For an AG code C = C(D, G) on a curve X /Fq , the structure of the hull (the intersection of a code with its dual) plays an important role in code equivalence, entanglement-assisted quantum codes, and efficient decoding. In general the hull need not be an AG code, but it becomes one when a certain divisor A associated to G and its dual divisor H is non-special. This reduces the problem to determining when effective divisors of the form A = \sum_{P in X} \max{v_P(G), v_P(H)} P are non special. In this talk I give explicit criteria for non-speciality of several classes of effective divisors of small degree on Kummer extensions. The characterization is obtained using the description of the Weierstrass semigroup at multiple points of these curves.

These results yield new families where the hull of an AG code can be written again as an AG code, and illustrate how the geometry of Kummer extensions controls the Riemann–Roch behaviour of low-degree divisors.

In this presentation, devoted to the one-dimensional nonlinear Schrödinger equation with non-zero boundary conditions at infinity, we will investigate different notions of stability in order to study certain special solutions known as travelling waves. These solutions appear in the large-time dynamics of general dispersive systems, and we will explain how such solutions, referred to as solitons, are crucial for understanding the overall behavior of solutions to these equations and how they are related to the notion of integrability of the system. Many different behaviors for these travelling waves have been highlighted, according to the shape of the nonlinearity. Nevertheless, we have been able to prove the existence of travelling waves with small momentum. Moreover, we shall dwell on the existence and uniqueness travelling waves with speed close to the speed of sound, the orbital stability of a well-prepared chain of such travelling waves, as well as the asymptotic stability of these special solutions.

English: I will briefly explain what an Algebraic Geometry code (AG code) on surface is, before talking about their duals.

French: Je vais présenter dans les grandes lignes ce qu'est un code de géométrie algébrique (AG), notamment sur les surfaces, et parler de leurs duaux.

Advances in single-cell sequencing have enabled high-dimensional profiling of individual cells, giving rise to single-cell data science and new statistical challenges. A key task is the comparative analysis of single-cell datasets across conditions, tissues, or perturbations, where traditional gene-wise differential expression methods often fail to capture complex, non-linear distributional differences. Perturbation experiments further amplify this challenge by introducing structured, high-dimensional responses that are poorly modeled by linear approaches.

We propose a kernel-based framework for differential analysis of single-cell data that enables non-linear, distribution-level comparisons by embedding data into a reproducing kernel Hilbert space. Our method quantifies differences between cellular populations through distances between mean embeddings and supports formal hypothesis testing in complex experimental designs, including perturbation studies via linear models in RKHS. The approach is robust to high dimensionality, sparsity, and noise, and is implemented in the Python package kaov, which provides visualization and interpretation tools. By offering a flexible, distribution-free alternative to classical methods, kernel-based testing facilitates the detection of subtle but biologically meaningful changes in single-cell data, enabling deeper insights into cellular regulation, disease mechanisms, and precision medicine.

A homogenized model is proposed for linear waves in 1D microstructured media. It combines second-order asymptotic homogenization (to account for dispersion) and interface correctors (for transmission from or towards homogeneous media). A new bound on a second-order effective coefficient is proven, ensuring well-posedness of the homogenized model whatever the microstructure. Based on an analogy with existing enriched continua, the evolution equations are reformulated as a dispersive hyperbolic system. The efficiency of the model is illustrated via time-domain numerical simulations. An extension to Dirac source terms is also proposed.

Time-frequency transforms are powerful signal processing tools. They enable us to analyse signals locally and extract key information, such as frequency or scaling components. The most popular transforms are the short-time Fourier transform (STFT) and the continuous wavelet transform (CWT). However, one issue with these transforms is that they have a fixed resolution; the time-frequency resolution is fully determined by the window function or the mother wavelet, regardless of the properties of the analysed signal. This can lead to a lack of precision when signals move from highly contained transient parts to highly contained harmonic parts, depending on the size of the window. This presentation introduces a solution to this problem: adaptive time-frequency transforms based on modifying the window function according to the analysed signal. We refer to this behaviour as the 'focus phenomenon' and it is linked to objects called 'focus functions'. We will define the adaptive transforms and introduce fundamental results such as the frame theorem, stability estimates, and focus function construction. We will also provide a few examples of applications of these transforms for audio processing.

Dans cet exposé, je donnerai un aperçu de la relation entre les géométries lorentziennes à courbure constante et la théorie de Teichmüller, un sujet initié par l'article fondateur de Geoffrey Mess en 1990. Dans cet esprit, je présenterai des résultats récents établissant une correspondance entre des champs de vecteurs sur le plan hyperbolique et des surfaces de type espace dans l'espace co-Minkowski, également appelé espace "half-pipe".

À l'aide de cette construction, j'étudierai le problème d'extension de champs de vecteurs définis sur le cercle au plan hyperbolique. Nous montrons que tout champ de vecteurs sur le cercle s'étend en un champ de vecteurs harmonique lagrangien sur le plan hyperbolique. J'expliquerai ensuite comment les propriétés de ces champs de vecteurs peuvent être interprétées en termes de géométrie des surfaces associées dans l'espace co-Minkowski, ainsi que du comportement de leurs bords asymptotiques.