10 janvier 2019

Tal Horesh (IHES)
Some counting and equidistribution results in geometry of numbers

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Lieu : Salle 2L8 (IMO, bâtiment 307)

Résumé : Geometry of numbers is the study of integer vectors and lattices in the n-dimensional space. I will discuss the equidistribution of certain parameters characterizing primitive integer vectors as their norms tend to infinity, such as their directions, the integral grids in their orthogonal hyperplanes, and the shortest solutions to their associated gcd equations. I will also discuss the equidistribution of primitive d-dimensional subgroups of the the integer lattice, Z^n.
The key idea is that these questions reduce to problems of counting SL(n,Z) points in SL(n,R), and in fact to the equidistribution of the Iwasawa components of SL(n,Z).

Notes de dernières minutes : Café culturel à 13h par Jean Lécureux

Some counting and equidistribution results in geometry of numbers  Version PDF

Frédéric Coquel (CNRS, CMAP (Ecole Polytechnique))
(SALLE CHANGEE) Schémas de relaxation de Jin et Xin avec correction par mesure de défaut

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Lieu : IMO, Salle 3L15

Résumé : We present a class of finite volume methods for approximating entropy weak so-lutions of non-linear hyperbolic PDEs. The main motivation is to resolve discontinuities aswell as Glimm’s scheme, but without the need for solving Riemann problems exactly. Thesharp capture of discontinuities is known to be mandatory in situations where discontinuitiesare sensitive to viscous perturbations while exact Riemann solutions may not be available(typically in phase transition problems). More generally, sharp capture also prevent discreteshock profiles from exhibiting over and undershoots, which is decisive in a many applications(in combustion for instance). We propose to replace exact Riemann solutions by self-similarsolutions conveniently derived from the Xin-Jin’s relaxation framework. In the limit of a van-ishing relaxation time, the relaxation source term exhibits Dirac measures concentrated onthe discontinuities. We show how to handle those so-called defect measures in order to exactlycapture propagating shock solutions while achieving computational efficiencies. The lecturewill essential focus on the convergence analysis in the scalar setting. A special attention ispaid to the consistency of the proposed correction with respect to the entropy condition. Weprove the convergence of the method to the unique Kruvkov’s solution. This is a joint workwith Shi Jin (Madison-Wisconsin Univ.), Jian-Guo Liu (Duke Univ.) and Li Wang (UCLA).

(SALLE CHANGEE) Schémas de relaxation de Jin et Xin avec correction par mesure de défaut  Version PDF