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Soit C une courbe algébrique réelle. Un morphisme $\C \to P^1$ est séparant si la préimage des points réels de P^1 est exactement la partie réelle de C. Le degré d'un tel morphisme est nécessairement supérieur au nombre de composantes de la partie réelle de C. Mais existe-t-il des morphismes séparants de degré égal au nombre de composantes ? Dans cet exposé on présentera une obstruction à l'existence de morphismes séparants de petit degré. Il s'agit d'un travail en cours avec A. Demory et A. Toussaint, basé sur des idées de M. Manzaroli.
J'expliquerai comment la géométrie de Hilbert des convexes peut être généralisée aux corps ordonnés non archimédiens et utilisée pour étudier les propriétés à grande échelle des géométries de Hilbert réelles et leurs dégénérescences. En étudiant le cas des polytopes, nous obtenons ainsi une description explicite des cones asymptotiques des géométries de Hilbert des polytopes réels.
Travail en commun avec Xenia Flamm
Dans le contexte des champs de vecteurs analytiques réels en un point singulier, Cano, Moussu et Sanz ont introduit et étudié la notion de pinceau intégral de trajectoires en ce point afin d'obtenir des informations sur les possibles comportements dynamiques. Nous prolongeons cette approche du côté formel, en utilisant la calculabilité explicite des transséries (réticulées i.e. grid-based au sens d'Ecalle - van der Hoeven) pour la résolution d'équations différentielles. Plus précisément, étant donné un champ de vecteur formel du plan, nous introduisons la notion de trajectoire transsérielle, et fournissons une description explicite des différents pinceaux transseriels possibles. Il s'agit d'une première étape dans l'étude en cours de la même question en dim 3. Travail en commun avec Olivier Le Gal, Daniel Pananzzolo et Fernando Sanz.
Entropy and Fisher information, where the latter is also known as the entropy production, play essential roles in fields such as physics, biology, and information theory, and they are also powerful tools in mathematics, for instance, in analyzing large time behavior, regularity, and asymptotic properties of solutions. In this talk, we investigate the time evolution of Fisher information for nonlinear diffusion equations on bounded domains with Neumann boundary conditions, extending classical results for the linear heat equation and the porous medium equation on the whole space. In particular, we introduce an alternative formulation of one-dimensional nonlinear Fisher information that reveals its time monotonicity. As an application, the global existence of solutions to the one-dimensional critical quasilinear fully parabolic Keller–Segel system with nonlinear diffusion and nonlinear sensitivity is studied. This is based on joint work with Tomasz Cieślak (IMPAN, Poland) and Kentaro Fujie (Tohoku University, Japan).
Tout polynôme multivarié P(x1,…,xn) peut s’écrire comme une somme de monômes, c’est-à-dire une somme de produits de variables et de constantes. En général, la taille d’une telle expression correspond au nombre de monômes ayant un coefficient non nul. Que se passe-t-il si l’on ajoute une autre couche de complexité et que l’on considère des expressions sous la forme de sommes de produits de sommes (de variables et de constantes) ? Dans ce cas, il devient difficile de démontrer qu’un polynôme donné P(x1,…,xn) ne possède pas de petites expressions de ce type. Nous présenterons le contexte de cette question, ses liens avec la complexité booléenne classique ainsi que quelques résultats de base dans ce domaine. Enfin, nous exposerons certains résultats récents sur ces objets.
Many natural phenomena, such as volcanic eruptions, involve complex multi-phase flows. These flows often feature a mix of compressible and incompressible behaviors, making their modeling particularly challenging. A widely adopted framework is provided by the Baer–Nunziato equations for compressible two-phase flow. We present a semi-implicit solver for this system, featuring a novel linearly implicit discretization for both the pressure fluxes and the relaxation source terms, while the nonlinear convective terms are treated explicitly. This formulation leads to a CFL-type stability condition on the maximum admissible time step only based on the mean flow velocity, rather than on the sound speed of each phase, so that the novel scheme works uniformly for all Mach numbers. Implicit terms are discretized with central finite differences on Cartesian grids, avoiding artificial numerical diffusion in the low-Mach regime, whereas shock-capturing finite volume schemes are employed for the convective fluxes to guarantee robustness at high Mach numbers. The discretization of nonconservative terms preserves moving equilibrium solutions, making the method well-balanced, while the asymptotic-preserving property ensure consistency in the low-Mach limit of the mixture model. Second-order accuracy in space and time is achieved through the IMEX time-stepping scheme combined with TVD reconstruction. The proposed method is validated through a series of benchmark problems spanning a wide range of Mach numbers, demonstrating both its accuracy and robustness.
Dans cet exposé nous évoquerons au moins une analogie structurelle entre le schéma des arcs et les opérateurs différentiels logarithmiques dans le cas des singularités des courbes planes homogènes.
This talk is an invitation to the algebraic side of category theory. The starting point is the empirical observation that algebraic structures of various sorts are closed under cartesian product. I'll explain how category theory makes this into an abstract result, namely: forgetful functors from monad algebras create binary products.
Constructive foundations have for decades been built upon realizability models for higher-order logic and type theory. However, traditional realizability models have a rather limited notion of computation, which only supports non-termination and avoids many other commonly used effects. Nonetheless, many recent works have shown how realizability models could benefit from side effects to provide computational interpretation for logic principles. In earlier work with Cohen and Tate [1], we addressed the challenge of finding a uniform and generic algebraic framework encompassing (effectful) realizability models, introducing evidenced frames for this purpose. These structures not only enable a factorization of the usual construction of a realizability topos from a tripos—evidenced frames are complete with respect to triposes—but are also flexible enough to accommodate a wide range of effectful models.
We pursued this along two main directions: - first, by extending the syntactic (typed) approach to realizability, following Kreisel’s tradition, where propositions from a ground logic (e.g., HOL) are translated into specifications and typed programs (realizers) that satisfy them. We introduce EffHOL [2], a new framework that expands syntactic realizability. It combines higher-kinded polymorphism—enabling typing of realizers for higher-order propositions—with a computational term language using monads to represent and reason about effectful computations. - second, by generalizing the traditional notion of Partial Combinatory Algebras (PCAs), which underpins classical realizability models. To better internalize a broad spectrum of computational effects, we propose the concept of Monadic Combinatory Algebras (MCAs) [3], in which the combinatory algebra is structured over an underlying computational effect captured by a monad. As we shall see, MCAs provide a smooth generalization of traditional PCA-based realizability semantics.
10h - 10h45 Gladys Narbona Reina "Asymptotic derivation of a depth-averaged model for bedload sediment"
10h45 - 11h30 Matthias Renaud "sedExnerFoam1.0: A 3D Exner-based sediment transport and morphodynamics model"
11h30 - 12h15 Maxime Kaczmarek "Ecoulements et transports sédimentaires en zone de swash"
14h-14h45 Yen Chung Hung "A New Breaking Wave Model and Preliminary Results on Morphodynamic Coupling"
14h45-15h30 Remi Chassagne "A frictional–collisional model for bedload transport based on kinetic theory of granular flows: discrete and continuum approaches"
The dynamics of the ocean can be described by the incompressible Euler equations, completed by a transport equation on the density. For large scale dynamics, the ocean is assumed stratified, meaning that it is layered by sheets of constant density, namely isopycnals. The aim of this talk is twofold. First, we present the physical context an describe the main result, that is the study of perturbations around shear flows, which are natural equilibria of this system. Second, we present a particular point of the proof, namely the use of Alinhac’s good unknown.
La géométrie lipschitz est une branche de la théorie des singularités qui étudie un germe d'espace analytique complexe $(X,0)\subset(\mathbb{C}^n,0)$ en le munissant d'une de deux métriques : sa métrique externe, induite par celle standard de l'espace ambiant, ou sa métrique interne, obtenue en mesurant la longueur des chemins sur $(X,0)$. Je ferai un panorama de certains résultats récents obtenus dans des collaborations ou travaux en cours avec André Belotto et Anne Pichon, me focalisant en particulier sur la structure métrique interne des surfaces. Cette étude fait intervenir l'entrelac de la singularité $(X,0)$, ainsi que deux de ses avatars, l'un non archimédien et l'autre logarithmique.
We consider the problem of reconstructing a graph G from a query Q which, for every k-subset of vertices S, provides some information Q(G)(S) about the induced subgraph G[S]. The vertices are labelled from 1 to n, so reconstruction up to isomorphism is not sufficient, we need to know which label is where. We have studied the case k=3. Given a query Q on triples, we are interested in two things: a structural characterization of all graphs G that are uniquely reconstructible from the function Q(G) (i.e. such that Q(H)=Q(G) if and only if H=G), and a polynomial-delay enumeration algorithm of all graphs that are consistent with some input query answers. In 2023, Qi and Bastide et al. respectively have managed this for the connectivity query (meaning that, for every triple S, Q(G)(S) indicates whether G[S] is connected). We have obtained the same results for all 13 other non-trivial queries on triples. This presentation will go into details of a select few of these queries. Joint work with Hoang La, Raphaëlle Maistre, Matthieu Petiteau and Dimitri Watel.
The accurate modelling of two-phase flows is a very active research field with significant implications for space applications, defense technologies, and geophysical phenomena. The derivation of models capable of describing the wide range of scales or the non-equilibrium effects that naturally arise in such flows, especially in the case of compressible high-velocity flows involving shocks, is still an open question.
The main goal of this presentation is to present a modelling strategy that allows for a systematic derivation of two-phase models within a variational framework. The strategy relies on two stages. First, the non-dissipative part of the model is derived by means of the Stationary Action Principle. A generic Eulerian framework, compatible with an underlying Lagrangian description, will be presented. The framework ensures that the resulting model admits a supplementary conservation law thus providing a coherent thermodynamic structure. The second stage introduces dissipative effects in accordance with the Second Principle of thermodynamics.
We will illustrate the application of this framework through several examples. After an introductory case to build familiarity, we will discuss the derivation of multi-fluid models which include a coupling between the relative velocity and velocity fluctuations. We will present a two-scale model which allows for the separated-to-disperse phase transition based on an interfacial energy budget that occurs in atomization processes. Finally, as a follow-up on the presentation of disperse two-phase flows, we will show how different levels of descriptions for the spray can naturally be integrated in the two-scale model.
The modeling and simulation of multiphase reacting flows covers a large spectrum of applications ranging from combustion in automobile and aeronautical engines to atmospheric pollution as well as biomedical engineering or pyroclastic flows. In the framework of this seminar, we will mainly focus on a disperse liquid phase carried by a gaseous flow field which can be either laminar or turbulent; however, this spray can be polydisperse, that is constituted of droplets with a large size spectrum; we have to deal with coalescence, break-up, droplet trajectory crossing and reduced order model for turbulent flows. Thus, such flows involve a large range of temporal and spatial scales, which have to be resolved in order to capture the dynamics of the phenomena and provide reliable and eventually predictive simulation tools. Even if the power of the computer resources regularly increases, such very stiff problems can lead to serious numerical difficulties and prevent efficient multi-dimensional simulations. The purpose of the presentation is to introduce to the Eulerian modeling of polydisperse evaporating spray for various applications, that is the disperse liquid phase carried by a gaseous flow field is modeled by "fluid" conservation equations. Such an approach is very competitive for real applications since it has strong ability for optimization on parallel architectures and leads to an easy coupling with the gaseous flow field resolution. We will show that all the necessary steps in order to develop a new generation of computational code have to be designed at the same time with a high level of coherence: mathematical modeling through Eulerian moment methods, development of new dedicated stable and accurate numerical methods, implementation of optimized algorithms as well as verification and validations of both model and methods using other codes and experimental measurements.
The Aronson-Benilan inequality, well known for the porous media equation $\partial_t \rho - \Delta \rho^m = 0$ provides a lower bound on the Laplacian of pressure: $\Delta \rho^{m-1} \geq C$. In this presentation, I will show that this estimate remains valid for another equation: the Keller-Segel system, which is a porous medium equation to which we add an aggregation term. Among other things, this provides a new demonstration of global existence for this system. I will focus on the case in dimension 2, with a linear diffusion and with a small initial mass but the result can be extended for any dimension, with the critical diffusion exponent and up to and including the critical mass. This work is in collaboration with Alejandro Jimenez-Fernandez (Oxford) and Filippo Santambrogio (Lyon).
Ordinals numbers, or transfinite numbers, are a set-theoretic notion allowing to extend the ordered structure of natural numbers beyond infinity. In fact, the collection of ordinals is bigger than any set, and thus contains more elements than we could ever describe with finite sentences.
Still, one can wonder how many ordinals can be expressed with finite sentences. Ordinal notations are systems of syntax that, similar to how one could describe mathematical objects with words, describe with formal terms a given set of ordinals.
In this talk, I will present ordinal notations, an example of their use and describe the notation of Buchholz's psi functions, allowing to define a lot of important countable ordinals.