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

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.

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.

Site web de l’évènement.
Programme du 1er jour – Salle de conférence :

• • 13h30 – 14h15 : Exposé de Benjamin Florentin – IECL
    
    Can one hear the shape of a Steklov drum ?

    Introduced at the beginning of the 20th century, the Steklov eigenvalue problem has attracted growing interest in spectral geometry over the last few decades and remains a major research topic in the field. In this talk, we will focus on the associated spectral inverse problem consisting in recovering a metric of a compact Riemannian manifold with boundary from knowledge of its Steklov spectrum, or equivalently the spectrum of its Dirichlet-to-Neumann map (DN map). In other words, can one hear the shape of a “ Steklov drum ” ? We will present some recent positive results obtained on a certain class of manifolds with negatively curved boundaries.

• • 14h15 – 15h00 : Exposé de Marie Abadie – Université du Luxembourg
    
    Hyperbolic surfaces and graphs

    The goal of this talk is to provide a brief overview of the interactions between hyperbolic surfaces and certain combinatorial objects. A given surface has a large space of hyperbolic metrics, called its Teichmüller space, which itself admits a natural metric, called the Weil-Petersson (WP) metric. Distances between two hyperbolic metrics with respect to this WP metric are hard to compute. We will describe Brock’s combinatorial approach for approximating that distance using the graph of pair-of-pants decompositions.

• • 15h00 – 15h30 : Pause café en salle 113

• • 11h10 – 11h50 : Exposé de Aurélien Minguella – IECL
    
    A brief introduction to stochastic partial differential equations.

    Stochastic partial differential equations (SPDEs) are the mathematical objects used
    to describe the random dynamics of infinite-dimensional objects. They take applications in a
    broad range of areas, from statistical and theoretical physics, to fluid mechanics. These objects
    display very rich mathematical behaviour and have known a gain of interest since Martin Hairer
    was awarded the Fields medal in 2014 for constructing a solution theory for a very broad class
    of parabolic singular equations, namely regularity structures. Staying far away from regularity
    structures, we will try to give a quick overview of some simple SPDEs, but where some
    essential phenomena already arise. Some emphasis will be given on the invariant measures for
    such equations. We will first give a review of basic stochastic calculus and continue with an
    example of a linear SPDE: the stochastic heat equation. If time permits, we will have a glimpse
    of the complications happening in the non-linear case. We will finish with an informal overview
    of the most recent theories and current challenges in the field.

• • 16h15 – 17h00 : Exposé de Alexandre Benoist – Université du Luxembourg
    
    The ternary cyclotomic polynomials $\Phi_{3pq}$

    Cyclotomic polynomials are a classical and fundamental topic in number theory, and still an active field of research. The aim of this talk is introducing results about the coefficients of cyclotomic polynomials. I will first speak about the family of binary cyclotomic polynomials, which is completely understood. Then, I will move on ternary cyclotomic polynomials. I will provide a formula for computing the coefficients of the ternary cyclotomic polynomials of the form $\Phi_{3pq}$, from which we can derive various properties and solve conjectures for this family.

Programme du 2ème jour – Amphi 8 :

• • 9h15 – 10h00 : Exposé de Carl-Fredrik Lidgren – Université du Luxembourg
    
    Reconstructions, complete invariants, and anabelian geometry.

    In topology, a basic invariant of interest is the fundamental group of a topological space. As an invariant, this is useful for distinguishing between two topological spaces, but is in general not useful for determining that two spaces are the same because radically different topological spaces can have the same fundamental group. In algebraic geometry, on the other hand, one can define an arithmetic analogue of the fundamental group, also related to absolute Galois groups in number theory, which turns out to be considerably more rigid. The aim of the talk is to discuss this phenomenon and the area which studies it: anabelian geometry.

• • 10h00 – 10h30 : Pause café, Hall de l’amphi 8

• • 10h30 – 11h15 : Exposé de Vidhi Vidhi – IECL
    
    Statistical approach for quantifying the evolution of tumor heterogeneity in chronic
    lymphocytic leukemia (CLL).

    Voir en plein ecran

    Aller au contenu PDF

• • 11h15 – 12h00 : Exposé de Tim Seuré – Université du Luxembourg
    
    Balancing Powers

    This talk explores surprising equalities between power sums arising from a
    binary-based partition of the integers.

• • 12h00 – 13h30 : Déjeuner

• • 9h15 – 10h00 : Exposé de Amine Iggidr – IECL
    
    From noise to harmony: Understanding the primes through waves

    Prime numbers appear scattered randomly along the integers, yet their distribution hides some sort of structure. This talk introduces how Fourier analysis ideas reveal periodic components inside the primes. This help us explain classical phenomena such as Chebyshev’s bias and reveal why complex zeros of L-functions govern the oscillations in prime-counting functions. In this talk we trace the historical development from Dirichlet to modern analytic number theory and show how harmonic analysis serves as a powerful tool which helps us understand the secrets of prime numbers.

• • 14h15 – 15h00 : Exposé de Lucia Celli – Université du Luxembourg
    
    Wide neural networks with general weights: convergence rate and explicit dependence on the hyper-parameters.

    Wide fully connected neural networks converge at initialization to a Gaussian process, but quantitative rates are not well understood. I present explicit, non-asymptotic bounds for this convergence in both one- and multi-dimensional settings under general weight assumptions. The results make all dependencies on depth, width, activation, and moments explicit, covering common cases such as ReLU and Gaussian initialization, and clarify when the limiting covariance remains non-degenerate.

• • 15h00 – 15h30 : Pause café, Hall de l’amphi 8

• • 15h30 – 16h15 : Exposé de Hugo Nouaille – IECL
    
    Rough paths: Integration beyond smoothness.

    Almost every theory of integration applies to solving certain ODE or PDE problems. We review some Cauchy problems with Hölder signals and their formulation in integral form. From there, we can observe how the idea of rough paths is motivated. Finally, we will try to provide some intuition about one object of this theory: the rough integral.

• • 16h15 – 17h00 : Exposé de Leolin Nkuete – Université du Luxembourg
    
    Hopf Galois extensions

    It is well known that for any Galois extension L/K one can associate an arithmetic object G:=Gal(L/K) called the Galois group of the extension L/K. This group gives rise to a group algebra H=K[G], which, is in particular a Hopf algebra. We called this group algebra a Hopf Galois structure associated with the extension L/K. The goal of this talk is to explain how this framework can be generalized to non-Galois extensions.

• • 19h30 : Dîner au Grand Café Foy (1 Pl. Stanislas, 54000 Nancy)

Programme du 3ème jour – Amphi 8 :

• • 8h30 – 9h15 : Petit déjeuner, Hall de l’amphi 8

• • 9h15 – 10h00 : Exposé de Simon Bartolacci – IECL
    
    Global Waiting: An Alarm Clock Optimization Perspective.

    What time should I set my alarm tomorrow morning? Like many, I have often
    wondered about this. And like many I first turned to deterministic constrained optimization
    theories, involving Lagrange multipliers and other classical tools. I soon realized that I cannot
    know exactly whether there will be traffic, at what time my colleagues will arrive, or whether I
    might fall back asleep after the first alarm. The optimization problem underlying this question
    is therefore inherently stochastic.
    While I cannot predict the future, I do have empirical knowledge: I frequently oversleep, often
    leave so late that I avoid traffic, and sometimes arrive well after my colleagues. This gives me
    an empirical understanding of phenomena whose laws I do not know. Can optimizing over such
    empirical phenomena lead to a truly optimal alarm time? This is precisely the kind of problem
    studied in stochastic optimization, and this talk will discuss the answer.
    Moreover, focusing solely on expected outcomes is not ideal: my primary concern is not just
    the average timing but reducing the risk of missing critical events, such as the coffee break. By
    incorporating risk measures beyond the expectation, stochastic optimization allows us to design
    alarm strategies that are robust to worst-case scenarios and better aligned with practical
    priorities.

• • 10h30 – 10h30 : Pause café, Hall de l’amphi 8

• • 10h30 – 11h15 : Exposé de Francesca Pistolato – Université du Luxembourg
    
    From Galton board to Fractional Brownian motion.

    In this talk, we will explore how simple experiments can illustrate fundamental ideas in probability. We revisit the Central Limit Theorem through the Galton board and then extend the intuition to fractional Brownian motion, highlighting how randomness can exhibit memory and long-range dependence.

• • 10h30 – 11h15 : Exposé de Kilian Lebreton – IECL
    
    Probabilistic approach to certain sums arising from number theory.

    Kloosterman sums, Gauss sums, Birch sums are examples of families of sums of fonctions \(\varphi : (\mathbb{Z}/n\mathbb{Z})^* \to \mathbb{U}_n \), where \(\mathbb{U}_n \) denotes the set of \(n\)-th roots of unity. They arise naturally in number theory. Once normalized by \(\sqrt{n} \), the complete sums (within their respective families) behave like bounded random variables. From this, we can deduce that their partial sums converge in law to a random Fourier series, that short sums converge to a Gaussian distribution, and that we can estimate the maximum of their partial sums.

Journée conviviale d’exposés mathématiques pour les doctorant.e.s de l’IECL.

Les exposés auront lieu en salle de conférence et les pauses en salle 313.

Programme de la journée :

• • 9h00 – 9h30 : Petit déjeuner

• • 9h30 – 10h10 : Exposé de Sophie Baland
    
    Un modèle de branchement pour la dynamique des longueurs de télomères dans les cellules sanguines.

    Dans les domaines de la biologie et de la médecine, la modélisation mathématique du développement cellulaire reste primordiale à étudier.

    Dans cet exposé, nous nous intéresserons aux télomères : ces petites structures situées aux extrémités des chromosomes eucaryotes et qui possèdent un rôle de capuchon protecteur permettant de préserver l’intégrité du génome.

    Dans une première partie, j’aborderai la structure et les fonctions des télomères, leur rôle dans les processus de vieillissement ainsi que dans les maladies résultant d’une modification de leur longueur, élément déterminant dans leur bon fonctionnement. De plus, je présenterai brièvement deux mécanismes biologiques : le processus de réplication de l’ADN et l’hématopoïèse, qui est le processus de fabrication des cellules sanguines, afin d’introduire les notions nécessaires à la compréhension d’un modèle décrivant la dynamique des longueurs de télomères.

    Dans une seconde partie, j’introduirai un modèle de branchement, qui va permettre de comprendre le mécanisme de l’hématopoïèse et qui reproduit les comportements des cellules lors de divisions cellulaires, en tenant compte de la longueur de leurs télomères. Il s’agit d’un modèle stochastique d’évolution d’une population de cellules et de leurs chromosomes, faisant intervenir plusieurs facteurs tels que l’attrition télomérique, l’action de la télomérase, les phénomènes d’autorenouvellement, de différenciation, et de mort cellulaire. Je présenterai ensuite deux résultats : le premier, appelé loi des grands nombres, lié au comportement du modèle en grande population, et le second, portant sur l’étude des fluctuations du modèle.

• • 10h15 – 10h55 : Exposé de Léo Delage
    
    Intro to graphs of groups.

    Groups acting on trees is a foundational topic in geometric group theory and topology, with an incredibly wide range of applications (graphs of spaces, JSJ theory, Outer spaces…) and generalizations (word-hyperbolic groups, CAT(0) cube complexes, real trees…). In this talk, I will sketch the classical correspondence between group actions on (simplicial) trees and the associated orbifold-like structures called graphs of groups that play the role of a quotient space. Some of my favorite examples will be provided.

• • 10h55 – 11h10 : Pause café

• • 11h10 – 11h50 : Exposé de Mabrouk Ben Jaba
    
    The human lung : an impossible-to-model organ ?

    The lung constitutes an essential exchange interface between ambient air and blood, playing a crucial role in the oxygenation of the latter and the elimination of carbon dioxide. Understanding its functioning therefore represents a major challenge.

    Various mathematical models have been developed to study its mechanisms, some involving complex partial differential equations. An alternative approach consists in considering models that integrate the bronchial tree as a whole, which is the perspective adopted here.

    Our approach is based on the hypothesis that gas exchanges are optimized to maximize the efficiency of the lung, in accordance with principles such as the theory of evolution. To explore this hypothesis and assess this optimality principle, we propose a model based on ordinary differential equations describing the evolution of oxygen concentration in the lung and its transport. Within this framework, we introduce, analyze, and study an optimal control problem aimed at characterizing the dynamics of the respiratory cycle.

• • 11h50 – 14h00 : Buffet déjeuner

• • 14h00 – 15h00 : Exposé de Aurélien Minguella
    
    The cutoff phenomenon for the Brownian motion on the torus.

    The cutoff phenomenon occurs in the study of the convergence of ergodic Markov chains towards their invariant measure. For a large class of these objects, we can expect that, when a size parameter (dimension, number of objects) becomes asymptotically large, convergence occurs abruptly. The aim of this presentation is to give an example of a natural Markov chain for which this phenomenon is relatively easy to prove.
    After reviewing discrete-time Markov chains, we will present their continuous-time counterparts. We will then define the Brownian motion on the torus and see how it fits into this framework. The end of the presentation will be devoted to concluding the proof of the cutoff.

• • 15h05 – 15h45 : Exposé de Marie Dautheville
    
    A quick tour inside the p-adic world

    The aim of this talk is to provide an introduction to several objects involved in the representation theory of p-adic groups, while keeping the presentation accessible to a broad audience. It will be divided into three parts. The first part will be devoted to the construction of the p-adic numbers, which form a totally disconnected number field that can be of characteristic zero or p, where p is a prime number. We will see some properties of this field and its ring of integers. Unlike the real numbers, compact subsets of \mathbb{Q}_p or \mathbb{F}_p((t)) can be both compact and open—this is the case, for instance, for the unit ball in \mathbb{Q}_p. This major difference leads us to the second part, concerning p-adic groups. In contrast with real Lie groups, p-adic groups may have several conjugacy classes of maximal compact subgroups. This section will be illustrated with examples of classical matrix groups. Finally, in the third part, we will briefly discuss representations of p-adic groups through the study of unramified characters of a group. We will determine those of SL(n, \mathbb{Q}_p) and, if time allows it, i will conclude by comparing the tempered duals of SL(2, \mathbb{Q}_p) and SL(2, \mathbb{R}).

• • 15h45 – 16h00 : Pause café – goûter

Le lieu d’annulation d’un polynôme en plusieurs variables peut être abordé sous plusieurs angles : l’étude de sa topologie et des ses points à coefficients entiers ou modulo p. Cela fournit donc des exemples d’espaces topologiques intéressants et permet en même temps d’aborder les équations arithmétiques sous un angle plus géométrique.

Nous chercherons dans un premier temps à voir pourquoi ces objets ont une topologie riche et restrictive, puis nous montrerons à travers un cas simple des conjectures de Weil comment la structure topologique restreint des données arithmétiques. Les bons objets derrière ce phénomène seront les motifs de Chow.

Les images numérisées sont généralement dégradées suite au processus d’acquisition ou de transmission. Quand la précision est cruciale, par exemple pour établir un diagnostique médical, il est préférable de filtrer ce bruit.

L’exposé porte sur un algorithme de filtrage du bruit reposant sur la dualité (Chambolle, 2004). L’information est extraite en minimisant une fonction construite afin de filtrer le bruit sans trop compromettre la netteté des contours de l’image sous-jacente.

Après avoir introduit la modélisation des images, la discussion s’articulera autour de trois outils : la variation totale, la conjuguée convexe et les sous-gradients. L’objectif est d’expliquer comment ils peuvent offrir un point de vue fructueux sur un problème d’optimisation. Si vous n’êtes pas convaincus, on me dit dans l’oreillette qu’il y aura moultes illustrations et animations !

Dans cet exposé nous regarderons l’apparition de la fonction exponentielle dans les ouvrages de Cauchy et discuterons, textes historiques à l’appui des critères de Cauchy, D’Alembert et d’Hadamard. Ce sera l’occasion de regarder quelques démonstrations d’analyse du 18-19ième siècle. La fonction exponentielle sera caractérisée via sa propriété de morphisme continu, comme l’a fait Cauchy dans son « Cours d’Analyse de l’Ecole Royale Polytechnique » paru en 1821. Son développement en série entière sera mis en lumière également par des textes d’époque. La suite de l’exposé est de constater l’émergence de la série exponentielle dans le contexte matriciel, puis dans le contexte des algèbres de Banach.  Dans la première moitié du 20-ième siècle, la norme matricielle de Frobenius,  sous-multiplicative, joue un rôle catalyseur poussant les mathématiciens comme Nagumo, Yosida, D.S. Nathan… puis Gelfand à étudier des structures algébriques générales – des C-algèbres (unitaires) – munies d’une norme sous-multiplicative rendant complet l’espace vectoriel sous-jacent : c’est l’émergence de l’étude des anneaux normés complets, connus aujourd’hui sous le nom d’algèbres de Banach. Toujours sous le prisme d’articles d’époque, nous donnerons quelques formules bien connues sur l’exponentielle, les sous-groupes à un paramètres fortement continus, et terminerons par la formule de Lie-Trotter dans le contexte des algèbres de Banach.

Les systèmes de Prony sont apparus dans de nombreux articles scientifiques issus de différents domaines sans que leur résolution ne devienne un classique en calcul scientifique. Cet exposé vise à mettre en avant ces systèmes en donnant deux applications (l’une en introduction pour motiver leur étude et l’autre en ouverture à la fin de l’exposé) ainsi qu’une étude de leurs principales propriétés. Un bon niveau licence est suffisant pour mener (et, je l’espère, apprécier) cette dernière. Des éléments de résolution numérique seront également présentés.

L’analyse topologique des données (Topological Data Analysis : TDA) est un domaine à l’intersection de la statistique et de la topologie algébrique qui a émergé au début des années 2000.

L’objectif est de tirer de nouvelles informations dans les données, en s’intéressant à des aspects géométriques et topologiques dans leur structure. Le cadre usuel consiste à étudier la structure d’un nuage de points, c’est-à-dire un ensemble de points dans un espace métrique, et un des outils le plus utilisé pour cela est ce qu’on appelle l’homologie persistante.

Dans cet exposé nous commencerons par introduire de manière pédagogique l’homologie persistante, puis nous discuterons de ces applications possibles dans un problème de neurosciences.