Doctorants

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

Qui des deux jumeaux, de celui qui reste sur terre ou de celui qui voyage à grande vitesse, vieillit le plus vite? Le chat de Schröndiger est-il mort ou en vie? Les objets quantiques sont-ils des ondes ou des particules? Pourquoi ne peut-on pas connaître à la fois la vitesse et la position d’un objet quantique? L’intuition et le langage naturels semblent impuissants face à ces questions et l’utilisation des mathématiques semblent incontournables. Je vous propose donc de découvrir, ou de redécouvrir, la théorie de la relativité restreinte et la mécanique quantique dont le mariage a donné naissance à la théorie des champs quantiques, le cadre formel du célèbre modèle standard de la physique des particules.

Je voudrais vous présenter dans cet exposé une notion très utilisée en géométrie asymptotique (Coarse Geometry) : les expanseurs.
Je partirai d’un problème concret de théorie des réseaux pour arriver à la notion d’expanseur. Nous exposerons ensuite quelques propriétés métriques remarquables de ces objets, notamment le fait qu’ils n’admettent pas de plongement uniforme dans l’espace de Hilbert, et les conséquences que cela a pour mon travail de thèse.

Dans cet exposé, je considérerai une équation cinétique collisionnelle qui dégénère en une équation de diffusion fractionnaire quand le nombre de Knudsen tend vers 0.
Cette limite est obtenue en considérant des particules dont l’équilibre est une fonction à décroissance polynomiale.

Dans cet exposé, je considérerai une équation cinétique collisionnelle qui dégénère en une équation de diffusion fractionnaire quand le nombre de Knudsen tend vers 0.
Cette limite est obtenue en considérant des particules dont l’équilibre est une fonction à décroissance polynomiale.

La classification de Killing-Cartan des algèbres de Lie complexes (semi-)simples implique aussi celle des algèbres de Lie réelles (semi-)simples ;
cependant ces dernières sont plus nombreuses.
Notamment il y a toujours une forme déployée ainsi qu’une forme compacte associées à chaque famille.
Après cette première partie, on va étudier la notion de graduation d’une algèbre de Lie, ainsi que deux graduations spécifiques :
les s-représentations et les graduations de Heisenberg.
Ces deux exemples permettent de construire bien explicitement la forme déployée et compacte d’une algèbre de Lie.

Lors de mon dernier exposé, j’avais présenté le théorème de dualité de Howe.
Ce dernier met en évidence une correspondance entre certaines représentations associées à une paire duale irréductible
(tilde{G}, tilde{G^{‘}}) dans le groupe métapléctique. Le but de cet exposé est d’obtenir une formule du caractère
pour les représentations pi^{‘} de tilde{G^{‘}} qui apparaissent dans la dualité dans le cas où le groupe tilde{G} est compact.
Pour cela, on commencera par quelques rappels assez généraux concernant la théorie des groupes de Lie compacts et de leurs représentations.
Ensuite, je présenterai la généralisation de la notion de caractère (en dimension infinie) établie par Harish-Chandra dans le milieu des années 50.
Je terminerai cette présentation par une détermination explicite des caractères dans la correspondance de Howe en utilisant les différents outils
vus précédemment.

Nous présenterons le formalisme hamiltonien de la mécanique classique, puis le formalisme, développé par Souriau, de la mécanique symplectique. La fin de l’exposé visera à introduire la mécanique quantique, via un exemple, et en insistant sur l’aspect historique.

La recherche en climatologie et en océanographie a conduit à résoudre des EDP de plus en plus complexes sur des domaines de plus en plus variés.
Un domaine de calcul naturel est celui de la sphère.
Nous proposons dans cet exposé une méthode basée sur les différences finies sur un maillage de type Cube-Sphère.
Nous verrons comment construire le maillage Cube-Sphère et comment sont calculées les dérivées.
L’ensemble sera utilisé pour le calcul du gradient sphérique.
Pour illustrer cela dans un cadre plus concret et si le temps nous le permet,
nous illustrerons ce calcul avec deux tests : le BUMP mobile et le vortex stationnaire.

La géométrie asymptotique, ou « coarse geometry », se propose d’étudier les propriétés à grande échelle des espaces métriques.
Nous présenterons dans cet exposé comment l’introduction de techniques asymptotiques en K-théorie amène à de nouvelles preuves
de la conjecture de Baum-Connes coarse pour de nouvelles classes de groupes, et de nouvelles preuves de la conjecture de Novikov.