PhD students

Wednesday 20/05 – MSA 2.240 :

• • 12:30 – 14:00 : Lunch + Poster Session

• • 14:00 – 14:45 : Javier Fernandez Piriz – University of Luxembourg
    
    Grassmannians and representations of Lie groups

    Grassmannians are objects endowed with rich geometrical structures that have been studied in algebraic geometry since the 19th century. A useful way to understand these spaces is through the seemingly unrelated theory of representations of Lie groups. The goal of this talk is to present a brief overview of the interplay between these fields and to motivate how computers are useful in answering many related questions.

• • 14:45 – 15:30 : Rodolphe Abou Assali – IECL
    
    Steklov problems and spectral inequalities in planar domains

    Classical spectral problems, such as the Dirichlet and Neumann problems, focus on the analysis of eigenvalues and eigenfunctions with applications to heat conduction, sound propagation, and vibrational modes in domains with boundaries. Other well-known problems are the Steklov and biharmonic Steklov problems with various boundary conditions. Kuttler and Sigillito established fundamental inequalities relating the eigenvalues of these problems in planar domains. These results were later extended to the scalar case on Riemannian manifolds by Hassannezhad and Siffert. We recently generalized these inequalities to the setting of differential forms. In this talk, we present these spectral problems and the Kuttler-Sigillito inequalities in planar domains, and briefly discuss their generalization.

• • 15:30 – 16:00 : Break

• • 16:00 – 16:45 : Quirijn Boeren – University of Luxembourg
    
    Cusps in the AdS/CFT correspondence

    The AdS/CFT correspondence is a powerful tool in theoretical physics, relating string theories on hyperbolic (Anti-de Sitter) manifolds to a conformal field theory on a boundary manifold. It provides some of the most promising models of quantum gravity. As often in theoretical physics the theory struggles with divergences. I will walk you through one such divergence, caused by a construction from hyperbolic geometry: a manifold with cusp—a puncture at infinite distance—can generate infinite summands to the relation, producing a divergence.

• • 16:45 – 17:30 : Valentin Clarisse – IECL
    
    General relativity and Gregory-Laflamme instability

    The Einstein equations are central to general relativity. They relate the geometry of spacetime to the distribution of matter within it. As we will see later, they form a particularly challenging system of partial differential equations to study. The first major breakthrough in mathematical relativity was achieved by Y. Choquet-Bruhat, who proved in 1952 the local-in-time existence of solutions to the Einstein equations viewed as an evolution problem. More recently, in 1993 and 1994, R. Gregory and R. Laflamme numerically demonstrated the instability of certain types of black string extensions in dimensions greater than or equal to $5$. In 2012, R.M. Wald and S. Hollands developed a fairly general method and criterion for studying the linear stability of black holes, which can be applied to establish Gregory–Laflamme-type instabilities. The article we will focus on, which is more accessible, comes from the doctoral thesis of Sam C. Collingbourne. It was submitted in 2020 and is entitled The Gregory-Laflamme Instability of the Schwarzschild Black String Exterior. It provides a direct mathematical proof of the Gregory–Laflamme linear instability in dimension $5$.

Thursday 21/05 – MSA 2.240 :

• • 9:00 – 9:45 : Katarzyna Szczerba – University of Luxembourg
    
    AI-informed Non-linear Cox Regression for Time-to-event Analysis

    The Cox proportional hazards model is the most commonly used method for multivariate survival analysis. Despite its many advantages, such as simplicity and interpretability, it has a serious drawback: it fails to capture non-linear relationships. In this study, we propose AI-informed Non-linear Cox Model, a method that uses insights from a highly predictive machine learning model, extracted with an interpretable machine learning tool, to integrate non-linear relationships into the traditional Cox model via means of splines. On simulated data with a deliberately introduced non-monotonic relationship between the predictor and the outcome variable, the AI-informed Cox model outperformed the traditional proportional hazards (PH) Cox model. Its concordance index (C-index) was also comparable to that of the best-performing machine learning model – gradient boosted Cox model. Similar results were observed when the models were applied to a prospective dataset in running.

• • 9:45 – 10:30 : Yingtong Hou – IECL
    
    Butcher series: from ordinary differential equations to Rough Path Theory and Regularity Structures

    In this talk, I will give a gentle introduction to Butcher series (B-series), Rough Path Theory, Regularity Structures, and their underlying Hopf algebras. Rough Path Theory and Regularity Structures provide pathwise frameworks for solving rough differential equations (RDEs) and singular stochastic partial differential equations (SPDEs), respectively. We will see that all these pathwise solution ansatz are obtained from iterating Taylor expansions. Therefore, Rough Path Theory and Regularity Structures can be viewed as generalisations of B-series designed for solving ordinary differential equations (ODEs). I will present the derivation of B-series-type solution ansatz for ODEs, RDEs, and SPDEs. Rooted trees and Hopf algebras appear naturally in encoding the expansions of solution ansatz. No prior background knowledge in rough analysis is required. Familiarity with Taylor expansions will be sufficient.

• • 10:30 – 11:00 : Break

• • 11:00 – 11:45 : Luís Maia – University of Luxembourg
    
    Fractional Brownian Fields at H=0: Constructions and Limit Theorems

    Fractional Brownian motion and fractional Brownian fields become singular at the endpoint H=0: the usual covariance degenerates. In this talk, I will explain two normalization that recover a meaningful object when $H=0$. The first, due to Neuman and Rosenbaum, treats one-dimensional fractional Brownian motion by subtracting a local average and rescaling. The second, due to Hager and Neuman, extends this idea to higher-dimensional fractional Brownian fields. In both cases, the normalized fields converge to log-correlated Gaussian distributions. I will then discuss results on Hermite functionals of these fields, both on fixed domains and on growing domains.

• • 11:45 – 12:30 : Juan Mardomingo-Sanz – IECL
    
    Slow-fast limits of stochastic particle systems arising in telomere biology

    The ends of linear chromosomes, called telomeres, shorten at each cell replication, eventually driving the cells to a senescent state when they become too short. The enzyme telomerase, present in cancerous cells and some unicellular organisms, elongates the telomeres and allows cells to continue replicating. Recent experiments show that if this enzyme is inactivated some rare survivors (ALT), which elongate their telomeres without telomerase, will appear and will eventually invade the cultures. I will present a simple stochastic particle system which accounts for the emergence and invasion of these ALT cells under an appropriate scaling with different speeds for each cell type.

• • 12:30 – 14:00 : Lunch

• • 14:00 – 14:45 : Szabolcs Buzogany – University of Luxembourg
    
    Galois and torsion-Kummer representations of elliptic curves

    The absolute Galois group $G_Q$ is the group of all isomorphisms from the field of all algebraic numbers to itself and remains a central object in contemporary number theory.
    A common way of studying $G_Q$ is to study its quotients, by the means of defining a group homomorphism between $G_Q$ and a well-studied group. Examples of these maps are Galois (respectively torsion-Kummer) representations, where the codomain is associated with n-torsion (respectively n-division points) of an elliptic curve. In this talk I will provide a gentle introduction to these representations.

• • 15:00 – 17:30 : Scavenger Hunt in the city (Luxembourg)

• • 19:00 : Social Dinner at Brasserie du Cercle (Luxembourg City)

Friday 22/05 – MSA 3.500 :

• • 9:00 – 9:45 : Gautier Schanzenbacher – IECL
    
    An Introduction to Hyperbolic Geometry: Surfaces, Geodesics, and Entropy

    For centuries, mathematicians tried to prove Euclid’s fifth axiom (the parallel postulate) using only the first four. In the 19th century, Gauss showed that replacing this axiom leads to a new, consistent geometry: non-Euclidean geometry. In particular, if we suppose that there are infinitely many lines parallel to a given line passing through a single point, we obtain Hyperbolic Geometry. In this talk, I will start from these foundations to define hyperbolic surfaces. We will then explore the world of curves, geodesics, and homotopy classes to understand the concept of entropy of the geodesic flow of a hyperbolic surface in the simplest way possible.

• • 9:45 – 10:30 : Francesco Tognetti – University of Luxembourg
    
    Who cares about coinduction?

    Everyone is familiar with the concept of (proof by) induction, though not as many are familiar with its dual. In this talk you will get an overview of what coinduction is, when it arises naturally and how it’s used throughout various areas of mathematics.

• • 10:30 – 11:00 : Break

• • 11:00 – 11:45 : Musbahu Idris – IECL
    
    Algorithmic Aspects of Newman Polynomials and Their Divisors

    A Newman polynomial is a polynomial with coefficients in ${0,1}$ and constant term $1$. We investigate which integer-coefficient polynomials divide a Newman polynomial, focusing on those with small Mahler measure. Using mixed-integer linear programming, we determine the divisibility status of all $8,438$ known polynomials with Mahler measure less than $1.3$. We further exhibit new polynomials that divide no Newman polynomial, improving the best known upper bound on a conjectural universal constant $\sigma$ to approximately $1.419$.

• • 11:45 – 12:30 : Francisco Pina – University of Luxembourg
    
    Statistics of Interacting Particle Systems

    Interacting particle systems can be seen as a system of N SDEs describing the evolution of a collection of agents whose behaviour depends not only on their own dynamics, but also on their interactions with the rest of the system. Such models arise in many different contexts, and a typical example is opinion dynamics, where the evolution of an individual’s opinion is influenced by the opinions of others.
    In this talk, we present the mathematical framework of interacting particle systems and discuss how statistical methods can be used to estimate the interaction law governing the system from observed particle trajectories. In particular, we introduce a nonparametric approach for estimating the underlying interaction function.

• • 12:30 – 14:00 : Lunch

This paper investigates the convergence time of log-linear learning to an $\epsilon$-efficient Nash equilibrium (NE) in potential games. In such games, an efficient NE is defined as the maximizer of the potential function. Previous literature provides asymptotic convergence rates to efficient Nash equilibria, and existing finite-time rates are limited to potential games with further assumptions such as the interchangeability of players. In this paper, we prove the first finite-time convergence to an $\epsilon$-efficient NE in general potential games. Our bounds depend polynomially on $1/\epsilon$, an improvement over previous bounds that are exponential in $1/\epsilon$ and only hold for subclasses of potential games. We then strengthen our convergence result in two directions: first, we show that a variant of log-linear learning that requires a factor $A$ less feedback on the utility per round enjoys a similar convergence time; second, we demonstrate the robustness of our convergence guarantee if log-linear learning is subject to small perturbations such as alterations in the learning rule or noise-corrupted utilities.

This work addresses the application of self-insurance in networks, where the network’s edges represent insured subjects facing losses. Each edge undertakes preventive efforts that influence the loss distribution, modeled as random variables. Insurance coverage is proportional, and a law-invariant coherent risk measure is considered to assess the network’s total risk. Furthermore, the work analyzes how preventive efforts impact the insurance cost and risk minimization. An optimization problem is proposed to determine the optimal levels of coverage and preventive effort, considering losses distributed according to a Pareto distribution. Through numerical techniques, specific cases, including global and local efforts, are studied to evaluate the model’s behavior in different scenarios.

• « Ô mon beau pendule ! » :

Si grand nombre d’entre vous ont déjà vu le Professeur Tournesol s’amuser avec son pendule, le pendule cycloïdal de Huygens vous sera sans doute étranger.

Le premier est simplement le système matériel composé d’un fil accroché à une extrémité et d’une masse accrochée à son autre extrémité. Le tout oscille sous l’effet du poids. On s’interrogera sur la période des oscillations du pendule i.e. le temps que met la bille pour faire un aller-retour. Dépend-t-il de la masse ? De la longueur du fil ? De l’angle initial auquel on lâche la masse ? À l’aide de théorèmes de la mécanique newtonienne, nous mettrons en équation le mouvement du pendule simple et répondrons théoriquement à ces questions.
Dans le cas de l’approximation des petits angles, l’équation de l’oscillateur harmonique fournit l’isochronisme des oscillations i.e. la période ne dépend pas de l’angle initial auquel on lâche la masse. On s’interroge : Quelle est donc exactement le temps pris par la masse pour effectuer une période ?

On s’interroge ensuite : Mais quelle courbe du plan (si elle existe !) devrait suivre la masse pour que le temps de parcours mis pour effectuer une période ne dépende pas de l’endroit où on la lâche ? Cette question occupât l’esprit de ceux qui, désireux de partir à la découverte du Monde, cherchaient un moyen de se repérer avec précision en mer. Le pendule devait alors permettre de mesurer le temps, sans se dérègler sous la houle. Surprise : cette trajectoire n’est pourtant rien de plus que la trajectoire  que fait  la valve d’une roue de vélo lorsque le ce dernier roule sur du plat.

• « Le niveau de distribution de la fonction somme des chiffres le long des progressions arithmétiques » :

Pour $q\geq 2, n\in \mathbb{N}$, soit $s_{q}(n)$ la somme des chiffres de $n$ écrit en base $q$. L. Spiegelhofer (2020) a prouvé que la suite de Thue-Morse a un niveau de distribution de 11, améliorant un ancien résultat de Fouvry et Mauduit. Nous généralisons ce résultat aux suites de type $\left\{\exp\left(2\pi i \frac{\ell}{b}s_{q(n)}\right)\right\}_{n\in\mathbb{N}}$ et fournissons un exposant explicite dans la borne supérieure.

• « Stonean spaces » :

In this talk we will give an introduction to stonean spaces. We will show that stonean spaces are exactly the projective objects in the category of compact Hausdorff spaces, and every compact Hausdorff space is a continuous image of a stonean space. As an application, we show that condensed sets (i.e. sheaves on the site of category of stone spaces) are equivalent to sheaves on the site of category of compact Hausdorff spaces, and to sheaves on the site of category of stonean spaces.

Cet exposé discute de quelques résultats originaux qui ne sont généralement pas enseignés dans un cursus classique du Supérieur en Mathématiques. Il s’agit en effet du lemme de Wielandt (1949), du théorème de Kleinecke-Shirokov (1957-1956) et du théorème de Fulglede-Putnam-Rosenblum (1950-1951-1958). Provenant historiquement de la théorie des algèbres d’opérateurs, il est en fait naturel de les traduire dans le langage des algèbres de Banach dont leur père, le mathématicien soviétique I. M. Gelfand, a démontré entre 1939 et 1941 la complémentarité singulière qui s’exprime entre algèbre et analyse.

C’est cette complémentarité qui est réaffirmée ici. Ces résultats gravitent tous autour de la notion de (non)-commutativité qui est le cœur de la mécanique quantique et de la théorie des opérateurs. Plusieurs démonstrations du théorème de Wielandt sont proposées dont l’une avec l’aide du théorème de Kleinecke-Shirokov. Les résultats ci-dessus sont mis en lumière par quelques réflexions dans l’algèbre $\mathcal{M}_n(\mathbb{C})$ des matrices carrées et par des questions personnelles sur les propriétés d’un couple $(a,b)$ d’éléments dans certaines $\mathbb{C}$-algèbres contraint à satisfaire une relation du type $[a,b]=\alpha a, ~ \alpha \in \mathbb{C}^*$. L’exposé est enrichi de remarques historiques et contextuelles sur la théorie des algèbres de Banach et des opérateurs.

Barycentres conditionnels de Wasserstein pour modéliser l’effet des conditions expérimentales sur la dégradation de batteries

Les performances des batteries électriques se dégradent au cours du temps. C’est le cas par exemple de la quantité d’énergie stockée qui diminue au cours du temps. Un enjeu important pour les constructeurs de batteries est de modéliser la dégradation caractéristique d’un nouveau modèle de batterie afin d’évaluer sa valeur.

Dans une présentation précédente, lors de la journée des doctorants, nous avions présenté des méthodes à base processus gaussiens pour modéliser la dégradation des batteries sous une condition de référence. Cela fournis un premier outil mais est souvent insuffisant en pratique. En effet, la dégradation des batteries dépend fortement de ses conditions d’utilisation, la température ambiante, le courant de charge ou de décharge… Nous avons donc besoin d’une méthode capable de prédire la dégradation en fonction des conditions expérimentales, et ce même pour des conditions jamais observées.

Face aux difficultés rencontrées à modéliser l’effet des conditions avec les méthodes utilisées précédemment, nous proposons une autre approche reposant sur la théorie du transport optimal. Dans cette présentation nous prendrons le temps d’introduire les éléments essentiels du transport optimal (problème de Monge, Kantorovitch …). Puis nous introduirons l’idée plus récente du barycentre conditionnel de Wassertein comme de méthode de régression lorsque les sorties sont des distributions de probabilités. La régression Fréchet, un type particulier de barycentre conditionnel, sera utilisée pour modéliser l’effet de la température sur le vieillissement des batteries.

Absorption d’un mouvement brownien réfléchi

Après avoir défini le mouvement brownien, j’essayerai de motiver l’étude d’une de ses très nombreuses généralisations : le mouvement brownien réfléchi (RBM) dans un cône. Nous verrons que ce processus est assez bien décrit par un petit jeu de paramètres. Nous discuterons en particulier de méthodes utiles au calcul de la probabilité qu’un RBM soit absorbé au sommet du cône. Cet exposé est issu d’un travail en collaboration avec Sandro Franceschi.

How cancer evolution can be modelled ? An example of a toy model giving insights on such evolutionary process.

Lors d’une palpitante partie de Qui-Est-Ce et des enjeux qu’une glorieuse victoire peut y représenter, il paraît fondamental d’y établir des stratégies solides : Quelles sont les questions qui optimisent nos chances de gagner ? Nous aborderons cette question sous le prisme de la théorie de l’information. Plus précisément, nous aborderons la notion d’entropie et en exhiberons quelques propriétés fondamentales pour mieux la cerner. Nous en profiterons pour introduire l’entropie relative, aussi appelée divergence de Kullback-Leibler, et présenter quelques résultats qui la font intervenir, notamment dans la théorie des grandes déviations avec le théorème de Sanov.