PhD students

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,
in which we obtain an almost sure path localization of the derivative
martingale.

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

L’objectif de cette journée est de rassembler les doctorants de Nancy et de Metz afin de faire plus ample connaissance autour d’exposés mathématiques.

Le programme est de 3 exposés le matin et 3 exposés le soir. La journée se passera entièrement à l’Amphi 7.

Organisateurs: Nicolas Frantz (Metz), Jimmy Payet (Metz) et Pierre Popoli (Nancy).

Les variétés de Siegel sont des variétés de Shimura qui paramètrent des variétés abéliennes avec une polarisation. Le premier exemple est la courbe modulaire dont l’importance est cruciale en théorie des nombres : elle intervient dans la preuve du théorème de Fermat-Wiles et plus généralement dans la correspondance de Langlands pour $GL_2$ sur $\mathbb{Q}$. Dans cet exposé, j’introduirai les variétés de Siegel en tant que variétés algébriques sur un corps fini et je décrirai les propriétés géométriques de certains fibrés vectoriels automorphes vivant dessus.

Les opérateurs de Schrödinger sont des incontournables dans la mécanique
quantique. J’exposerai d’abord des motivations physiques de l’étude
spectrale de ces objets. Plusieurs auteurs ont obtenu des bornes en
norme $L^p$ sur les quasi-modes des opérateurs de Schrödinger. On verra
ensuite comment se généralisent de telles estimées à des systèmes
orthonormés de fonctions. L’idée de l’exposé est de donner un avant-goût
des jolis outils sous-jacents.

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.