# Séminaire Teich

Vendredi 26 février, à 11h,

Khaydar Nurligareev nous parlera des contages combinatoires des différents objets, notamment, des surfaces à petits carreaux.
Le thème sera :
Probabilité dêtre irréductible

Asymptotics for the probability of labeled objects to be irreducible

Abstract

There are a number of combinatorial structures that admit, in a broad sense, a notion of irreducibility, including connected graphs and surfaces, irreducible tournaments, indecomposable permutations and so on. We are interested in the probability that a random labeled object is irreducible, as its size tends to infinity. We will show that for some classes of objects it is possible to obtain the asymptotics for these probabilities in a common manner and that asymptotic coefficient have a combinatorial meaning. More precisely, it is so when the considered combinatorial class can be described as a set or a sequence of the corresponding irreducible class, and its counting sequence grows sufficiently fast. Moreover, we will show how to obtain the asymptotic probability that a random labeled object has a given number of irreducible components, and we will indicate the combinatorial meaning of the coefficients involved in the asymptotic expansions.

This is an ongoing work joint with Thierry Monteil.

# Séminaire singularité

Jeudi 25 février 14h

Title: On zeta functions, weighted blow-ups and some applications for quasi-homogeneous surface singularities.

Abstract: The Denef-Loeser topological and motivic zeta functions are analytic invariants of holomorphic map germs $f:{\mathbb C}^n\to {\mathbb C}$, which are usually computed from embedded resolutions of $f$.

They codify some information about the topology of the Milnor fiber of the zero locus. More concretely, the Monodromy Conjecture predicts that any pole of these zeta functions is related with an eigenvalue of the monodromy at some point of $f^{-1}(0)$.

In this talk, we introduce some recent techniques that we have developed for the study of these zeta functions for $\mathbb{Q}$-divisors over orbifold varieties: a change of variables formula from relative canonical divisors, as well as a closed formula using compositions of weighted blowing-ups. Finally, we present some work in progress about applications on the study of the Monodromy Conjecture for quasi-homogeneous surface singularities.

This is a joint work Edwing LEON-CARDENAL (UNAM), Jorge MARTIN-MORALES (UNIZAR-CUD) y Wim VEYS (KU Leuven).

# Séminaire Teich

vendredi 19 février: 11 h 00
Mario Shannon continue de nous parler des flots d’Anosov.
Le thème sera :
Décompositions en livre ouvert d’une suspension d’un flot Anosov et des structures affines.

Open book decompositions of a suspension Anosov flow and affine structures.

Despite the good comprehension that we have nowadays about the asymp- totic dynamical behaviour of a general Anosov flow, classification of the different orbital equivalent classes rest a major subject nowadays. In the particular case of dimension three, a lot of different examples of Anosov flows can be constructed using surgery methods, which shows that the set of different equivalence classes is not at all simple to describe.

If we restrict to the special subfamily of transitive 3-dimensional Anosov flows, each flow has (many) associated open book decompositions of the 3- manifold with pseudo-Anosov monodromy. Since pseudo-Anosov homeomor- phisms can be classified by terms of a combinatorial invariant, there is a hope to produce combinatorial invariants of the orbital equivalence class of these Anosov flows in terms of those available for pseudo-Anosov. The problem to solve is : Given two open book decompositions with pseudo-Anosov monodromy, how to determine if both correspond to the same flow?

We study a simpler question related to the previous one: Given an open book decomposition with pseudo-Anosov monodromy, can we determine whether or not the corresponding Anosov flow is a suspension Anosov flow ? In this talk we will provide a criterion for this question based on the existence of some affine incomplete structures associated with the pseudo-Anosov monodromy. In turn, we can provide a natural bijection between the set of genus one Birkhoff sections of a suspension Anosov flow and these affine structures.

# Séminaire Teich

Vendredi 12 février  : 11 h
Mario Shannon parlera des iModèles hyperboliques des flots topologiques d’Anosov transtitifs.

Hyperbolic models of transitive topological Anosov flows.

A topological Anosov flow on a closed 3-manifold is a non-singular flow that resembles very much a smooth Anosov flow: It is expansive and satisfies the (global) shadowing property. Moreover, it preserves a pair of transverse sta- ble/unstable foliations that intersect along the orbits of the flow. The main difference with a (smooth) Anosov flow is the lack of a global uniformly hyper- bolic structure.

We investigate the question of weather or not every topological Anosov flow in a 3-manifold is, actually, orbit equivalent with some smooth Anosov flow. Apart from its own theoretical interest, this question appears related with some techniques for the construction of Anosov flows on 3-manifolds, notably with the so called Fried surgery.

Our work consists in show that, under the hypothesis of transitivity, every topological Anosov flow is orbitally equivalent with a smooth Anosov flow.

In this talk we are going to present the main ideas and difficulties behind the construction of these smooth hyperbolic models associated with transitive topological Anosov flows. As well, we will give a flavour of how this study can be reduced to pseudo-Anosov dynamics on surfaces with boundary.

# Séminaires Teich à venir

Voici la liste des exposés :
Vendredi 12 février: Mario SHANNON, titre à préciser

Vendredi 19 février: Mario SHANNON, titre à préciser
Vendredi 26 février: Khaydar NURLIGAREEV, Asymptotics for the probability of labeled objects to be irreducible

Vendredi 5 mars: Vacances du séminaire

Vendredi 12 mars: Clémence PERRONNET, titre à préciser
Vendredi 26 mars: (pas de Teich mais) N. Pythéas Fogg invite Pierre GUILLON, odd shift
Vendredi 2 avril: Pierre DEHORNOY, titre à préciser

Vendredi 9 avril:  Gabriela WEITZE-SCHMITHÜSEN, titre à préciser