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

Cet exposé tourne autour de trois problèmes distincts mais fortement reliés. Tout d’abord, l’étude de la limite hydrodynamique de systèmes de particule soumises à des dynamiques de branchement et de sélection, qui est la question centrale que je me suis posée pendant ma thèse.

Ensuite, les équations de réaction-diffusion faisant intervenir une frontière libre contrôlant la masse totale, connus depuis une vingtaine d’année pour être reliés aux systèmes de particule en interactions.

Et enfin, le problème inverse du premier temps de passage pour un processus markovien, que l’on peut interpréter comme une reformulation probabiliste des problèmes à frontière libre.

Mon but sera de vous présenter ces trois problèmes et de vous expliquer l’état de la littérature sur ce qui les relient.

A first simplification of the gene expression mechanism considers that a gene is transcribed into messenger RNA, which in turn is translated into protein. Single-cell data have revealed the presence of biological variability between cells of identical genome and environment, highlighting not only epigenetic aspects but also the stochastic nature of gene expression.

In the context of regulatory networks underlying cell states and types, we need to build a model that takes into account both stochasticity and the interaction of genes with each other. Here we focus on a dynamical model of gene expression, formulated as a piecewise-deterministic Markov process (PDMP) and describing an arbitrary number of interacting genes. This stochastic model is able to reproduce the biological variability measured experimentally, but remains mathematically complex and difficult to study. This is why, in the litterature, a simplified model with only proteins is considered. 

During this talk, we provide insights on construction and use of semigroups and infinitesimal generators for PDMPs. Afterwards we present both models and use coupling methods to explicitly upper bound the error made when substituting the full model with its simplified version.

A fundamental question in geometry consists in understanding the effect of curvature on the shape of geometric spaces. In the case of surfaces, the Gauss-Bonnet Theorem establishes a link between the curvature of a surface and its topology. For example, it allows us to understand the topology of surfaces whose curvature is positive at every point.

When considering higher-dimensional geometric objects, called manifolds, we can define different notions of curvature. Scalar curvature is the weakest of these notions, and for this reason it is difficult to extract topological or geometric information from it. In particular, can we describe the topology of a manifold with positive scalar curvature?

In this talk, I will explain why this is an interesting question, and I will present a classification result for 3-dimensional manifolds with positive scalar curvature. This is a collaborative work with F. Balacheff and S. Sabourau.

In this talk, we begin by examining the application of curvature flows to a broad range of geometric problems. Following this, we introduce the essential geometric concepts required to understand these flows. Thereafter we focus on the mean curvature flow and its singularities. In particular, we give an intuitive and accessible proof of why singularities must occur if the initial surface is compact. After conducting a graphical analysis of various types of singularities, we describe how these singularities can be modeled by self-similar solutions of the mean curvature flow. Motivated by this, we conclude the presentation by exploring a current area of research: investigating the behavior of solutions that are in the proximity of such self-similar solutions.

Cela fait déjà plus de 150 ans que la recherche mathématique se casse les dents sur ce fameux problème appelé “Hypothèse de Riemann”. Portant sur les zéros non triviaux de la fonction Zêta de Riemann, elle est étroitement liée à la répartition des nombres premiers.

Journée conviviale d’exposés mathématiques pour les doctorants de l’IECL.

Programme :

Matin :

• 8h50 : Café d’accueil ;
• 9h20 : Karim Ramdani : Edition scientifique : un rapide survol des évolutions en cours ;
• 10h15 : Rodolphe Abou Assali : The Biharmonic Steklov Operator ;
• 10h55 : Pause ;
• 11h25 : Jérémy Dousselin : Arithmetic: from elementary statements to complex tools ;
• 12h15 : Pause repas

Après-midi :

• 14h : Aurélien Minguella : A brief introduction to stochastic partial differential equations ;
• 15h : Nathan Toumi : The level of distribution of the sum-of-digits function in arithmetic progressions ;
• 15h40 : Pause ;
• 16h10 : Valentin Schwinte : A minimization problem in the lowest Landau level, and centrosymmetric matrices ;
• 17h10 : Fin de la journée