Exposés passés

29 juin 2022
Ioanna-Maria Lygatsika (LJLL)
The Hartree-Fock problem in electronic structure and Gaussian discretisations
The behaviour of electrons in atoms and molecules is governed by the N-body Schrödinger equation. We provide simple examples of systems under study, issued from the field of quantum chemistry. The Hartree-Fock model, that replaces the N-body problem by a mean field one-electron approximation, will be the main topic of this talk. We focus on its Galerkin approximation over a Gaussian-type orbital (GTO) basis set. The discretised Hartree-Fock equations will be presented in detail. As an attempt to illustrate why GTOs are widely used in today's quantum chemistry codes, we demonstrate the advantages and common issues of such a basis set, throughout numerical examples on atoms and small molecules, using the PySCF electronic structure code (https://pyscf.org/).
22 juin 2022
Alexis Leculier (LJLL)
Propagation phenomena in a homogeneous field, how it happens ? how can we prevent it ?
In the end of the thirties, the muskrat starts to invade France starting from a few amount of individual in the north east of the France. The same happens for the tiger mosquitoes since 2008 (only during the summer) from the south east of the France. It is remarkable that in both cases, the invasions occur with a constant speed and just a few of individuals at the begining. We propose to study a general model that allow to capture this dynamics. In the final part of the talk, we will see some new results that allow to prevent a such invasion phenomena. This final part is a joint work with Nga Nguyen.
15 juin 2022
Break (LJLL)
8 juin 2022
Rémi Robin (LJLL)
Optimization of a stellarator: from an inverse problem to shape optimization.
In this introduction talk, we will describe an inverse problem related to the optimal positions of the coils on a nuclear fusion reactor known as stellarator. This talk aim at provided a simple introduction to the classical tools used in this context.
2 juin 2022
Charlie Hérent (LIGM - Université Paris-Est)
Brownian motion and Pitman's theorem
Pitman's theorem discovered in 1975 gives a transformation of a brownian motion into a Bessel process. The Bessel process obtained can be seen as a brownian process conditioned to be positive. In this presentation, we will introduce the original theorem and then its generalizations for vector space and Lie algebra due to Biane-Bougerol-O'Connell.
25 mai 2022
Darryl Ondoua (LJLL)
Fast-slow system of symbiotic interactions between a host and its mutualistic bacteria
Symbiotic interactions between species, in particular between a host and its colony of microorganisms, are one of the essential characteristics of the evolution of biological diversity.

However, it is not known to what extent mutualistic interactions between a host and its bacteria influence the host's fitness. Conversely, what impact does this have on the bacterial population? To try to answer these questions, we introduce a theoretical mathematical model of the symbiotic interactions between the host and its mutualist bacteria.

In this talk, we first present a model showing the cooperative interactions and coexistence of bacteria in the absence of the host. We then extend the model by including host health status. Taking into account that bacteria and host have different time scales (host generation time and bacteria cell division), this leads us to study the host-bacteria system as a singular perturbation problem.
18 mai 2022
Elias Drach (LAMA, Université Gustave Eiffel)
A well-balanced entropy scheme for a shallow water type system describing two-phase debris flows
In the context of modeling two-phase debris flows involving a layer of fluid, and a layer of a mixture between grains and fluid, some shallow water systems arise with internal variables.
Our work focus on such a shallow water system with two internal variables and a topography which adds a nonconservative term.

For numerical purposes, it is desirable to deal with a system where the mathematical entropy (the physical
energy of the system) is convex with respect to the chosen conservative variables including the internal variables.

Then at the numerical level, we can look for a scheme satisfying a semi-discrete entropy inequality.

Moreover a crucial point in modeling debris flows is to well describe the stopping of the flow to assess the
maximum velocity, lifetime and runout extent. This means in particular that the flow should stop when
it is a steady state at rest. This is the so called well-balanced property.

Writing the system with conservative variables for which the energy is convex, we derive a well-balanced
scheme satisfying a semi-discrete entropy inequality.
10 mai 2022
Jakub Skrzeczkowski (University of Warsaw / LJLL)
Fast reaction limit with nonmonotone reaction function
We consider a reaction-diffusion system with nonmonotone nonlinearity F and one non-diffusing component. Surprisingly, as the speed of reaction tends to infinity, the concentration of the non-diffusing component exhibits fast oscillations. We identify precisely its Young measure which, as a by-product, proves strong convergence of the diffusing component, a result that is not obvious at all from a priori estimates! Moreover, it shows why the oscillations are a consequence of the non-monotone character of F. Our work is based on the analysis of regularization for forward-backward parabolic equations by Plotnikov. We also refine the method of Plotnikov by application of classical Radon-Nikodym theorem. The talk is based on two papers: first that appeared in CPAM (doi: https://doi.org/10.1002/cpa.22042, joint with B. Perthame) and another one in Comptes Rendus (doi: https://doi.org/10.5802/crmath.279).
4 mai 2022
Jérôme Carrand (LPSM)
Cats, Billiards and Anisotropic Banach spaces
The aim of this talk is to present the construction of equilibrium states for the billiard map, using appropriate transfer operators acting on anisotropic Banach spaces. We first present the general motivations in Dynamical Systems -- statistics of an orbit, invariant measures, etc. -- with many toy examples. Then we introduce the notions of entropy, pressure, the variational principle and equilibrium states. From all this, we present the dynamics of a Sinai Billiard, as well as recent results. Finally, we introduce the construction of the appropriate anisotropic Banach spaces and their use through a transfer operator to construct equilibrium states. If time permits, we present a work in progress concerning the measure of maximal entropy for the Billiard Flow and its relation to a specific equilibrium state for the billiard map.
27 avril 2022
Elena Ambrogi (LJLL)
On the Leaky I&F Model. A generalization of Doeblin’s method for the long time convergence.
The understanding of the dynamics of neural networks is a chal- lenging problem, and although the significant progresses achieved in the past years, it still offers several stimulating questions. The Integrate and Fire (I&F) model is a class of mean-field evolutionary equations which describe the activity of a population of neurons via their membrane potential.
This presentation will be divided in two moments. On first I will introduce the I&F neuron model: its biological motivation and some qualitative properties on its solutions which have been studied by several authors. In the second part I will focus on a proof scheme for the long time behaviour of the solutions to this problem. In a previous publication C ́aceres, Carrillo and Perthame already proved a result of exponential convergence by means of the General Relative Entropy. We have developed an approach that is inspired by Doeblin’s theory and generalize it. My aim is to illustrate what is the novelty of the approach we propose; what are the difficulties that prevent us to directly apply Doeblin’s the- ory and give you an idea of the proof scheme. This talk is based on a joint work with Delphine Salort (Laboratory of Computational and Quantitative Biology) and Nicolas Torres (Laboratoire Jacques-Louis Lions).