Lundi après midi
Noisette Florent - Low regularity solution of the Euler equation with entering flux
During this talk, i will shorlty describe the Euler equation. Then, i will discuss what is an energy method, as well as its downfall in the case of an open domain (when there is flux entering and exiting the domain). Finally, the goal is to introduce Kreiss symetrizer and hidden boundary regularity.
Gouthier Bianca - Infinitesimal birational actions on curves
I will start by talking about a conjecture of Dolgachev on the Cremona group in positive characteristic, which motivates our interest in studying the actions of infinitesimal group schemes. I will then give a flavour of what an infinitesimal group scheme is and conclude with a result about infinitesimal actions on curves.
Lanabi Ibtissem - Behavior of the Compressible Euler equations in the low Mach number limit
Cosserat Oscar - Dynamique tourbillonnaire : du cas planaire aux trois dimensions
La dynamique tourbillonnaire de n nourbillons sur un plan est bien connue : son système d'équations, ses propriétés géométriques... Je les rappellerai et donnerai quelques problèmes ouverts associés. Un domaine actif est la recherche d'un analogue en dimension 3. Si le temps le permet, j'aborderai quelques pistes de la littérature.
Untrau Théo - Equirépartition de certaines sommes d'exponentielles courtes
Dans cet exposé, je commencerai par présenter quelques applications des sommes d'exponentielles à des problèmes concrets de théorie des nombres. Ces applications servent de motivation pour une étude plus fine des propriétés de ces sommes. Je présenterai quelques résultats d'équirépartition de sommes d'exponentielles obtenus pendant ma thèse, et je montrerai des images qui illustrent ce type de phénomène.
Robert Florian - Impact de l'organisation interne des tissus hépatiques tumoraux
La microscopie électronique à balayage en série (SBF-SEM) permet d'obtenir une reconstruction digitale 3D d'un tissue par un empilement d'images à l'échelle micrométrique. Cela permet notamment de visualiser des capillaires sanguins, des zones hémorragiques, des cellules, leur noyaux et de nombreux autres
organelles. Grâce à cette technologie nous souhaitons étudier l'architecture tridimensionnelle de tumeurs du foie. En étudiant un stack de 246 images, de précédents travaux ont mis en avant certains paramètres bio-architecturaux et ont établi des hypothèses sur l'architecture de cette tumeur (alignement des
cellules avec un capillaire sanguin, proximité avec ce dernier des cellules les plus volumineuses...). En utilisant les segmentations de ce précédent stack, nous poursuivons cette étude avec dix stacks de 1000 images (2 zones différentes de 5 tissus /patient derived xenografts /différents). Cela nous permettra de mener une étude comparative inter et intra-tumorale. Le deuxième axe de cette thèse est trouver des marqueurs bio-architecturaux caractérisant le type cellulaire (tumorale, endothéliale ou immunitaire), puis d'automatiser cette classification en utilisant les algorithmes d'intelligence artificielle.
Monti Martina - Calabi-Yau manifolds quotient of Abelian manifold: the 3-dimensional cases
It is possible to construct Calabi-Yau manifolds as quotient of complex tori via certain action of finite groups. We are interested in the study of the geometric properties behind this construction, in particular if we can deduce geometry of the Calabi-Yau manifold from the geometry of the complex tori. In this talk I will focus on the 3 dimensional case, presenting my results about the geometry of Calabi-Yau 3-folds as quotient of Abelian 3-folds.
Zorkot Ahmad - Approximation for mean field games
This talk is devoted to the numerical approximation of mean field games problems. We consider two cases : a first order problem, i.e the diffusion is null, and a second order problem. For the first one, we propose a Lagrange-Galerkin method to approximate the solution of a class of continuity equation, coupled with a semi-Lagrangian discretization of an Hamilton-Jacobi-Bellman equation, in order to obtain an approximation method for a first order Mean Field Games system. We prove a convergence result and we show some numerical simulations. For the second order case, we consider a Newton iterations approach for the continuous mean field game system, and we prove that the rate of convergence is quadratic. Finally we propose a semi Lagrangian scheme to approximate the continuous Newton iterations, and we show some numerical results.
Mardi matin
Zelada Rodrigo - Shape Optimization for heat exchangers with ventcell conditions
In this talk, we will study equations with non-standard transmission problems (discontinuous at the interface, Fourier-Robin and/or Ventcell conditions).
If the coefficients in the interface boundary conditions are too small (or too high depending on the case), the problem becomes ill-conditioned, hence we propose a Nitsche method to handle it. Then, we consider some models where we want to minimize/maximize some cost functional, with the interface as the unknown.
We compute the shape derivatives using the Hadamard method and we use the level-set method to capture the boundary changes. To solve the standard equations we have used FreeFem++ and to solve the non-standard equations we have implemented our own code. Finally, we will give an application to heat exchangers (that are devices that allow the heat exchange between two or more fluids without mixing of fluid), maximizing the heat exchanged and keeping the drop pressure bounded. We consider the framework of two fluids separated by a thin layer (the wall of the pipes) and we perform and asymptotic development in order to obtain a domain that does not depend on the thickness of the layer.
Hong Haojie - On primes of a class of algebraic numbers in quadratic fields
Let r be an algebraic number of degree 2 and not a root of unity. We can show that there exists a prime ideal P of Q(r) satisfying $\nu_P(r^n-1)\ge 1$, such that the rational prime underlying P grows quicker than n
Wang Yunlei - Spectral inequalities and their application to control of PDEs
Spectral inequalities are quantitative estimates for finite combination of eigenfunctions of differential operators. These inequalities can be applied to PDEs and give the corresponding controllability results. We give a short introduction about spectral
inequalities of particular type operators and their controllability results.
Carrel Tristan - Chimiométrie pour l'analyse du Skydrol
L’analyse de grande quantité de données est un domaine mathématique en expansion qui suscite de l’intérêt dans divers domaines tels que le renseignement, la médecine, l’écologie, la sociologie, etc. Des méthodes mathématiques ont été créées afin de résoudre ce problème tout en transformant les données en de nouvelles informations, dont leur nombre est réduit. Lorsque ces méthodes sont appliquées à des données psycho-chimiques, elles font partie de la chimiométrie. Dans notre cas, nous effectuons une étude de spectre infrarouge du Skydrol. Les données que l’on récupère par la spectroscopie sont donc des points discrétisés d’une fonction. Il nous est alors nécessaire d’avoir une méthode d’analyse de données pour pouvoir déterminer certaines informations physiques et chimiques, invisibles à l’œil nu sur le spectre d’un échantillon de Skydrol. Nous parlerons tout d’abord des méthodes existantes et classiques, l’Analyse en Composantes Principales (ACP) et la PLS (Partial Least Squares), puis d’une méthode plus spécifique : l’ACP pénalisée. Nous comparerons leurs applications sur le cas du Skydrol. L’analyse topologique de données est un domaine en expansion que nous pouvons également appliquer dans la chimiométrie. Nous verrons une nouvelle méthode de lissage qui s’adapte à la forme des données étudiées.
Saba Chadi - The littlewood problem and non-harmoic Fourier series
In this short talk, i will give a direct quantitative estimate of L 1 norms of nonharmonic trigonometric polynomials over large enough intervals.
Gouasmi Aimene - Flux reconstruction for interface problems and applications to a posteriori error analysis
The importance of reconstructing conservative local fluxes from a primal discrete solution is widely recognized in the literature. One major application is in a posteriori error analysis: the difference between the numerical flux and a recovered equilibrated flux provides an a posteriori error indicator, with a reliability constant equal to 1, which is further used in adaptive mesh refinement.
In this talk, we consider an elliptic diffusion problem with discontinuous coefficients, approximated by conforming and nonconforming finite elements of arbitrary polynomial degree. We recover a conservative flux in the Raviart-Thomas space following the approach proposed by Becker, Capatina and Luce in 2016 for the Poisson equation. The idea is to introduce an equivalent hybrid mixed formulation and to use the Lagrange multiplier, which is defined on the sides of the mesh, as correction of the degrees of freedom of the flux. The main difficulty lies in the choice of the multiplier's space, which should allow to establish a uniform inf-sup condition and to compute the multiplier locally. Thus, we do not solve the mixed formulation but only use it for the error analysis. Contrarily to other approaches, no local nor global mixed problem needs to be solved. In addition, the previous approach provides a unified framework for several standard finite element methods and differential operators.
This study generalizes the methodology described above to diffusion problems, the main focus being on the robustness of the reconstruction and of the a posteriori error bounds with respect to the diffusion coefficients.
Firstly, we consider a conforming finite element approximation on triangular meshes. We express the inf-sup constant of the equivalent mixed formulation in terms of the coefficients and we also establish a local bound for the multiplier, with a constant whose dependence on the coefficients is given explicitly. This allows us to deduce, besides the usual sharp reliability of the a posteriori error indicator, its local efficiency with an explicit constant. This result is new, at the best of our knowledge; for quasi-monotone coefficients, we retrieve the complete robustness, which already exists in the literature in the quasi-monotone case.
Secondly, we consider a nonconforming finite element approximation of arbitrary polynomial degree, based on the space introduced by Matthies and Tobiska in 2005. The standard nonconforming space of odd degree rises no difficulty and the reconstruction of conservative fluxes in this case is well-known. Meanwhile, this is no longer true for even degree, due to the loss of unisolvence. Our contribution is the extension of the previous approach to the spaces of Matthies and L. Tobiska, which are well-defined for any degree and also inf-sup stable for the Stokes problem, in a robust way with respect to the diffusion coefficients.
Finally, we will present several numerical experiments illustrating the theoretical results.
Safi Rouba - Discrete Embedding Of Hamiltonian Systems And Variational Integrators
An introduction to the discrete embedding of Lagrangian and Hamiltonian systems using a discrete differential and integral calculus of order one. This theory is compared with the seminal work of J-E. Marsden and M. West on variational integrators.
Ghantous Joyce - Numerical analysis of a diffusion equation with Ventcel boundary condition using curved meshes
We define a higher-order finite element method for numerically approximating the solution of a scalar diffusion equation on a bounded domain with the so-called Ventcel boundary conditions.
Mardi après-midi
Coiffard Théo - Lattice Boltzmann model for fluid flow in porous media : a probabilistic and partial bounce-back method
La gestion des ressources en eau en zone littorale nécessite d'appréhender les mécanismes physiques d'intrusion d'eau salée dans les aquifères donc en particulier la dynamique d’un mélange de fluides dans un milieu poreux en fonction de contraintes imposées par la nature (des marées au climat) ou l’Homme (exploitation). Les schémas de Boltzmann sur réseau sont connus pour leurs performances dans l’approximation de fluides libres (Navier-Stokes). Une première étape consiste à les adapter pour approcher la loi de Darcy qui permet de modéliser l’écoulement à une échelle adaptée à l’aquifère. Un schéma D2Q9 incluant un transfert partiel des distributions (lors de l'étape de streaming) est construit à cette fin.
Dor Dieunel - On the hyperbolic relaxation of the Cahn-Hilliard equation with a mass source
Nous considérons la relaxation hyperbolique de l’équation de Cahn-Hilliard avec un terme de prolifération, qui a des applications en biologie. On associe ce modèle avec des conditions aux limite de Dirichlet. D’abord, nous étudions le caractère de bien posé et la régularité des solutions de notre système. Ce qui nous a permis ensuite d’étudier la dissipativité et l’existence de l’attracteur global. Enfin, nous donnons des simulations numériques qui confirment les résultats théoriques.
Dabo Issa Mbenard - Introduction to Free Probability
While classical Probability relies on Measure theory, Free Probability studies randomness from an algebraic point of view.
This theory based on non-commutative algebra is used in various domains such as random matrix theory, combinatorics, quantum information theory.
After a brief introduction to Free probability, we will see how the theory is used to tackle random matrix problems.
Bentbib Rachad - Groupes en théorie des modèles
Le but est de présenter ce qu'est la théorie des modèles en logique mathématique; évoquer la notion de rang de Morley et l'intérêt de son intérêt; évoquer les travaux faits durant ma thèse à ce sujet
Lamberti Giuseppe - Interpolation in spaces of holomorphic functions
The study of interpolating sequences for analytic functions in one or more complex variables is one of the main research areas in complex analysis. It has plenty of applications in fields such as signal theory, control theory, operator theory, etc. For many spaces, like Hardy spaces, these sequences are well understood while for others, like Dirichlet spaces, there exists a characterization which is not very easy to verify. In other circumstances, a characterisation does even not exist. In this scenario it is useful to consider a random setting, which can help us to understand when interpolation is “generic”, in particular we are interested in a radio model, where points’ radii are fixed, while the arguments are uniformly distributed.
Artusa Marco - Exploring Geometric Intuition in Number Theory: the study of the Weil group
Geometric intuition can help us to solve many problems in number therory. Sometimes imagining the geometric shape of the set of solutions of an equation is easy, thanks to real numbers. But what happens when we go beyond reality ? In this talk I well present the type of questions motivating my research : the study of the Weil group of a field can be seen as the attempt to "draw" a picture of the field itself.
