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Résumé : In classical mechanics physical systems are described using Hamiltonian or Lagrangian formulations. In this settings the classical observables are functions defined on the phase space. On the other hand, in quantum mechanics observables are operators acting on quantum states. We will briefly introduce geometric quantization which is a method to try to connect this two physics using symplectic manifolds and line bundles.

Résumé : The study of swimming at the microscopic scale is of increasing interest, driven by advances in theoretical modeling and experimental techniques. Inspired by the locomotion strategies of natural microorganisms, artificial micro-swimmers are designed to replicate their motion, showing promising potential in biomedical applications. In this presentation, we model both rigid and soft micro-swimmers in Newtonian fluids, taking into account their interactions with rigid obstacles such as the boundaries of the fluid domain. A particular focus is placed on the magneto-swimmer, an elastic swimmer driven by an external magnetic field, whose motion results from the interaction of its tail with the surrounding fluid. Our numerical approach relies on the finite element method within the Arbitrary Lagrangian–Eulerian framework. Rigid contact interactions are modeled using a repulsive lubrication force, while elastic collisions are handled using a Nitsche method, enforcing the Signorini contact conditions. For magneto-swimmers, we employ a partitioned implicit scheme, alternating between monolithic resolutions of a fluid-rigid and a fluid-elastic sub-problem. After presenting the modeling and numerical strategy, we will illustrate some applications of both rigid and soft swimmers.

Résumé : Oda montre l'existence de représentation pro-l universelle du groupe fondamental de l'espace des modules de courbes de genre g avec m points marqués. Suite à cela il définit une extension algébrique de Q associé à cette représentation et demande la dépendance en g et m des corps ainsi obtenus.

Dans cet exposé, j'introduirai ce problème classique de l'école japonaise en commençant par les objets analogues dans le cas, plus habituel, des courbes. Je présenterai ensuite une version analogue pour les lieux spéciaux, des sous-champs obtenus des courbes admettant une action de Z/lZ. Les idées principales de la preuve de l'indépendance des corps dans ce contexte.

Résumé : Consider closed holomorphic curves in symplectic $G$-manifolds, with respect to a $G$-invariant almost complex structure. We should not expect the moduli space of such curves to be a manifold (after all, transversality and symmetry are famously incompatible). However, we can hope for a clean intersection condition: the moduli space decomposes into countably many disjoint strata which are smooth manifolds, whose dimensions are explicitly computable.

I present this decomposition for simple curves, and indicate how to extend this to multiple covers. These are the first steps towards a well-behaved theory of equivariant holomorphic curves. This has applications to the 3-body problem and real Gromov-Witten theory.

Résumé : TBA

Résumé : Résumé : Le groupe pro-unipotent DMR0, qui sous-tend le schéma des relations de double mélange satisfaites par les nombres zêtas multiples (Racinet 2002), est un sous-groupe de celui des automorphismes de l'algèbre de Lie libre en deux générateurs e0 et e1, envoyant e1 sur lui-même et e0 sur un conjugué. Nous montrons que DMR0 est en fait contenu dans le sous-groupe des automorphismes spéciaux, c'est-à-dire envoyant également einfty:=-e0-e1 sur un conjugué, et est globalement invariant sous l'involution du groupe des automorphismes spéciaux induit par l'échange de e0 et einfty (travail commun avec H. Furusho).

Résumé : In this work, we are interested in the mathematical modeling of the impact of apoptosis on the dynamics of cellular tissues, and in particular on cell tissue fluidity. Indeed, cell apoptosis corresponds to the programmed cell death: when they leave the tissue, they induce local contractions but also enable cells rearrangements.

We first propose an individual-based model that provides the dynamics of the positions, velocities and polarities of the cells, idealized as hard spheres. Cells interact with each other through contact forces, smooth attraction, and polarity alignment. The model also involves a microscopic description of apoptotic and proliferation events. The present work is an extension of the model proposed in [3] and validated by experiments on cellular rings. Numerical simulations are performed and several indicators of fluidity are analyzed.

Next, we derive a macroscopic description, following the methodology proposed in [1], [2]. We start from a mean-field dynamics of the kinetic distribution function in phase-space (position, polarity, radius), where contact forces have been replaced with repulsion forces. We then introduce a specific time and space rescaling and identify the equilibria distribution functions, which are parameterized by two macroscopic quantities: the density and the mean polarity. Based on the Generalized Collision Invariant (GCI) method [2], we are then able to identify their dynamics: the resulting description can be seen as a modified Self-Organized Hydrodynamics (SOH) model. We finally discuss the obtained model and highlight the effect of the apoptotic events on the dynamics.

This is a joint work with Laurent Navoret (Université de Strasbourg) and Marcela Szopos (Université Paris Cité). It has also been carried out in collaboration with Romain Levayer (Institut Pasteur) and Daniel Riveline (IGBMC, Université de Strasbourg) in the context of the ANR project MAPEFLU.

References

[1] Degond, P., Dimarco, G., Mac, T. B. N., and Wang, N. Macroscopic models of collective motion with repulsion. Communications in Mathematical Sciences, 13(6), 1615–1638 (2015).

[2] Degond, P., and Motsch, S. Continuum limit of self-driven particles with orientation interaction. Mathematical Models and Methods in Applied Sciences, 18(supp01), 1193–1215 (2008).

[3] Vecchio, S. L., Pertz, O., Szopos, M., Navoret, L., and Riveline, D. Spontaneous rotations in epithelia as an interplay between cell polarity and boundaries. Nature Physics (2024).

Résumé : Online