Séminaires doctorant en 2023

Resolving subcategories are a kind of subcategories widely used in representation theory. They offer strong results in the context of exact, triangulated and even extriangulated categories. The major drawback is that, given a category of these previous types, it is difficult to find all its resolving subcategories. It is not even easy to find non trivial examples of these subcategories. The interest of the gentle algebras is that they have a combinatorial description of their module category, which is exact. A small modification of the model gives a combinatorial description of the derived category which is triangulated and a last one 2-term silting complexes whose category is extriangulated. This talk will be targeted on the module category over a gentle algebra and the combinatorics associated which uses basic comprehension about smooth orientable surfaces and curves on these surfaces.

Tilings of the plane or infinite grid have been a rich field of study for many years. Through tiling with local rules, it is even possible to create aperiodicity and embed computation in the plane. But what happens when we begin changing the underlying structure? We will explore how different underlying groups and their geometries influence what we can obtain through the use of local rules and take a look at the state of the art of the problems of emptiness and aperiodicity.

We are interested in the following system :

where is the initial data and is the control. Our goal is to prove the null controllability of the previous system : we want to have a solution that vanishes in finite time :

Here we suppose that the coefficient may vanish strongly at the point . To do this, we don’t use some Carleman estimates, but the flatness approach which gives an explicit solution. We first investigate the associated elliptic problem to study its spectral elements and thus the generating functions in order to express the solution and the control in terms of these spectral elements, these generating functions and the initial data.

In this talk we will discuss about an example of modelisation for coagulation in head and neck cancers. For this kind of model, the easiest way to obtain our differential system is to use the law of mass action on the more appropriate system of enzymatic reactions. Then we will study our model from a theoretical point of view also with some numerical simulations. Finally, we study our model using coagulation-fragmentation equations to determine the possible apparition of a blood clot.

Inspired by the the application of level set method in the medical field particularly in human health, a new approach in plant health is developed using the same method. Along with its numerical schemes, the method is used to analyze the Spatiotemporal dynamics of host-pathogen interactions when the pathogen causes growing lesions on host tissues. In this talk, some initial results will be discussed as well as the biological context of the study.

Persistent homology is a computational tool which was created in the end of the 20th century for applied algebraic topology. The main idea is to understand the topological structure of a starting object by progressive approximations: for that we use simplicial theory and more precisely simplicial complex and homology, which we will begin by remind the basics. In practice, we extract from our starting object a point cloud and we change it into a filtered simplicial complex by using a method called the Vietoris-Rips filtration. Persistent homology then encodes the evolution of homology classes and more precisely their lifespan during the new created filtration: we will represent all this information on a family of graphs called barcodes, from which we will be able to analyze or even compare further starting objects. We called this process Topological Data Analysis. As an illustration, we will see how to apply this process to classification of musical style.

Through examples, I will start by talking about the main actors to understand the dynamics of a holomorphic map on the Riemann sphere. I will introduce a particular type of rational maps which are the hyperbolic ones. Those have been studied a lot since the beginning of the 20th century and one has a good understanding of their dynamics. Their importance is due to their potential density in the set of rational maps of a certain degree. This question is still open even in the case of the polynomial family . I will talk about some results that have been done to give us hope to answer positively to that question. Finally, I will present a result from my thesis work which gives a class of rational maps that can be approached by hyperbolic ones.

Systems of differential equations are central to the modeling of various phenomena, whether in biology, chemistry or physics. It is therefore crucial to find solutions to such systems. Their resolution leads to a formulation of the solution in the form of a continuous fraction, which we naturally find in graph theory. In this talk, we are interested in the link between graph theory and the solution of such systems. This link leads us to define a product, which we will call ★-product, between distributions, we show that it is well defined and study the algebraic structures that it allows to define.

In this talk, I study the non-linear Poisson's equation . The aim is to determine some qualitative and geometric properties of the solution such as the monotocity of the solution. To achieve this, I will need a recent method called "moving plane" introduce by J.Serrin, this method is based on the maximum principle. So after a quick reminder of the maximum principle, I will apply the moving plane in two cases: the ball and the half space. Finally, I will present other results with other domains like an epigraph.

Brauer algebras were introduced by Richard Brauer in 1937 as the dual object to orthogonal and symplectic groups in the context of Schur-Weyl duality. This original form of Brauer algebras was a natural extension of the algebra of the symmetric group. It took until 1988 for their structure to be completely described by Wenzl. Since then, many efforts have been made to define corresponding algebras for other types of Coxeter groups but also for complex reflection groups. In 2011, Chen gave a uniform definition of a Brauer algebra associated to every complex reflection group, encompassing many of the already existing algebras. We will review, in this talk, the background that led to this general Brauer-Chen algebra and discuss some results concerning its structure.

In fluid mechanics, the Boussinesq system is an approximation of the Euler equations for incompressible irrotational free surface flows. Many authors studied this system. One of them proved the existence and uniqueness of the solution to the regularized Boussineq system. We will prove theoretically the existence of an asymptotic expansion of this regularized solution with respect to the regularized parameter. Then, we will verify numerically this existence up to order 2. Lastly, we will explain the Littlewood-Paley theory and prove some basic estimates with it.

This talk will be an introduction to the dynamics on the moduli spaces of translation surfaces. As a motivation, we will first consider a simple dynamical system -- a billiard in a rational polygon. We will see how its study corresponds to the study of the geodesic flow on some singular flat surface of a special type, called translation surface. Then we will explain how dynamical properties of one individual translation surface can be deduced from the properties of another dynamical system -- this time on the set of all translation surfaces, called the moduli space. Some classic results (dating from the 80's) as well as some recent advances will be presented, hopefully with sketches of (some) proofs.

Meanders are mathematical objects that can be defined both geometrically and combinatorially. Combinatorially, they are pair of noncrossing partitions of {1,...,n} made of blocks of size at most two. Even though they are relatively simple objects, various problems about them are still open. In this talk, after defining them and giving motivations, I will enumerate some of those open problems. Then we will focus on a subfamily of meanders called bi-rainbow meanders, which have more structure, and for which we can develop tools to help us in answering some questions. If time allows, I will discuss a possible generalization of meanders by replacing the line by a tree. This talk is in part based on a joint paper with Mélodie Lapointe, Yann Palu, Pierre-Guy Plamondon, Christophe Reutenauer and Hugh Thomas which can be found in arXiv:2301.07222, and in part based on a separate individual project.

Models of biological interactions are in full swing. They come in different flavors, one of them being systems of partial differential equations, which are then studied by mathematicians. Here, we will be interested in the establishment of stationary solutions to problems of the form , with a computer-assisted method. Starting from a known approximate solution, we get back to study a fixed point problem to solve our initial problem. This fixed point then becomes our existing and unique theoretical solution in a neighbourhood of our approximate solution. The difficulty lies in the choices we make when reducing the problem. In particular, the non-linearity of the equations is a significant obstacle. More precisely we will look at a chemotaxis model where , . In this type of model the search for patterns and the study of their stability is interesting, which justifies our numerical to theoretical approach. We will look at different results according to the chosen function : rational fraction, decreasing exponential, power series, ...

This talk presents my thesis' main results. It focused on the study of minimal subshifts via -adic sequences. First, we investigate automorphisms and factors of minimal subshifts generated by -adic sequences with alphabets of bounded cardinality. As a result, we prove that these subshifts have virtually automorphism groups, finitely many infinite symbolic factors (up to conjugacy), and we give a fine description of symbolic factor maps. In the second part, we consider the -adic conjecture, an old problem asking for a structure theorem for linear-growth complexity subshifts. We completely solve this problem by proving an -adic characterization of this class of subshifts. Our methods extend to nonsuperlinear-growth subshifts. We show how this provides a unified framework and simplified proofs of several known results, including Cassaigne's Theorem.

Il s'agit d'une répétition pour ma soutenance. On y parle de dynamique holomorphe sur la sphère de Riemann, de certains types d'expansivité de fractions rationnelles et de sélection de paramètres. Après une brève introduction et une remise en contexte de mes travaux, j'essaierai de vous expliquer quelques bouts de preuve.

A metric space is said to be if every non-expansive bijection is in fact an isometry ("non-expansive'' means "-Lipschitz''). In this talk, we aim to explore the plasticity of the unit ball of some spaces, motivated by the fact that the unit ball of the space of all convergent sequences of real numbers is plastic. Specifically, we will investigate two cases: when is a compact metrizable space with a finite number of accumulation points, and more generally, when is a zero-dimensional compact Hausdorff space with a dense set of isolated points.

As a model of red blood cell, we study equations describing the motion and deformation of a non-spherical microcapsule with an incompressible interface in steady shear flow. The unstressed shape is assumed to coincide with a slightly ellipsoidal shape, for which the revolution axis is parallel to the flow vorticity. Firstly, we find that the equations can be mapped onto those describing the time evolution of the vector orientation of a (rigid) spherical microswimmer in fictitious external and shear fields, for which the fluid vorticity direction and the external field, which varies with time, are secant. An exact analytical solution is found showing, as it is well known, that the microcapsule never tumbles and always attains a stationary tank-treading shape in an off shear plane for which an exact analytical expression is derived.

The purpose of this talk is to use circular groups as a pretext to investigate the following question: how to check wether or not two (infinite) groups are isomorphic ? Circular groups are defined by a group presentation depending on two integer parameters, they are always infinite, and even without torsion. Furthermore, we can explicitely describe their elements, along with the centralizers of arbitrary elements. From these descriptions, we can compute several group theoretic invariants of circular groups: abelianization, center, integral homology, periodic elements. From these invariants, we can then deduce the complete classification of circular groups up to isomorphy.

Yield-stress fluids are a type of non-Newtonian fluids : under a certain constraint they have a rigid behaviour, and above it they flow like a classical fluid. We can find them for instance in industry (concrete), geophysical phenomenas (dense avalanches, mud flow,...) and even eat them (mayonnaise,ketchup). As a consequence, this field has known many developments in the last decades. In this talk we will present simple examples of such flows, the theoretical framework behind these problems and some numerical methods to solve them.

Introduced in 1945, category theory become indispensable in various mathematical fields. Working with categories is adopting a universal view on objects: they are no longer characterized by a property or their internal structure, but by their position among others. This abstract viewpoint finds applications in other scientific domains such as computer science and even raises philosophical questions. The purpose of this presentation is to provide an intuitive introduction, with examples, to the categorical language (categories, functors, natural transformations...) and some other categorical constructions (limits, colimits). The main objective is to familiarize novices with categories and their manipulation.

In this talk, I will speak about the Non-linear Poisson equation in unbounded domain. In the literature, there exists few results about this problem. We can quote monotocity results for the case of the half space or for a coercive epigraph. The proof of these results is based on the "moving plane" method introduce by J.Serrin. I will present this method with the example of the half space and consequences of this result on the classification of the solution. Finally, by taking the same ideas, we can show the monotocity of the solution in the case of an epigraph. However, here we must take into account the geometry of the domain which is more complex.

Cluster algebras are a powerful tool from algebraic combinatorics. It appeared firstly in physics before being axiomatized. In this talk I will give a short introduction to the combinatorics of cluster algebras and to some of their incarnations in different topics such as geometry and representation theory.

The (strong) Bruhat order first appeared in a geometric context by describing the containment ordering of Schubert varieties in flag manifolds. It is instrumental in many aspects of representation theory, such as our understanding of Verma modules. Recent work shows that the cardinality of the intervals associated to this partial order, play a crucial role in the computations of indecomposable Soergel bimodules and Kazhdan-Lusztig polynomials. In this talk we will discuss finite real reflection groups and their combinatorics. Then I will briefly show some results regarding the cardinality of Bruhat intervals on affine Weyl groups.

In 2007, Achar & Aubert introduced a family of groups called J-groups generalizing rank two complex reflection groups. In this talk, I will introduce the notions of complex reflection groups and J-groups and explain the link between these two notions via a Theorem of Achar & Aubert. Moreover, I will expose the existing results on classification of J-groups and present a generalization of these results.

Soit Q un carquois (= graphe orienté) à n sommets, R un ensemble de relations et K un corps algébriquement clos. Notons I l'idéal engendré par R dans KQ. Fixons X un KQ/I-module. Rappelons qu’un KQ/I-module peut être vu (via une équivalence de catégorie) comme une substitution de chaque sommet par un K-espace vectoriel, et une substitution de chaque flèche par une transformation linéaire, de façon à ce que les compositions des transformations linéaires respectent les relations imposées par R. Avec cette traduction, un endomorphisme de X peut se voir comme une collection d’applications linéaires, un pour chaque sommet. Pour cet exposé, nous allons nous intéresser aux endomorphismes nilpotents de X. Dans l’ensemble de tels endomorphismes, il existe un ensemble ouvert dense O tel que, si M et N appartiennent à O, alors pour chaque sommet q de Q, les formes de Jordan de Nq et Mq sont les mêmes. Ainsi, nous appelons la forme générique de Jordan de X la collection de partages encodant les formes de Jordan de Nq obtenues pour N dans O. Une sous-catégorie C de KQ/I-modules est dite retrouvable de Jordan si nous pouvons retrouver X, à isomorphisme près, dans C, connaissant sa forme générique de Jordan. Le but de cet exposé, qui est avant tout une initiation à la théorie de la représentation de carquois, est d’introduire cette notion plus en détails. Après avoir mis en lumière les difficultés que nous pouvons rencontrer pour montrer qu’une catégorie est retrouvable de Jordan, je présenterai un raffinement de cette notion, appelée retrouvabilité de Jordan canonique. J'amenerai mon exposé jusqu'à vous exprimer les résultats importants que j'ai pu établir tout au long de ma théèse. Si le temps me le permet, je discuterai des pistes de recherches que j'ai pour aller plus loin. Il s’agit d'une répétition de ma soutenance de thèse de doctorat, fruit de mon travail effectué sous l'encadrement de Hugh Thomas.

We introduce a mathematical model based on mixture theory intended to describe the tumor-immune system interactions within the tumor microenvironment. The equations account for the geometry of the tumor expansion, and the displacement of the immune cells, driven by diffusion and chemotactic mechanisms. They also take into account the constraints in terms of nutrient and oxygen supply. The numerical investigations analyze the impact of the different modeling assumptions and parameters. Depending on the parameters, the model can reproduce elimination, equilibrium or escape phases and it identifies a critical role of oxygen/nutrient supply in shaping the tumor growth. In addition, antitumor immune cells are key factors in controlling tumor growth, maintaining an equilibrium while protumor cells favor escape and tumor expansion.

In this talk, we will introduce the variational approach to the study of equations. To get started, we will review the spectral theory of symmetric real matrices and the min-max principle for eigenvalues. We will then move on to the spectral theory of the Laplacian on bounded open sets of ℝⁿ with Dirichlet boundary conditions and use the min-max characterisation of eigenvalues to deduce Courant's nodal domain Theorem. We will see that a rich and somewhat surprising behaviour may be observed when one replaces smooth domains of ℝⁿ by unidimensional domains called metric graphs, showing the role of unique continuation principles in the classical theory. In the last part of the talk, we will see how variational methods may also be applied to study solutions of some nonlinear elliptic PDEs, using suitable constrained minimisation problems.

The intersection algebra of a smooth, compact and oriented manifold corresponds to the singular cochain complex endowed with the cup product. Poincaré duality implies the existence of algebraic structures (Gerstenhaber, Batalin-Vilkovisky) on the Hochschild cohomology of this algebra. The aim of this thesis is to study the case of spaces with singularities. In general, for these spaces, Poincaré duality is not satisfied. In order to restore it, Goresky and MacPherson have introduced the intersection complexes. In this thesis, we show that the Hochschild cohomology of the blown-up intersection cochain complex (defined by Chataur, Saralegui and Tanré) of a pseudomanifold (a certain type of space with singularities) with coefficients in , can be endowed with a Batalin-Vilkovisky algebra. More generally, we show that the Hochschild cohomology of a perverse differential graded algebra is well defined and that it is a Gerstenhaber algebra. Furthermore, when the algebra verifies some form of duality, we also get a Batalin-Vilkovisky algebra structure. This talk is a rehearsal for my PhD defense.

Dans cette présentation, nous aborderons deux grandes parties : Première partie : nous nous sommes penchés sur la dynamique et le transport des micro nageurs sphériques (rigides) soumises à divers écoulements tels que le Poiseuille, le cisaillement et le turbulent. Les équations régissant cette dynamique sont non linéaires, rendant les résultats analytiques rares. Nous avons introduit une méthode générale basée sur l'approche de Bretherton pour obtenir une expression exacte du vecteur d'orientation de la particule. Cela nous a permis d'identifier différents régimes tels que le "Run", où l'angle d'orientation reste constant dans le temps, et le "Tumbling", où l'angle d'orientation évolue de manière cyclique. Cette étude a révélé diverses trajectoires, notamment paraboliques, elliptiques et hélicoïdales. Deuxième partie : notre attention s'est portée sur la dynamique et la déformation d'une micro capsule dont la membrane est considérée comme une surface incompressible. Ce modèle, proposé et étudié par Vlahovska et al., représente une particule déformable initialement ellipsoïdale à paroi mince. La particule est exposée à un écoulement de cisaillement défini par la vitesse , où est le taux de cisaillement. Nous avons identifié deux types de mouvements distincts : le mouvement stable "Tank-treading" où l'angle d'orientation oscille autour d'une valeur constante, pouvant être crucial dans des applications telles que la conception de micro robots capables de maintenir une trajectoire stable dans des environnements complexes et changeants. Le mouvement oscillatoire "Vacillating breathing", caractérisé par des oscillations périodiques de l'angle d’orientation, démontre l'exceptionnelle capacité d'adaptation de la micro capsule aux variations des forces environnementales.

Dans cette présentation, nous aborderons deux grandes parties : Première partie : nous nous sommes penchés sur la dynamique et le transport des micro nageurs sphériques (rigides) soumises à divers écoulements tels que le Poiseuille, le cisaillement et le turbulent. Les équations régissant cette dynamique sont non linéaires, rendant les résultats analytiques rares. Nous avons introduit une méthode générale basée sur l'approche de Bretherton pour obtenir une expression exacte du vecteur d'orientation de la particule. Cela nous a permis d'identifier différents régimes tels que le "Run", où l'angle d'orientation reste constant dans le temps, et le "Tumbling", où l'angle d'orientation évolue de manière cyclique. Cette étude a révélé diverses trajectoires, notamment paraboliques, elliptiques et hélicoïdales. Deuxième partie : notre attention s'est portée sur la dynamique et la déformation d'une micro capsule dont la membrane est considérée comme une surface incompressible. Ce modèle, proposé et étudié par Vlahovska et al., représente une particule déformable initialement ellipsoïdale à paroi mince. La particule est exposée à un écoulement de cisaillement défini par la vitesse , où est le taux de cisaillement. Nous avons identifié deux types de mouvements distincts : le mouvement stable "Tank-treading" où l'angle d'orientation oscille autour d'une valeur constante, pouvant être crucial dans des applications telles que la conception de micro robots capables de maintenir une trajectoire stable dans des environnements complexes et changeants. Le mouvement oscillatoire "Vacillating breathing", caractérisé par des oscillations périodiques de l'angle d’orientation, démontre l'exceptionnelle capacité d'adaptation de la micro capsule aux variations des forces environnementales.

Cette présentation est une introduction à la typographie. J'y présente les principes généraux, illustrés par diverses polices de caractères. Dans une seconde partie j'aborde la typographie numérique et ses dernières fonctionnalités, et je conclus par un passage sur la typographie mathématique.