Institut de recherche mathématique avancée

Résumé : The spectral theory of Toeplitz operators originates in Szegő’s classical work on the eigenvalue distribution of Toeplitz matrices and was later extended by Boutet de Monvel and Guillemin to general complex manifolds. This talk has two objectives. First, we present a generalization of these results in the framework of arbitrary Bernstein-Markov measures on complex manifolds. Second, we investigate the asymptotic behavior of the smallest eigenvalue of Toeplitz operators, a problem that — unexpectedly — turns out to be closely related to Mabuchi geometry originally introduced in connection with constant scalar curvature Kähler metrics. Our aim is to show how techniques from Kähler geometry and pluripotential theory emerge organically in the asymptotic analysis of Toeplitz operators.

Résumé : TBA

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.