Maël Chantreau, IDP-Tours: Isolated systems and asymptotically flat space-times Room: F123 In physics, the notion of isolated system is a crucial one. However, the definition of this concept in the context of general relativity contains some subtleties. One generally proceeds by asking the boundary of the space-time (obtained through conformal compactification) to share some similarities with the one of the Minkowski space-time, $\textit{i.e.}$ the space-time of an empty and flat universe. This powerful idea from Penrose, which dates back to the 60's, enables computations at infinity and has been extensively used in the litterature. However, while it works very well on some part of infinity called null infinity, it is unable to describe correctly some other parts, space and time-like infinity. After having reviewed Penrose's formalism and the structures it induces on null-infinity, we will see how one can construct analogues of these structures on space and time-like infinity.

Paguiel Hossie, IDP-Orléans: Multifluid modelling of intestinal transport under the effect of intestinal peristalsis Room: F123 We have developed a mathematical and asymptotic model to simulate the transport of an intestinal mixture in a deformable cylindrical conduit representing the colon. The radius of the colon varies in both time and space, mimicking peristaltic movements. The mixture consists of mucus, luminal fluid, dietary fibres and bacteria, whose volume fractions change under the influence of variable viscosity, peristaltic wall motion and physiologically relevant boundary conditions. The governing equations combine incompressible Stokes equations coupled to transport equations for each component, where the effective viscosity depends strongly on the mucus concentration. We consider physiological boundary conditions: secretion of mucus by the wall, absorption of luminal fluid, and confinement of fibres and bacteria... After non-dimensionalisation and asymptotic expansion in the small radius versus length regime, we derive a reduced system that captures the first-order behaviour of the mixing dynamics. The analysis reveals how wall deformation and mucus-induced viscosity gradients shape the axial and radial flow profiles, directly influencing the efficiency of intestinal content transport. These simulations explore the impact of different wall deformations (e.g., peristaltic patterns) on the spatial distribution and transport efficiency of the mixture. However, in this presentation, we only focus on a preliminary stage of this work in which only the mucus component is considered. These results lay the groundwork for future extensions including other mixture components precised above.

Alandra Zakkour, LMA: Généralisations de la q-ex-gaussienne par la méthode de Tsallis Room: F123 Bien que la distribution ex-gaussienne nous permette de mieux analyser les temps de réaction afin de proposer une analyse approfondie du traitement cognitif humain, dans cette présentation, je souhaite introduire l'utilisation du calcul quantique à la place du calcul classique dans cette distribution. Cela nous permet de généraliser la distribution q-ex-gaussienne et de comparer les résultats afin de déterminer si le calcul quantique améliore notre modèle ou non.

Jad Abou Yassin, IDP-Tours: Catalan combinatorics from the perspective of Coxeter sortable elements of the symmetric group Room: F123 A large range of mathematical objects are enumerated by the famous $\text{Catalan}$ numbers : triangulation of polygons, non-crossing partitions, number of ways to correctly parenthesize an expression, $\text{Dyck}$ paths, binary trees, monotonic lattice paths, ... I will introduce a less famous object that is also enumerated by the $\text{Catalan}$ numbers : the $\text{Coxeter}$ sortable elements of the symmetric group. They are specific permutations that are defined $\textit{a priori}$ in a way completely unrelated to the $\text{Catalan}$ combinatorics. I will show explicit bijections from the $\text{Coxeter}$ sortable elements to the non-crossing partitions and to binary search trees and motivate their importance. Then, if time is permitted, I will present a generalization of the binary trees that emerges from the choice of specific $\text{Coxeter}$ elements and discuss.

Rim Mheich, LMA: Comportement Asymptotique du Système Couplé d'Allen-Cahn/Cahn-Hilliard avec Terme de Prolifération Room: F123 Dans cette présentation, nous étudions les équations couplées d'Allen-Cahn/Cahn-Hilliard avec un terme de prolifération, permettant de modéliser la croissance de tumeurs cancéreuses et d'autres entités biologiques. Nous nous concentrons sur l'établissement de l'existence, de l'unicité et de la régularité des solutions, ainsi que sur l'analyse de leur comportement asymptotique, en accordant une attention particulière à l'existence d'attracteurs de dimension finie. Le système est considéré sous des conditions aux limites de Dirichlet, et nous introduisons des hypothèses sur le terme de prolifération afin de garantir la dissipativité.

Emma Grugier, IDP-Orléans: Small eigen values of non-reversible metastable diffusion processes with Neumann boundary conditions Room: F123 Let $\Omega \subset \mathbb{R}^d$ be a bounded smooth domain and $b : \Omega \rightarrow \mathbb{R}^d$ be a smooth vector field. We focus on the associated overdamped Langevin equation : $$\dot{X_t}=b(X_t) + \sqrt{h} \dot{B_t} $$ in the low regime temperature where $h \rightarrow 0$ and in the case where $b$ admits the decomposition $b = - \nabla f - l$ with : $$\bullet ~ l\text{ a smooth vector field ; }$$ $$\bullet~ f\text{ a Morse function on }\overline{\Omega}\text{ admitting several local minima ; }$$ $$\bullet~ \nabla f \cdot l =0\text{ on }\overline{\Omega}.$$ In this framework, minima of the function $f$ correspond to metastable states for this Langevin dynamics. In this context, we analyse the spectrum of the infinitesimal generator of the dynamics : \begin{align*} L_h = - \frac{h}{2} \Delta + \nabla f \cdot \nabla + l \cdot \nabla \end{align*} with Neumann boundary conditions. In this case, moving particles will remain trapped inside the domain. More specifically, we will consider additional hypotheses ensuring that the measure $e^{-\frac{2f}{h}}dx$ is invariant.

Zeina Rammal, LMA: Étude mathématique et numérique d'une navigation optimisée dans des eaux stratifiées. Room: F123 Lors de la navigation dans des eaux stratifiées, des ondes peuvent apparaître à l'interface entre deux couches de fluide de densités différentes. Ces ondes, appelées ondes internes, se propagent et peuvent influencer les performances du bateau en générant une résistance à son avancement, connue sous le nom d'eaux mortes. Le phénomène est modélisé sous forme d'équations aux dérivée partielle résolue dans l'espace de Fourier. L'équation sera étudiée et un schéma numérique sera introduit. En particulier, nous mettrons en évidence l'existence d'une vitesse critique de navigation qui joue un rôle centrale sur la propagation des ondes à l'interface entre les deux fluides. Des simulations ont été réalisées et seront présentées.

Ibrahim Hmamou, IDP-Orléans: Fluid-particle suspension flows dominated by sedimentation. Room: F123 Sedimentary flows involve different types of fluids/sediments, with varying particle concentrations. The dynamics of these extreme events is very difficult to observe: the deposit is the only quantity that can be measured. We aim to model these flows in order to predict their duration, the distance covered, and their deposition geometry. In this talk, we derive a simplified model from the Navier-Stokes equations, with several thin layers, of shallow water type. Thanks to an approximation based on a finite volume scheme, this model can be compared to real data measured in the laboratory.

Greta Lamonaca, IDP-Orléans: Mean Field Models in Spatial Ecology Room: F123 In this presentation, we will qualitative explore the critical role of human decisions in biodiversity decline, focusing on the problem of fishing in a bounded domain using Mean Field Game (MFG) models in Spatial Ecology. We will combine reaction diffusion equations and MFG theory to model and analyze two fishing systems. We will focus on existence, uniqueness and long-time behavior.

Mathilde Rousselot, LMA: Détection de rupture dans un processus autorégressif multivarié sous hypothèse de rang faible. Room: F123 Dans ce court exposé, je commencerai par introduire la notion de statistique en grande dimension avec un rappel sur les tests statistiques. On observera ensuite les réalisations d’un processus auto-régressif multivarié (VAR) de dimension $p$ défini par l’équation $X_{t+1}= \Theta X_{t} +Z_{t}$ où $\Theta$ est une matrice réelle de taille $p$ et la suite $(Z_t)_t$ est le bruit blanc gaussien. On suppose que la dimension $p$ de $(X_t)_t$ est assez grande. Sous les hypothèses de rang faible de la matrice $\Theta$, le but est de prédire si la matrice $\Theta$ subi un changement au cours du temps. Pour détecter cette rupture, on propose un test statistique dont la performance est vérifiée à l’aide des simulations numériques.

Dimitri Faure, IDP-Orléans: A quick introduction to singular stochastic PDEs Room: F123 The same way one can randomize an ODE by adding a finite-dimensional noise term, one can randomize a PDE by adding an infinite-dimensional noise term. Stochastic PDEs are, however, most of the time not well-posed due to the presence of ill-defined non-linear terms in their expressions. To have a fruitful notion of solution for singular stochastic PDEs, one needs to add diverging counter-terms to the equations, a procedure called renormalization. In this presentation, we will give a quick introduction to all these concepts.

Chloé Weckel, IDP-Tours: Spatiotemporal modeling of signaling pathways: impact of endosomal compartmentalization and application to gonadotropin receptors Room: F123 Cells communicate with each other by sending extra-cellular ligands such as hormones. The ligands of interest are reproductive hormones which, once recognized by their receptors at the plasma membrane, trigger a signaling cascade within the cell. These receptors are internalized in different compartments called endosomes, which produce a cellular response with a different physiological role. This notion of compartmentalization is mathematically modeled using two approaches: a structured model based on systems of ordinary differential equations, and an individual-centered model using piecewise deterministic Markov processes. This modeling enables us to study the effect of internalization and recycling on the quantity of cellular response. The ultimate aim of the project is to further characterize the pharmacological action of ligands, contributing to a better understanding of gonadotropin signaling pathways.

André Lapuyade, LMA: Généralisation d'un théorème de Kawakita sur les contractions divisorielles Room: F123 On partira d'un théorème de Kawakita qui classifie les contractions divisorielles en dimension 3, puis on expliquera comment on essaie de faire grandir la dimension en se plaçant de la cadre particulier des variété horosphérique.

Marion Meutelet, IDP-Tours: Nonlinear reaction-diffusion problem with membrane conditions Room: F123 The heat equation is a classical linear partial differential equation. I will illustrate its diffusive and regularizing effects through numerical simulations. To move toward the framework of my subject, I will then introduce an interface within the domain, which splits the medium into two subdomains. At this interface, appropriate transmission conditions must be imposed to describe heat transfer between the regions. In particular, I will focus on the Kedem-Katchalsky conditions, which model semi-permeable membranes and are widely used in biological contexts. Next, I will discuss a nonlinear extension of the heat equation, motivated by biological models, especially those describing tumor invasion through biological tissues. This leads to a reaction-diffusion system with nonlinear interface conditions. We will see that the study of an associated elliptic (stationary) problem provides results, such as existence and uniqueness of a weak solution, that can be extended to the evolution problem thanks to nonlinear semigroup theory.

Anwar El Rhirhayi, IDP-Orléans:Langevin Approach to the Landau Equation through Large Deviations. Room: F123 Our goal is to study the Landau equation through the Langevin approach. We will show that this method recovers the same large deviation Hamiltonian as the one derived by Landau and Boltzmann. Then, we will attempt to connect it with Rostoker's principle by studying a three-body system. Our initial objective is to demonstrate the equivalence of both methods in the limit of an infinitesimal scattering time. Finally, we aim to investigate energy conservation within the Langevin framework. - References: https://arxiv.org/abs/2410.12079 https://arxiv.org/abs/2002.10398 https://arxiv.org/abs/2101.04455

Tom Maitre, LMA: Inégalités de concentration non commutatives Room: F123 La concentration de la mesure est au cœur de la recherche actuelle, en raison de la richesse des informations qu’elle fournit en géométrie des espaces de Banach et en analyse fonctionnelle. Après un bref rappel de quelques inégalités classiques, je présenterai mes premiers résultats portant sur des sommes d'intégrales stochastiques à valeurs matricielles. Enfin, je présenterai mon projet de recherche pour l'année à venir, qui vise à établir des inégalités de concentration analogues dans le cadre des processus stochastiques libres.