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

Résumé à venir

In modern algebraic geometry, the study of moduli spaces plays a central role in the problem of classifying certain geometric objects (e.g., Riemann surfaces, vector bundles), up to a fixed notion of isomorphism. The foremost question arising is whether we can construct a moduli space which, roughly speaking, parametrizes the isomorphism classes of such objects. The moduli space will be endowed with a natural geometric structure, which is often a scheme or an algebraic stack. In this talk we give an introduction in the theory of moduli spaces, with special emphasis on some classical examples: the Grassmannian, the Hilbert scheme, the moduli space of sheaves etc.. We will formulate the moduli problems using the categorical language of representable functors, and introduce the notions of fine and coarse moduli spaces.

Stochastic Approximation is a useful tool for Machine Learning techniques such as Stochastic Gradient Descent. These algorithms are applied to a lot of different fields, improving the transportation times, helping doctors diagnosing with medical images, automatically translating text, detecting spam and more. Most of the time, the model traditionally lies in a vector space. However, some problems present non-linear constraints, that can be translated into a manifold. This framework ensures the conservation of key properties. As an introduction to geometric machine learning, we study the gradient descent algorithm, and its adaptation to Riemannian manifolds. Finally, we compare the performance of the two, introducing new non-asymptotic bounds.

À venir

En France la liste d’attente pour la greffe d’organe est nationale. La question de la décision autour de l’attribution d’un greffon est donc très importante. Dans cet exposé je vous présenterai l’approche théorique d’un tel problème à l’aide des modèles d’appariement avec impatience et je détaillerai l’évolution des simulations de ce problème, au fur et à mesure des interactions avec l’agence de la biomédecine.

Cette présentation concerne l’étude des courses de polynômes irréductibles unitaires dans les corps de fonctions à (3) compétiteurs ou plus. Plus concrètement, soit (m in F_{q}[T]) un polynôme unitaire (avec (F_{q}) un corps à (q) éléments et (q) une puissance d’un premier (>2)) de degré (M geq 2), (r) un entier (geq 3). Pour (a in F_q[T]) premier avec (m) et pour (N in mathbb{N^{*}}), on désigne par (pi(a,m,N)) le nombre de polynômes irréductibles unitaires congrus à (a ) et de degré (N). On considère (A_{r}(m) ) l’ensemble des (r)-uplets des différents éléments ((a’_1,..,a’_r) in F_{q}[T]) modulo (m) qui sont premiers avec (m.) Pour ((a_1,..,a_r) in A_{r}(m)), on définit :
begin{align*}
P_{m;a_1,..,a_r} &:= left{ X in mathbb{N}^{*} : hspace{0,2cm}
sumlimits_{N=1}^{X} pi(a_1,m,N) > …> sumlimits_{N=1}^{X} pi(a_r,m,N)
right}
end{align*}
Ainsi, sous l’hypothèse LI, pour réaliser cette étude, il suffit d’étudier la densité naturelle suivante :
begin{align*}
delta_{m;a_1,..,a_r} :&= limlimits_{X longrightarrow +infty} frac{# left( P_{m;a_1,..,a_r} cap left{1,2,.., Xright} right)}{X}
end{align*}

Il s’agit d’analyser les différentes densités afin de déterminer l’équipe gagnante.

In statistical analysis, change point detection aims to identify the times when the probability distribution of a stochastic process or a time series, or the parameter of the time series models changes. In general, the problem concerns both detecting the changes and identifying their locations. My goal is not only to detect the big breakpoint, but also, the detection of the small changes. The likelihood ratio test is used to detect these changes (small and big changes). The distribution
under the null and the alternatives hypothesis of the test was did by the LAN property (Locally asymptotic normal) and the Le Cam’s third lemma. The optimality of the test was proved at the end of the job.

Les travaux de Charles Darwin sur l’évolution ont motivé de longues recherches sur les effets des mutations génétiques et de la sélection naturelle. Cependant, les avancées techniques ont récemment permis aux biologistes de s’apercevoir qu’à l’échelle individuelle et sur une échelle temps plus courte que l’échelle évolutive, l’expression des gènes impliqués dans le métabolisme bactérien est hétérogène.

Nous proposons dans cet exposé quelques approches de modélisation plus ou moins simples soutenues par des hypothèses biologiques, partant d’une formalisation des mécanismes majeurs qui ont lieu à l’intérieur de la cellule bactérienne à une description des dynamiques globales pour des cultures en grande population.

Hydrogeochemical data may be seen as a point cloud in a multi-dimensional space. Each dimension of this space represents a hydrogeochemical parameter ( i.e. salinity, solute concentration, concentration ratio, isotopic composition…). While the composition of many geological fluids is controlled by mixing between multiple sources, a key question related to hydrogeochemical dataset is the detection of the sources. By looking at the hydrogeochemical data as spatial data, this work presents a new solution to the source detection problem that is based on point processes. Results are shown on simulated and real data from geothermal fluids.

La théorie des jeux coalitionnels est la partie de la théorie des jeux qui s’intéresse à la formation de coalitions. Son but est de proposer des concepts de solutions qui satisfont plusieurs propriétés : anonymat, symétrie, efficacité entre autres. En 1944, von Neumann et Morgenstern propose le concept des “ensembles stables”, définis comme l’ensemble des solutions desquelles nous n’allons pas dévier. En 1959, Gillies propose le concept de “cœur”, défini comme l’ensemble des solutions qui donnent à chacun au moins ce qu’il mérite, en fonction des rapports de forces qui s’appliquent au sein du jeu. Chacun de ces concepts a ses inconvénients : les ensembles stables ne sont pas uniques et sont très difficiles à calculer, le cœur quant à lui ne propose pas un ensemble de solutions stables. L’idéal serait d’avoir un cœur stable: dans ce cas il serait unique, facile à calculer et chaque solution satisferait tous les joueurs qui ne vont pas dévier de celle-ci. Cependant, savoir si un jeu admet un cœur stable ou non est un problème très complexe.