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Résumé : Après un bref survol des avancées récentes sur le problème de comptage des points rationnels au voisinage de courbes dans la plan, nous nous intéresserons tout particulièrement au cas du mouvement brownien. Des résultats récents en collaboration avec Evgenyi Zorin et Volodymyr Pavlenkov permettent en effet de résoudre un problème posé par Sprindzuk (1979).

Résumé : Cette présentation porte sur la simulation des milieux granulaires denses composés de particules macroscopiques en suspension dans un fluide visqueux, comme les boues. Ces systèmes présentent des comportements macroscopiques extrêmement complexes, incluant des phénomènes tels que les instabilités de concentration et les blocages d’écoulement. Au cœur de ces comportements se trouvent les interactions physiques microscopiques entre particules voisines, en particulier les effets de lubrification induits par le fluide interstitiel, ainsi que les contacts solides. Ces interactions sont essentielles non seulement pour comprendre les écoulements granulaires immergés, mais aussi dans le cadre plus large des suspensions de particules. Malgré leur importance, l’influence de ces mécanismes microscopiques sur la dynamique globale du système reste mal comprise. Une difficulté majeure provient du caractère singulier de ces effets lorsque la distance entre particules tend vers zéro, introduisant des singularités temporelles. Nous présenterons dans cet exposé de nouveaux modèles de type dynamique des contacts, prenant en compte la lubrification ainsi que le contact solide, avec ou sans friction. Nous montrerons que l’on peut construire, à partir de ces modèles, des schémas numériques stables et robustes. Ces schémas mènent, à chaque itération en temps, à la résolution d’un problème d’optimisation sous contrainte. Nous illustrerons l’exposé par des résultats numériques.

Résumé : Standard neural network optimization treats all parameters equally, frequently ignoring the inherent separable structure of models that are linear in output weights and nonlinear in hidden weights. This presentation explores Variable Projection (VarPro), a method that explicitly eliminates these linear variables, reducing the task to optimizing only the hidden parameters. Consequently, this reformulates the process into optimizing a subspace on the Grassmannian Manifold, a geometric landscape mathematically proven to be free of spurious local minima.

To resolve practical challenges such as rank deficiency—which occurs when the feature matrix becomes singular during training—a regularized projector manifold is introduced. This regularization ensures the continuous smooth mapping of subspaces while preserving benign critical points. The theoretical framework is validated through the Deep VarPro algorithm, demonstrating robust numerical efficiency in capturing oscillatory behaviors in both low and high-dimensional regressions, as well as successfully solving the heat equation using Physics-Informed Neural Networks (PINNs).

Résumé : Let f, g1, . . . , gd : Rd −→ R be continuous functions. When the functions g1, . . . , gd are of class C1, identifying the d-form f dg1 ∧ · · · ∧ dgd with the continuous function f det(dg) allows one to define the integral ∫_Ω f dg1 ∧ · · · ∧ dgd = ∫_Ω f(x) det(dg(x)) dx, for a bounded Borel set Ω ⊂ Rd. If the functions g1, . . . , gd are not differentiable, it is not clear how to give a meaning to the object f dg1 ∧ · · · ∧ dgd, nor even how to define certain integrals of the form ∫ f dg1 ∧ · · · ∧ dgd. Under regularity assumptions of the type introduced by Züst, we adopt a distributional viewpoint to give a meaning to the object f dg1 ∧ · · · ∧ dgd itself. This approach allows one to define the corresponding integrals, by duality, over more general domains, including sets with fractal boundaries, and to extend integrability results previously obtained by Züst, Alberti–Stepanov–Trevisan, and Bouafia. This talk is based on the preprint available at https://arxiv.org/abs/2510.20427.

Résumé : Les connexions du type Painlevé V sont une classe particulier de connexions irrégulières de rang deux sur la sphère de Riemann, avec un précise diviseur des poles. Leur espace des modules est doté d'un feuilletage en courbes dont les feuilles contiennent les connections avec même monodromie. Ce feuilletage est induit par une équation différentielle classique, l'équation Painlevé V. Le but de cet exposé est de présenter une compactification de l'espace des modules qui nous permettra d'étudier le comportement asymptotique des feuilles isomonodromiques sur les composants du bord.

Résumé : Continuing from last year, on my side quest into the land of cobordisms, I come to the question of why we study cobordisms? Interestingly, cobordisms are closely related to maps between spheres. In this talk I will speak about what are homotopy and homology groups and how they are really just two definitions of holes. I will speak about the homotopy groups of spheres and how studying them we may come to framed cobordisms. Finally, should there be time I will explain how the homotopy of spheres and cobordisms may motivate more advanced homotopy topics such as stable homotopy and rational homotopy.

Résumé : In this presentation, I will introduce a hybrid finite volume method on general polygonal and polyhedral meshes for the modeling of an anisotropic cross-diffusion system arising from a mesoscopic stochastic process describing diffusion in solids, under a size-exclusion constraint.
This system possesses an entropy structure, which is exploited to define the numerical scheme in terms of (discrete) entropy variables, and is thus preserved at the discrete level.
This structure makes it possible to prove the existence of nonnegative discrete solutions satisfying the size-exclusion constraint, as well as mass conservation, and to establish the convergence of the scheme under mesh refinement.
To the best of our knowledge, this is the first work proposing and analyzing a structure-preserving hybrid finite volume scheme for anisotropic cross-diffusion systems on general polygonal and polyhedral meshes.

The preprint associated with this presentation is:
V. Ehrlacher, A. Massimini, J. Moatti. Structure-preserving hybrid finite volume scheme for an anisotropic cross-diffusion system, 2026. Preprint, HAL : hal-05589824

Résumé : In nuclear fusion, simulations are essential for understanding and controlling tokamak instabilities, phenomena that can severely damage reactors. Neural approaches for solving partial differential equations (PDEs) are gaining interest due to their mesh-free nature, flexibility, and scalability. These methods rely on neural networks as approximation spaces instead of classical polynomial bases, and this work investigates the efficiency of several neural techniques applied to plasma simulations. We first study stationary elliptic equations, with particular attention to the Grad–Shafranov equation, solved using Physics-Informed Neural Networks (PINNs). We then address time-dependent problems such as anisotropic diffusion, relying on adapted neural schemes, including Discrete PINNs and Neural Galerkin methods. In both cases, the Natural Gradient method is employed to significantly accelerate and stabilize the optimization process during training.

Résumé : TBA

Résumé : In this talk we discuss the convergence of adaptive FEM for strongly monotone nonlinear partial differential equations arising from the minimization of a convex energy functional. Rather than focusing on specific algorithmic implementations, we establish a generalized theoretical framework that identifies sufficient conditions for an adaptive scheme to contract the energy difference at an optimal rate relative to the number of degrees of freedom. The presented abstract point of view is subsequently shown to include many existing error estimation approaches (local residuals, flux equilibration, linear residual liftings, energy descent), thereby providing a unified proof for their optimal convergence. A key contribution of our recent work is the introduction of computable local equivalence factors yielding a computable a posteriori bound on the contraction rate which can then be used for an improved local refinement procedure. Finally, we will discuss technical challenges posed by higher-order discretizations and show how the arising oscillations can be managed through local higher-order flux-equilibration.

Résumé : Thèse de doctorat en VAE.

Résumé : The Mordell-Weil group of abelian varieties over complex function fields can be studied using techniques from differential geometry and algebraic topology. Specifically, each rational point corresponds to an algebraic section of the associated abelian scheme. While such sections admit local logarithms, the obstruction to the existence of a global logarithm is encoded in a lattice known as the relative monodromy of the section. Intriguingly, this object appears to be deeply linked to the arithmetic properties of the section itself. For example, under certain hypotheses, the non-triviality of the relative monodromy is equivalent to the section being non-torsion. Furthermore, it is conjectured that the rank of the relative monodromy relates to the dimension of the minimal abelian subscheme containing the section’s image. In this talk, I will present partial results toward this conjecture, along with applications to transcendence problems. This is joint work with F. Tropeano (Università Roma Tre), extending earlier work of Corvaja and Zannier in the setting of elliptic surfaces.

Résumé : TBA

Résumé : TBA