3 avril 2018

Leonid Polterovich (Tel Aviv)
Persistence barcodes in geometry and analysis

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Résumé : While originated in topological data analysis, persistence modules and their barcodes provide an efficient way to book-keep homological information contained in Morse and Floer theories. I shall describe applications of persistence barcodes to symplectic topology and geometry of Laplace eigenfunctions. Based on joint works with Iosif Polterovich, Egor Shelukhin and Vukasin Stojisavljevic.

Persistence barcodes in geometry and analysis  Version PDF

Andras Vasy (Stanford)
Boundary rigidity and the local inverse problem for the geodesic X-ray transform on tensors

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Lieu : IMO ; salle 3L8.

Résumé : In this talk, based on joint work with Plamen Stefanov and Gunther Uhlmann, I discuss the boundary rigidity problem on manifolds with boundary (for instance, a domain in Euclidean space with a perturbed metric), i.e. determining a Riemannian metric from the restriction of its distance function to the boundary. This corresponds to travel time tomography, i.e. finding the Riemannian metric from the time it takes for solutions of the corresponding wave equation to travel between boundary points.
This non-linear problem in turn builds on the geodesic X-ray transform on such a Riemannian manifold with boundary. The geodesic X-ray transform on functions associates to a function its integral along geodesic curves, so for instance in domains in Euclidean space along straight lines. The X-ray transform on symmetric tensors is similar, but one integrates the tensor contracted with the tangent vector of the geodesics. I will explain how, under a convexity assumption on the boundary, one can invert the local geodesic X-ray transform on functions, i.e. determine the function from its X-ray transform, in a stable manner. I will also explain how the analogous result can be achieved on one forms and 2-tensors up to the natural obstacle, namely potential tensors (forms which are differentials of functions vanishing at the boundary, respectively tensors which are symmetric gradients of one-forms vanishing at the boundary).
Here the local transform means that one would like to recover a function (or tensor) in a suitable neighborhood of a point on the boundary of the manifold given its integral along geodesic segments that stay in this neighborhood (i.e. with both endpoints on the boundary of the manifold). Our method relies on microlocal analysis, in a form that was introduced by Melrose.

Boundary rigidity and the local inverse problem for the geodesic X-ray transform on tensors  Version PDF

Florian Herzig (University of Toronto)
Vers la partie ’pente finie’ pour GL_n sur un corps p-adique

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Lieu : IMO Bât. 307, salle 3L15

Résumé : Soit F/Q_p une extension finie et rho une représentation continue cristalline de Gal(\bar F/F) de dimension n sur un corps p-adique. On construit une représentation Pi(rho)^pf localement analytique de GL_n(F) dont tous les sous-quotients sont des sous-quotients de séries principales (donc ’de pente finie’). Si rho est la restriction d’une représentation galoisienne globale r convenable, et sous des hypothèses de Taylor-Wiles, on démontre que Pi(rho)^pf est contenu dans la partie r-isotypique de la cohomologie p-adique completée d’un groupe unitaire convenable. Il s’agit d’un travail en commun avec C. Breuil.

Vers la partie ’pente finie’ pour GL_n sur un corps p-adique  Version PDF