020 Katie Vokes Geometry of graphs of multicurves
Lien Zoom

Résumé : Given a compact, connected, orientable surface, we can define many associated graphs whose vertices represent curves or multicurves in the surface. A first example is the curve graph, which has a vertex for every simple closed curve in the surface and an edge joining two vertices if the corresponding curves are disjoint. A key property of the curve graph is that it is Gromov hyperbolic, but this is not the case for all such graphs. I will introduce some graphs of multicurves, and will present joint work with Jacob Russell classifying when such graphs are hyperbolic, when they are relatively hyperbolic (a generalisation of hyperbolicity), and when they are neither of these.

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Lundi 15 juin 2020 Adrien Boulanger Grandes déviations de la vitesse de fuite des marches aléatoires hyperboliques
Lien Zoom

Résumé : La première partie de l'exposé sera introductive et se cantonnera aux marches simples sur les groupes. Dans la deuxième partie on discutera un énoncé de type grandes déviations pour la vitesse de fuite des marches aléatoires non élémentaires sur des espaces Gromov hyperboliques. Travail en commun avec Pierre Mathieu, Cagri Sert et Alessandro Sisto.

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