La semaine de l’IMB

Exceptionnellement, l'accueil de la Cellule Informatique aux bureaux 225 et 270

- sera fermé vendredi 5 décembre

- sera ouvert de 10h à 12h et de 14h à 16h de lundi 1er décembre au jeudi 4 décembre

en raison des Journées Mathrice en région parisienne où participe une partie de l'équipe informatique.

Pensez à anticiper les retraits de matériel de prêt.Merci de votre attention,

The Boundary Control method is one of the main techniques in the theory of inverse problems. It allows to recover the metric or the potential of a wave equation in a Riemannian manifold from its Dirichlet to Neumann map (or variants) under very general geometric assumptions. In this talk we will address the issue of obtaining stability estimates for the recovery of a potential in some specific situations. As it turns out, this problem is related to the study of the blow-up of quantities coming from control theory and unique continuation. This is based on joints works with Lauri Oksanen.

The study of Galois representations attached to elliptic curves is a highly fruitful branch of number theory, leading to the resolution of deep problems such as Fermat’s Last Theorem. In 1972, Serre proved his foundational Open Image Theorem, which states that for every non-CM elliptic curve defined over a number field, the image of the adelic Galois representation on its torsion points has finite index. This result soon inspired Mazur to propose his famous Program B, aiming to classify all possible images of such representations.

In recent years, substantial progress has been made toward Mazur’s Program B, with several authors undertaking a systematic classification of all possible images of $p$-adic Galois representations attached to elliptic curves over $\mathbb{Q}$. At present, the classification is complete only for $p \in \{2, 3, 13, 17\}$. The main obstacle for other primes arises from the difficulty of understanding elliptic curves whose mod-$p^n$ Galois representations are contained in the normalizer of a non-split Cartan subgroup. Equivalently, this amounts to determining the rational points on the modular curves $X_{ns}^+(p^n)$.

In this talk, we focus on the case $p=7$ and show that the modular curve $X_{ns}^+(49)$, which has genus $69$, has no non-CM rational points. To achieve this, we establish a correspondence between the rational points on $X_{ns}^+(49)$ and the primitive integer solutions of the generalized Fermat equation $a^2 + 28b^3 = 27c^7$, the resolution of which can be reduced to determining the rational points on several genus-three curves. Furthermore, we reduce the complete classification of $7$-adic images to the determination of the rational points on a single plane quartic.

This is joint work with Davide Lombardo.

In this talk, we present a stabilization result for a simplified one-dimensional fluid-particle interaction system. First, without any smallness assumptions, we establish a non-collision property: the particle never reaches the fluid boundary in finite time, which in turn yields global well-posedness of the interaction system. Next, we study a stabilization problem of this model. In the absence of feedback control, the particle converges to an unknown limit position $h^\ast$ that cannot be described solely by the initial data. To regulate the final position of the particle to an arbitrary target $h_1\in(-1,1)$, we employ a proportional feedback control on the particle, $u(t)=K(h_1-h(t))$ with $K>0$, and prove that both the position error $h(t)-h_1$ and the total kinetic energy of the feedback system decay exponentially to zero. 

Les transports évoluent vers des systèmes toujours plus connectés, capables de répondre à des enjeux majeurs tels que la sécurité, la fluidité du trafic et l’impact environnemental. Les Systèmes de Transport Intelligents Coopératifs (C-ITS) reposent sur la communication en temps réel entre véhicules, infrastructures et usagers, grâce aux avancées en réseaux, capteurs et traitement de données.

In this talk, we will focus on boundaries of multiply connected Fatou components, from a topological, measure-theoretical and dynamical point of view. The main tool in our analysis is the universal covering map (and its boundary extension), which allows us to relate the dynamics on the boundary with the dynamics of the radial extension of the so-called associated inner function. This way, we can deal with all Fatou components (invariant or wandering, with all possible internal dynamics) simultaneously. 

This is joint work with G. R. Ferreira.

The multimodal nature of clinical assessment and decision-making, and the high rate of healthcare data generation, motivate the need to develop approaches specifically tailored to the analysis of these complex and potentially high-dimensional multimodal datasets. This poses both technical and conceptual challenges: how can such heterogeneous data be analyzed jointly? How can modality-specific information be identified from shared information? Variational autoencoders (VAEs) offer a robust framework for learning latent representations of complex data distributions, while being flexible enough to adapt to different data types and structures, and having already been successfully applied for latent disentanglement of multimodal (multi-channel) data. We aim at tackling multi-channel disentanglement from a causal perspective, and seek at identifying causal relationships between channels, beyond simple statistical associations. To do that, we propose Multi-Channel Causal VAE (MC²VAE), a novel causal disentanglement approach for multi-channel data, whose objective is to jointly learn modality-specific latent representations from a multi-channel dataset, and identify a causal structure between the latent channels. Each channel is projected into its own latent space, where a causal discovery step is integrated to learn the hidden causal graph. Finally, the decoder takes into account the discovered graph to predict the data. Covariate of interest can be integrated as well when available, and accounted in the causal graph structure. Extensive experiments on synthetically generated multi-channel datasets demonstrate the ability of MC²VAE in effectively uncovering the underlying latent causal structures across multiple channels, hence making it a strong candidate for real-world multi-channel causal disentanglement. Application to multi-channel data on neurodegeneration extracted from the Alzheimer’s Disease Neuroimaging Initiative highlights the existence of a biologically meaningful latent causal structure, whose pertinence is supported by multiple previous experimental and modelling work, and provides actionable insight for disease progression.

On peut associer à une transformation birationnelle du plan projectif un 'type homaloidal' qui est une suite d'entiers positifs (d;m_1,...,m_r) où d est un entier, le degré, et m_i des multiplicités. Après avoir expliqué le contexte, et comment l'algorithme de Hudson donne une structure d'arbre enraciné à l'ensemble des types homaloidaux, on donnera une estimée de la croissance du nombre de types homaloidaux de degré d. Travail en collaboration avec A. Calabri, S. Cantat, A. Massarenti et M. Mella.

L'étude des marches confinées dans un cône est un sujet central en combinatoire énumérative. En effet, ces classes combinatoires sont en bijection avec de nombreuses classes d'objets en mathématiques discrètes (permutations, arbres, cartes planaires,...) en physique statistique (polymères,...) ou encore en probabilité (marches aléatoires, mouvements browniens,...). Ces dernières années, les travaux de nombreux chercheurs en combinatoire, calcul formel, probabilité et théorie de Galois différentielle ont permis de classifier entièrement les marches à petits pas et d'élaborer des stratégies robustes pour étudier celles à grands pas.

Dans cet exposé, je montrerai comment l'étude des marches à petits pas est liée à celle de courbes elliptiques sur des corps de fonctions et à leur réseau de Mordell-Weil. Si le temps le permet, je montrerai comment on peut attaquer les problèmes de classification à grands pas grâce à la théorie de Galois classique.