Institut de recherche mathématique avancée

Résumé : We will present a pair of flexible and degenerate constructions of objects related to Thurston’s Lipschitz metric on Teichmüller space. In particular, we will explain how to construct sums of Fuchsian representations of surface groups whose limit cones are finite-sided polyhedra and how to construct irregular geodesics for Thurston’s Lipschitz metric on Teichmüller space. Both constructions are “as degenerate as possible” in appropriate senses. We will emphasize the close relationship of these constructions with a counterintuitive theorem of Thurston on the generic simplicity and stability of solutions to a length-ratio optimization problem on hyperbolic surfaces.

Résumé : $\varphi$-FEM is a new finite element method, proposed to solve partial differential equations on complex domains, using simple non-conforming meshes. The method relies on the use of a level-set function $\varphi$, which defines the domain and its boundary. In this presentation, I will introduce the method in the simple case of the resolution of the Poisson equation with Dirichlet boundary conditions. Then I will present the extension to the case of mixed Dirichlet/Neumann boundary conditions. I will also present results for the resolution of the Heat equation with Dirichlet boundary conditions or linear and non-linear elasticity problems. I will finally present different evolutions of the method, including its combination with Neural Operators or the use of the finite difference method. I will also discuss perspectives and future challenges for $\varphi$-FEM.