Séminaires doctorant en 2022

The notion of Garside monoid goes back to F.A Garside’s PhD thesis, where it is introduced for the study of the classical braid group in a combinatorial way. These tools later were generalized (and formalized !) in order to study Artin groups, complex braid groups and a lot of other combinatorial topics. This talk’s aim is to give a quick introduction to the theory of Garside monoids, and to investigate its consequences in the case of the usual Artin monoid. This will first require some reminders on monoids in order to properly state the definition of a Garside monoid. I will then explain the basics of the theory of Coxeter groups, mostly the combinatorial results needed to study their Artin groups. Lastly, I will describe the "classical" Garside structure on the Artin monoid associated to a finite Coxeter group, and its first consequences in the study of the Artin groups if time permits.

This talk links ergodic properties on probability dynamical system to spectral ones on the transfer operator. On the particular toy model of the subshift of finite type, the transfer operator provides limit theorems (such as central limit theorems) and thus enabling to state it on many different systems ( doubling map, Sinai billiard through young towers,...). Transfert operator may also leads to local limit theorems providing mixing and conservative ergodicity in non measurably finite dynamical systems such as Z-extensions.

In this presentation, we will see step by step why and how to build a numerical scheme that preserves the steady states of a hyperbolic system. The common thread will be the shallow water model for which we will add source terms to model different physical situations. We will mention the difficulties that arise in preserving moving steady states when the Froude number is close to 1 and briefly discuss the case of 2D stationary states.

I would like to introduce my field of research : geometric and algebraic combinatorics, in particular the study of polytopes. These are geometric objects defined very easily as the convex hull of a finite number of points in a Euclidean space of finite dimension. However, they provide beautiful surprises and connections to various areas of mathematics : combinatorics and geometry obviously, but also algebra (group theory, homotopy theory, …), optimization, theoretical physics. I chose to focus on a special and very interesting polytope : the permutahedron. Depending on the time and the interest of the participants, I may also present its cousin the associahedron (which appeared in homotopy theory) or other variations related to Coxeter groups or quantum physics.

On s’intéresse à un problème de variation de la somme des chiffres quand on ajoute un entier r fixé : à quel point cette variation va prendre une valeur d ? Ce problème dépend évidemment de la manière d’écrire les nombres et a beaucoup été étudié en base entière (en particulier en binaire). On se penchera sur un autre système d’écriture semblable au binaire et liée à la suite de Fibonacci : la représentation de Zeckendorf. Pour cela, on introduira l’odomètre associé à cette écriture. Grâce à lui, on construira un espace de probabilité adapté à ce problème. On proposera également une méthode pour répondre à la question initiale.

Le but de cet exposé est de donner une introduction à l’étude des représentations des algèbres et des carquois ainsi qu’aux outils les plus utilisés dans le domaine. En particulier, on introduit les espaces moduli de carquois, leurs propriétés et leurs liens avec autres domaines comme la géométrie algébrique, la combinatoire ou les algèbres amassées.

Le résultat principal de ce travail est la dérivation d’expressions analytiques exactes pour l’orientation et la trajectoire d’un micro-nageur sphérique soumis à des écoulements linéaires (cisaillement, hyperboliques, rotation et de stagnation) et à un champ de force externe. Les équations d’évolution de l’orientation du nageur et de sa position sont non linéaires et les résultats analytiques sont rares. La plupart des résultats disponibles pour l’orientation et les trajectoires des cellules sont obtenus numériquement. La solution pour l’orientation du nageur est inspirée d’une méthode due à Bretherton, initialement développée pour une équation non linéaire différente. Nous montrons ici que cette méthode peut être généralisée à notre équation d’évolution. Nous avons montré que le nageur sous l’effet de l’écoulement présente à la fois des régimes de « run » (un mouvement où l’angle d’orientation est maintenu constant avec le temps) et de « tumbling » (l’angle d’orientation est cyclique avec le temps), et avons obtenu une riche variété de trajectoires analytiquement, telles que paraboliques, elliptiques et hélicoïdales.

The Möbius function is a multiplicative function that appears in number theory as the coefficients of the inverse of the Riemann Zeta function. We can actually define the Möbius function on any poset (partially ordered set) and see it as the reduced Euler characteristic of some topological space naturally associated to the poset, namely the order complex. It also appears in combinatorics as in certain cases, the Möbius function counts interesting objects.

Hecke algebras arise very naturally when we study representations of p-adic groups or finite groups of Lie type, there are various Hecke algebras. In this talk, I will start from very basic facts about Hecke algebras for finite groups and their representations, then turn to p-adic cases with finite group cases at hand as analogs. If time permits, I will also talk about the duality operator for representations of the Hecke algebra of a Weyl group or of an affine Weyl group in terms of a certain involution on this algebra which is introduced by Shin-ichi Kato and later constructed for representations of p-adic groups by Anne-Marie Aubert.

The aim of this talk is to introduce the mathematics of general relativity, focusing on the concept of black hole that naturally emerges from Einstein’s field equation. The precise goal is to apply numerical methods to the geodesic equations in order to produce a picture of a black hole as realistic as possible, using only simple mathematical softwares, such as Scilab. We first quickly review the basics of special relativity and formalize them via the Lorentz transforms and the Minkowski metric on the 4-dimensional space. This is the first reasonable geometrization of spacetime. The striking idea of Einstein is that gravitation too is a geometric feature (a metric), and not a force. Next, we introduce the notion of Lorentzian manifold, which is the good framework for relativity. In particular, we define geodesics, which are ``straight lines in a curved space’’ and the differential equations they satisfy. Then, we state Einstein’s field equation which says what kind of geometry a spacetime with a gravitation field should carry. After that we review some exact solutions of Einstein’s equation, like the Schwarzschild and Kerr metrics, in which the concept of a black hole appears naturally. We explain some strategies to numerically solve the geodesic equations in order to draw trajectories of particles around a black hole. We finish by explaining the backward ray tracing method that we used to draw black hole shadows.

We first introduce the SEIR (Susceptible, Exposed, Infected, Recovered) model. We then present a generalization of these model into metapopulations. The population is first subdivided into age classes and then into social classes. We then study numerically the impact of different containment strategies according to these classes on the spread of the epidemic.

Dengue is the most common mosquito-borne viral infection transmitted disease. It is due to the four types of viruses (DENV-1, DENV-2, DENV-3, DENV-4), which transmit through the bite of infected Aedes aegypti and Aedes albopictus female mosquitoes during the daytime. The first globally commercialized vaccine is Dengvaxia, also known as the CYD-TDV vaccine, manufactured by Sanofi Pasteur. I will present a Ross-type epidemic model to describe the vaccine interaction between humans and mosquitoes, accounting for the life cycle. We present different control strategies : vaccination, pesticide, and copepods. We use Pontryagin’s minimum principle to characterize optimal control and apply numerical simulations to determine which strategies best suit each compartment. Results show that vector control requires shorter time applications in minimizing mosquito populations. Whereas vaccinating the primary susceptible human population requires a shorter time compared to the secondary susceptible human.

We generalize the Reduction Theorem of Kessar-Stancu so it can be applicable to exotic fusion systems over the maximal nilpotency class of rank two 3-groups. This is an essential step towards proving that these exotic fusion systems are also block exotic.

In this talk we will present some recent results about the normalizer group of topological dynamical systems, which is a group extension of the automorphism group. We will concentrate in substitutive symbolic systems generated by constant-shape substitutions. These are a multidimensional analogue of the so-called constant-length substitutions. Subtitutions have been extensively studied in the past years (such as criteria of ergodicity, entropy, mixing and spectral properties).

The main goal of this talk is to present a friendly introduction to regular braids. We will first need some reminders on complex reflection groups and braid groups. Then we describe the (classical) theory of regular elements in complex reflection groups, most notably the behavior of regular elements towards conjugation and centralizers. Regular braids then appear as a natural lift in the braid group of regular elements in the reflection group. This will lead to a remarkable theorem explaining that regular braids behave in the same nice way than regular elements. If time permits, we will finish by presenting some consequences of this on central elements in the braid group.

Using derived equivalences we can compare ordered sets, and more generally quivers with relations, in non trivial ways. In this talk we will see how this abstract concept can be manipulated in a very concrete and combinatorial way on the lattice of order ideals of grids, which are a familly of ordered sets related to Dyck paths, having interesting properties.

We will introduce the notion of confined extensions from an example of a compact extension on the torus. We will show one use of that notion through some related lifting results. Then we will see some of the constraints on the strucure of confined extensions in relation with the notions of super-innovations and standard extensions. Finally, we will present one more object: dynamical filtrations, and the related standardness problem.

The study of the statistical and dynamical properties of infinite sequences of symbols is a classic topic in mathematics. In this context, the class of "sublinear complexity sequences" is of particular relevance as it occurs in a variety of areas, such as geometric dynamical systems, informatics, number theory, and numeration systems, among others, and thus it is important to have tools to study it. Inspired by the great success of Kakutani-Rohlin partitions in the study of dynamics on the Cantor set, Bernard Host asked in the '90s if the symbolic analog of these partitions, the so-called S-adic sequences, could be used to describe sublinear complexity sequences. This problem was named the "S-adic conjecture" and inspired several influential results in symbolic dynamics. In this talk, I will present an S-adic characterization of sublinear complexity sequences and some of its applications, which in particular give a solution to this conjecture.

Finitely generated groups can be studied from a geometrical point of view via their Cayley graphs. In this introductory talk, we will see examples of different notions of regularity (and irregularity) properties of these discrete metric spaces, and discuss results relating them with the algebraic nature of the groups defining them. We will discuss results about the language of geodesic words, the asymptotic behavior and rationality of growth, and the problem of extending geodesic paths to larger ones.

Algebraic topology is a field which uses algebraic tools in order to classify topological spaces up to continuous deformation. We will begin this talk by defining one such tool : homology theory. Various examples will be presented, in particular, we will focus on the homology of the loops of a topological space X. Chas and Sullivan showed that when X has nice properties, this homology has interesting loop products known as Batalin-Vilkovisky (BV) structures. An algebraic interpretation is given by Hochschild cohomology. If time permits, I will explain my thesis work which is about extending these stuctures to spaces with singularities.

Using derived equivalences we can compare ordered sets, and more generally quivers with relations, in non trivial ways. In this talk we will see how this abstract concept can be manipulated in a very concrete and combinatorial way on the lattice of order ideals of grids, which are a familly of ordered sets related to Dyck paths, having interesting properties.

Talking during the phd seminar is the opportunity to train me for my thesis’defense, which is about thermal rotating shallow water equations : theoritical aspects and numerical schemes. I will mainly focus on the numerical part of my work, i.e fully well-balanced schemes. As I will speak french on D-Day, I will also do this training in french. I apologize to non-french speakers.