Talks

Oussama Bensaid : Coarse embeddings and homological filling functions

 Introduced by Gromov in the 80's, coarse embeddings are a generalization of quasi-isometric embeddings when the control functions are not necessarily affine. We will be particularly interested in the obstructions to the existence of such embeddings between symmetric spaces of noncompact type, Euclidean buildings, CAT(0) spaces and mapping class groups. We show that, like in the quasi-isometric case, the rank (the maximal dimension of a flat/quasi-flat) is monotonous under coarse embeddings, provided that there is no Euclidean factor in the domain, or a Euclidean factor of dimension 1. The proof involves higher homological filling functions.

Yassine Guerch : Croissance de l'homologie et nombres de Betti l² du groupe des automorphismes extérieurs d'un groupe de Coxeter universel

Soient n ≥ 3 et Out(W_n) le groupe des automorphismes extérieurs d'un groupe de Coxeter universel W_n de rang n, produit libre de n copies d'un groupe cyclique d'ordre 2. Dans cet exposé, nous présentons des résultats sur la croissance de la dimension des groupes d'homologie (à coefficients dans un corps quelconque) de sous-groupes d'indice fini distingués du groupe Out(W_n).

Dans un travail en commun avec Damien Gaboriau et Camille Horbez, nous montrons que, pour tout degré au plus égal à ⌊n/2⌋-1, la croissance de tels nombres de Betti est au plus sous-linéaire avec l'indice du sous-groupe. Ceci implique, par le théorème d'approximation de Lück, que les nombres de Betti l² de Out(W_n) s'annulent jusqu'au degré ⌊n/2⌋-1.

Nous présentons également des éléments de la démonstration, qui repose sur une méthode introduite par Abért, Bergeron, Fraczyk et Gaboriau. L'action par isométries de Out(W_n) sur un complexe simplicial appelé le complexe des W₂-bases partielles de W_n joue un rôle clé.

Antoine Meddane : Morse Inequalities for Axiom A flows

 Axiom A flows are flows introduced by Smale in the 60' to generalise the geodesic flows on negatively curved manifolds (and more generally the Anosov flows) and the Morse gradient flows. The dynamics of the latest are well-known to be related to the topology of the underlying manifold, notably through Morse inequalities. After presenting the Axiom A flows, we will explain how tools from microlocal analysis can be used in order to obtain general Morse inequalities for Axiom A flows (verifying the strong transversality assumption) which extend the previous ones. This work constitutes a progress towards the links between the dynamics of Axiom A flows and the topology of the underlying manifold.

Neige Paulet : Building Anosov flows with hyperbolic plugs

In this talk, I will present Anosov flows and their interest in the theory of hyperbolic dynamics and the topology of manifolds of dimension 3. Recently, Béguin, Bonatti and Yu have developed a very general and powerful technique which allows to construct Anosov flows in dimension 3 by gluing together building blocks along their boundary. I will present a method that generalizes Béguin-Bonatti-Yu gluings, and allows to enrich the class of examples built in that way. On the other hand, the building blocks that we use are pieces of an “optimal” decomposition of an Anosov flow along incompressible surfaces, and allow us to better understand the interactions with the topology of the manifold that carries surch a flow.

 Bernhard Reinke : The Weierstrass root-finder is not generally convergent

We will give an overview of root-finding methods and their interpretation as complex dynamical systems. The main focus will be the Weierstrass/Durand-Kerner method and its similarities and differences to the Newton and the Ehrlich-Aberth methods.

In particular, we will show how to use methods from computer algebra to investigate (and/or establish) the existence of attracting periodic cycles, as well as diverging orbits, and present explicit examples of both phenomena for the Weierstrass method.

 Rym Smai : Globally hyperbolic maximal conformally flat spacetimes with complete photons

 A conformally flat spacetime is an oriented and time-oriented Lorentzian manifold which is locally conformal to the Minkowski spacetime. The natural geometrical objects in a conformally flat spacetime are the non-parametrized isotropic geodesics, called photons. In this talk, I will define a notion of completeness of photons. Then, I will give some "classification" results on conformally flat spacetimes satisfying some nice assumptions of causality and containing complete photons. I will end up with some open questions around this topic.

Mireille Soergel : A generalized Davis complex for Dyer groups

First I will introduce Dyer groups. This family of groups has the same solution to the word problem as Coxeter groups and right-angled Artin groups. We will see that every Dyer group is a finite index subgroup of some Coxeter groups. I will also construct actions of Dyer groups on CAT(0) spaces which extend the actions of Coxeter groups on Davis complexes and of right-angled Artin groups on Salvetti complexes.

Marie Trin : Thurston's compactification via geodesic currents: the case of non-compact finite area surfaces

Paula Truöl : Strongly quasipositive knots are concordant to infinitely many strongly quasipositive knots

 Knots are smooth embeddings of the (oriented) circle S¹ into R³ (or into the 3-sphere), usually studied up to an equivalence relation called ambient isotopy. A natural generalization in dimension 4 of the question whether certain knots are isotopic to the trivial knot is the concept of concordance, another equivalence relation on the set of knots.

 We show that every non-trivial strongly quasipositive knot is smoothly concordant to infinitely many pairwise non-isotopic strongly quasipositive knots. In contrast to our result, it was conjectured by Baker that smoothly concordant strongly quasipositive fibered knots are isotopic. Our construction uses a satellite operation whose companion is a slice knot with maximal Thurston-Bennequin number -1.

In the talk, we will define all the relevant terms necessary to understand the theorem in the title, and explain the context of this result.