Séminaires doctorant en 2025

Tout est dans le titre.

In this talk, we will discuss coagulation and fragmentation phenomena through a mathematical model, obtained using the law of mass action, consisting of an infinite ordinary differential system of equations. Close to the well-known Becker-Döring model, ours introduces production and irreversible fragmentation, which entails among other things, the loss of mass preservation. We will first study our system of equations theoretically; we will establish well-posedness under some suitable conditions, and then we will focus on the long-time behaviour of our solution. Then, we will present a numerical scheme and some numerical simulations. We will compare our numerical scheme to well known scheme; qualitatively, quantitatively and in computation time.

The nonlinearity of free surface fluid mathematical models presents a significant challenge in predicting the effects of ground topography on the evolution of a wave at the surface. The use of IA should be an alternative. However, AI learning requires a huge amount of synthesized data. My thesis and internship aim to address these two needs: cheap simulation and cheap analysis so as to generate these data in a reasonable amount of time. It starts by simplifying the general equations, which are too computationally expensive, through asymptotic derivations and proposing numerical schemes of theses simpler equations. Then, i'll explain the symbol approximation method for interpreting the evolution of the wave in the frequency domain.

How do computers compute?
We will first consider this question and will discover floating-point numbers, the most common way to perform computations involving continuous quantities with a computer. We will see that those computations necessarily lead to numerical errors, which we will illustrate by several examples, sometimes spectacular.
A second question then arises: how to obtain rigorous mathematical results based on numerical computations performed on a computer?
We will introduce interval arithmetic, based on the simple idea to replace real numbers by couples of real numbers corresponding to a lower bound and an upper bound of the quantity of interest. We will mention a few famous examples of problems whose solution used interval arithmetic.
Finally, we will present briefly the application of such methods to the analysis of symmetries of solutions for a nonlinear Schrödinger equation set on the "tetrahedron graph", in an "almost linear" regime. We will see that the computer will be very useful in the analysis of the problem. This last part comes from a collaboration with my PhD supervisors, Prof. Colette De Coster (CERAMATHS/DMATHS, Valenciennes, France) and Prof. Christophe Troestler (UMONS, Mons, Belgium).

Le théorème de Perron-Frobenius est un résultat important sur les matrices positives avec de nombreuses applications en particulier en dynamique symbolique pour décrire les mesures invariantes. Nous exposerons dans un premier temps son rôle dans le cas des chaines de Markov et des shifts substitutifs puis nous exposerons une généralisation du théorème dans le cadre de la théorie spectrale pour l'étude ergodique des shifts engendrés par des substitutions généralisées.

In this talk, I will be interested in the space charges problems associated with the transmission of electric current on High Voltage Direct Current (HVDC) lines. I will start by explaining the chosen non-linear physical model that couples electric potential and space charge densities. In an annulus, the explicit expression of all radial solutions will be given. In the general case, an existence result will be discussed. Since the solutions are of course no longer explicit, I will propose a numerical algorithm and associated simulations. It will be validated by comparing, in the radial case, the exact solution with the numerical one. I will also present some numerical result for a half-disk with a single radial cable.

In a previous joint work with Castillo, Libedinsky and Plaza, we established a counting formula for the cardinalities of Bruhat intervals (in affine Weyl groups) in terms of the volume of the permutohedron. Motivated by this I obtained a polynomial formula of the volume of this polytope (in type A) in terms of Dyck paths, which will be the focus of this talk.

In my previous talk, I stated news monotonicity results for solution of the nonlinear Poisson's equation in epigraph. In this talk, I will present somes consequences of these theorems. After a recall of my results, I will establish symmetry results which holds true even if the solution is not increasing. Finally, I will present classification results for specific nonlinearities .

In this work, we derive a periodic model from a one dimensional nonlocal eikonal equation set on the full space modeling dislocation dynamics. Thanks to a gradient entropy estimate, we show that this periodic model converges toward the initial one when the period goes to infinity. Moreover, we design a semi-explicit numerical scheme for the periodic model that we introduce. We show the well-posedness of the scheme and a discrete gradient entropy inequality. We also prove the convergence of the scheme and we present some numerical experiments.

The extrusion process plays a crucial role in both mixing and shaping materials during battery manufacturing. However, identifying the optimal formulation and process parameters requires extensive and time-consuming trial-and-error experimentation. Due to the complex and enclosed nature of extruders, in-situ observation of material behavior is not feasible. Making post-process analysis of the microstructure has become the primary evaluation method. In this seminar, I will present our efforts to build a reliable simulation framework that captures the complex material–process–microstructure relationship in extrusion, with the goal of providing insights into the process within a reasonable computational time. Specifically, I will introduce: 1-The background of our experimental work: a solvent-free extrusion process for producing filaments used in 3D-printed battery electrodes. 2-The modeling approach: the Discrete Element Method (DEM), a particle-based numerical technique suited for simulating granular material behavior. 3-The design of our model and key findings (If you are interested, here is the link of our preprint paper:https://chemrxiv.org/engage/chemrxiv/article-details/67b6e90afa469535b920d329). 4-The limitations of our current work and our future plan to integrate AI models for optimization. This work aims to bridge the gap between experimental constraints and predictive simulation tools, enabling more efficient development of advanced battery materials.

In this talk, we will explore different topics from symbolic dynamics from the original idea that gave life to the area, to the state of the art of the study of SFTs on groups. We will begin by exploring two parallel lines: one-dimensional symbolic dynamics and tilings of the plane. We will then see how these two lines converge to create the modern study of SFTs on groups, particularly the existence of aperiodic tilings and the Domino Problem.

The two-headed snake is a standard example of a non-Hausdorff groupoid. We study the Steinberg algebra, a convolution algebra of linear combinations of continuous functions from the groupoid to a field K, for the two- and three-headed snake groupoids. We are interested in elements of this algebra that are no longer continuous, known as singular functions. The set of singular functions forms an ideal of the Steinberg algebra. We are interested in whether the characteristic of the field K impacts the structure of this ideal. In the two-headed snake, has no proper subideals regardless of the choice of K, but in the three-headed snake, field characteristic plays a major role.

L'histoire des groupes quantiques est pavée d'allers-retours entre la physique et les mathématiques. C'est au milieu des années 80' que ceux-ci, sous l'impulsion des travaux de Drinfeld et Jimbo, sont formalisés algébriquement et ouvrent la voie à nombre de résultats et d'applications qui continuent de se développer aujourd'hui : invariants de noeuds, topologie en basse dimension ou théorie des représentations des groupes algébriques en caractéristique non nulle.
Dans cet exposé, je donnerai le contexte historique dans lequel les groupes quantiques sont apparus, en discutant notamment de leurs inspirations physiques. Nous verrons ensuite de quelle manière on peut construire des groupes quantiques en déformant l'algèbre enveloppante d'une algèbre de Lie, en détaillant explicitement la construction de la déformation quantique de l'algèbre de lie . Si le temps le permet, nous discuterons rapidement de la manière dont on se sert des groupes quantiques (quasi-triangulaires) pour construire des invariants de noeuds.

Locally stationary processes (LSPs) provide a robust framework for modeling time-varying phenomena, allowing for smooth variations in statistical properties such as mean and variance over time. In this paper, we address the estimation of the conditional probability distribution of LSPs using Nadaraya-Watson (NW) type estimators. The NW estimator approximates the conditional distribution of a target variable given covariates through kernel moothing techniques. We establish the convergence rate of the NW conditional probability estimator for LSPs in the univariate setting under the Wasserstein distance and extend this analysis to the multivariate case using the sliced Wasserstein distance. Theoretical results are supported by numerical experiments on synthetic datasets, demonstrating the practical usefulness of the proposed estimators.

Despite the availability of effective treatments, Tuberculosis (TB) remains a significant global health concern, affecting millions of people each year. Individuals with Diabetes Mellitus (DM) are particularly vulnerable, as their compromised immune systems increase the likelihood of progressing from latent, non-infectious TB to active, contagious TB. With the global prevalence of diabetes on the rise, DM has emerged as a major contributing factor to TB incidence—tripling the risk of developing active TB. This study explores the interplay between TB and DM through a mathematical model designed to capture the influence of diabetes on TB transmission dynamics. The model also provides a framework for evaluating the potential impact of various intervention strategies, such as TB chemoprophylaxis and improved glycemic control. In this talk key concepts necessary to understand the model and the epidemic threshold known as the basic reproduction number and stability of equilibria will be introduced and discussed.

Nous présentons ici un problème inverse ayant pour objectif la reconstruction des propriétés électriques (permittivité et conductivité) des tissus biologiques à partir de données d'imagerie par résonance magnétique (IRM). Après avoir défini le problème direct correspondant, nous utiliserons une méthode de contraste de source (CSI) reformulant le problème de reconstruction de paramètres en un problème de reconstruction de source par la minimisation d'une fonctionnelle de coût appropriée.

Adaptive mesh refinement is used to perform precise flux and current computations on relatively small mesh. In the case of conforming Cartesian mesh, the refinement cannot be optimal. Indeed, one can not refine only a single cell in a conformal Cartesian mesh, as the latter would no longer be conformal. In this work, we investigate the adaptive mesh refinement applied to a domain decomposition (DD) method, where non-conformities can arise at the interface between subdomains. Since the refinement is more optimal, the number of mesh elements required to reach a given precision is lower in the case of the DD method than in the monodomain method.