Institut de recherche mathématique avancée

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.