Doctorants

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
σ to approximately 1.419.

Wednesday 20/05 – MSA 2.240 :

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

• • 14:00 – 14:45 : Talk
    
    Title

    Abstract

• • 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 : Talk
    
    Title

    Abstract

• • 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 : Talk
    
    Title

    Abstract

• • 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 : Talk
    
    Title

    Abstract

• • 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 : Talk
    
    Title

    Abstract

• • 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 : Talk
    
    Title

    Abstract

• • 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 : Talk
    
    Title

    Abstract

• • 12:30 – 14:00 : Lunch

We consider branching Brownian motion (BBM), a random process
that describes the evolution of a particle population, reproducing and
moving independently. Beyond obvious biological motivations and its link
with the F-KPP equation, BBM can be seen as a toy model for spin
glasses, such as the Sherrington-Kirkpatrick model. In this perspective,
we will introduce the Gibbs measures of BBM. We will study some of their
properties, including their connection with the so-called additive
martingales. We will also study the maximal particle of BBM (or, from
the perspective of statistical physics, the ground state of the system).
A new martingale then appears, that is, the derivative martingale. If
time allows, we will briefly present an ongoing work with Gabriel Flath and Julien Berestycki,
in which we obtain an almost sure path localization of the derivative
martingale.

We study a model of random permutations, which we call random standardized permutations, based on a sequence of i.i.d. discrete random variables. This model generalizes others, such as the riffle-shuffle and the major-index-biased permutations. We first establish an exact result on the joint distribution of the number of cycles of given lengths, involving the notion of primitive words. Thanks to this result obtained via combinatorial methods, we obtain convergence in distribution as the size of the permutation tends to infinity . This talk will be an opportunity to introduce (or recall) the method of moments, a very useful tool for proving convergence in distribution, particularly for combinatorial objects. We will present a few limit results on the distribution of « small » and « large » cycles of the permutation, as well as on the total number of cycles.

La géométrie birationnelle est la branche qui étudie les espaces « presque partout isomorphes ». Dans ce cadre, j’introduirai une opération fondamentale: l’éclatement, qui permet de modifier localement un espace en remplaçant un point par un ensemble de directions. Je présenterai ensuite une version plus flexible de cette construction, les éclatements à poids, où les différentes directions sont prises en compte de manière non uniforme.

J’introduirai brièvement les variétés horosphériques, qui fournissent un cadre particulièrement bien adapté aux actions de groupes et à une description combinatoire de la géométrie. Je terminerai par un aperçu des contractions divisorielles horosphériques, qui s’avèrent être, dans ce contexte, toutes données par des éclatements à poids.

The Wasserstein space $\mathscr{P}(M)$ associated with a closed Riemannian manifold is defined as the space of probability measures on the manifold, endowed with the so-called Otto metric, which provides it with the structure of a formal infinite-dimensional Riemannian manifold. In this talk I will describe the geometric features of this space, emphasizing its connections with optimal transport theory and some classical PDEs. I will then introduce the group of diffeomorphisms $\mathscr{D}(M)$, viewed as an Inverse Limit Hilbert Lie group, and present the Riemannian submersion structure that relates $\mathscr{D}(M)$ and $\mathscr{P}(M)$. The space $\mathscr{P}_\infty(M) \subset \mathscr{P}(M)$ of smooth positive measures is of particular interest. The geodesic convexity of such a space highly depends on the geometry of the base manifold. I will review some significant developments on this topic, mainly due to Ma, Trudinger, Wang, Loeper and Villani. If time permits, I will try to introduce some of the topics of my PhD thesis which focuses on the study of random paths on $\mathscr{P}(M)$ and its tangent bundle $T\mathscr{P}(M)$.

It is well known about the Riemann zeta function that $\zeta(2i) \in \mathbb{Q}\pi^{2i}, i\geqslant 1$, and thus all these numbers are linearly independent over $\mathbb{Q}$, since $\pi$ is transcendental.
However, very little is known about the numbers $\zeta(2i+1), i\geqslant 1$. It was proved in 1978 that $\zeta(3)$ is irrational, and in 2000 that infinitely many of these numbers are irationnal.

The aim of this talk is to introduce the core ideas used to prove such a result, and to elaborate on the new ideas that allowed many generalizations since the 2000s (better bound on the proportion of irrational numbers, broader class of functions rather than just $\zeta$, …)

We study the residual stream of multi-head Transformers in which the attention weights are i.i.d.\ random matrices across layers and heads. We identify critical scaling laws linking the depth $L$, the residual scale $\eta$, and the number of heads $H$, and show that different joint limits yield distinct homogenized effective models. To formalize these limits, we leverage the theory of stochastic modified equations. We apply this framework to Transformers at initialization and derive effective dynamics that clarify the roles of additional parameters, including the inverse temperature $\beta$, the embedding dimension $d$, and the context length $n$.

In this talk, we’ll consider various situations (favorable, semi-favorable and unfavorable) for the computation of the partition function. For each situation, an overview of the techniques used to simulate and run the inference for the models will be discussed. More specifically, we will briefly discuss : the inverse of the cumulative distribution function, maximum likelihood, Gibbs sampler, maximum pseudo-likelihood, Metropolis-Hastings algorithms and maximum likelihood MCMC.

Je ne vois pas l’avenir. Et c’est bien là le souci : les problèmes d’optimisation liés à la prise de décision concernent bien trop souvent des décisions futures.
Optimiser l’espérance mathématique en fonction des événements envisageables ? Encore faut-il en connaître les probabilités.

Nous avons toutefois connaissance du passé. Une approche consiste alors à résoudre, dans un premier temps, le problème empirique construit à partir de ces données. La solution que nous obtiendrons sera-t-elle proche d’une solution optimale pour le problème de départ ? Combien de données sont nécessaires pour réaliser cette approximation ? Nous verrons, dans un premier temps, comment l’optimisation stochastique traite ces questions.

Nous discuterons ensuite des limites du critère de l’espérance, notamment dans les cas où un risque de grande perte est compensé par l’espoir de grands bénéfices. Ces limites motivent l’introduction de mesures de risque comme critère dans les problèmes d’optimisation stochastique. Nous en aborderons, pour finir, une généralisation multivariée et présenterons les premiers résultats associés.