Institut de recherche mathématique avancée

Résumé : Traditionally, Langlands functoriality refers to the identification of automorphic forms whose parameters take a special shape. In this talk, we explain how to ask analogous questions on the Hitchin moduli space using perspectives from the relative Langlands program. We gain, in this setting, the advantage of working with a version of Langlands duality which is readily computable, and if time permits we will discuss the ramifications of these calculations for automorphic periods and L-functions.

Résumé : Non-associative algebras arise in many mathematical settings, with the Lie algebra being a significant example. A powerful tool to study a Lie algebra is its universal enveloping algebra, which is an associative algebra, whose representation coincides with the representation of that Lie algebra. The classical Poincaré-Birkhoff-Witt (PBW) theorem plays a role by giving a vector space basis of the enveloping algebra. We will generalise the notion of enveloping algebra to arbitrary (non-associative) algebra and present the generalised PBW theorem via giving examples. If time permits, we could talk about a more generalised perspective — universal enveloping Operads.

Résumé : In particle methods, also known as Lagrangian methods, the discretization of the simulation domain is realized on a set of numerical particles that follow the dynamics of the system and are displaced with respect to the flow velocity. Vortex methods belong to particle methods and allow to discretize in a Lagrangian way the vorticity-velocity formulation of the Navier-Stokes equations, for the simulation of incompressible flows. This work is focused on a semi-Lagrangian variant of the Vortex Methods, the so-called « Remeshed Vortex Methods » (RVM). In this approach, the Lagrangian particles are regularly re-located on a Cartesian grid (through a « particle-remeshing » procedure) in order to avoid one of the major drawback of purely mesh-free Lagrangian schemes, namely the distorsion of particle distribution, leading to convergence loss. These methods constitute an extension of the « Particle-In-Cell » methods and can be seen as a hybrid Lagrangian-Eulerian approach; they take advantage from both the low numerical dissipation brought by the Lagrangian treatment of the flow advection and the efficiency of Eulerian schemes to handle the Poisson equation of the problem, the viscous term or the external forces. The present talk will present the numerical characteristics of the RVM, through a comparison of Direct Numerical Simulations (DNS) obtained with other numerical methods (like a Lattice Boltzmann method (LBM), i.e. an Eulerian method based on a mesoscopic flow description). We will then highlight the interest of RVM to afford Large Eddy Simulations (LES) and present the sub-grid scale models suited to RVM for the LES of homogeneous isototropic turbulents flows and of turbulent flows over solid walls.

Résumé : La majoration des sommes de Kloosterman obtenue par Deligne est un résultat crucial, exploité notamment en théorie analytique des nombres. La borne qu'il obtient est de l'ordre de la racine carrée des bornes "naïves", ce qui est comparable aux gains obtenus en admettant l'hypothèse de Riemann. L'objectif de cet exposé sera d'expliquer comment ces sommes d'exponentielles s'expriment dans un langage géométrique/algébrique, puis d'expliquer les avantages d'une telle traduction.