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H 1 Geometry and analysis of locally symmetric spaces of infinite volume

Auteurs : Ji, Lizhen (Auteur de la Conférence)
CIRM (Editeur )

Résumé : For any symmetric space $X$ of noncompact type, its quotients by torsion-free discrete isometry groups $\Gamma$ are locally symmetric spaces. One problem is to understand the geometry and analysis, especially the spectral theory, and interaction between them of such spaces. Two classes of infinite groups $\Gamma$ have been extensively studied:
$(1) \Gamma$ is a lattice, and hence $\Gamma$ $\backslash$ $X$ has finite volume.
$(2) X$ is of rank $1$, for example, when $X$ is the real hyperbolic space, $\Gamma$ is geometrically finite and $\Gamma$ $\backslash$ $X$ has infinite volume.
When $\Gamma$ is a nonuniform lattice in case $(1)$ or any group in case $(2)$, compactification of $\Gamma$ $\backslash$ $X$ and its boundary play an important role in the geometric scattering theory of $\Gamma$ $\backslash$ $X$. When $X$ is of rank at least $2$, quotients of $X$ of finite volume have also been extensively studied. There has been a lot of recent interest and work to understand quotients $\Gamma$ $\backslash$ $X$ of infinite volume. For example, there are some generalizations of convex cocompact groups, but no generalizations yet of geometrically finite groups. They are related to the notion of thin groups. One naturally expects that these locally symmetric spaces should have real analytic compactifications with corners (with codimension equal to the rank), and their boundary should also be used to parametrize the continuous spectrum and to understand the geometrically scattering theory. These compactifications also provide a natural class of manifolds with corners. In this talk, I will describe some questions, open problems and results.

Codes MSC :
53C35 - Symmetric spaces (differential geometry)
58J50 - Spectral problems; spectral geometry; scattering theory

 Informations sur la Vidéo Langue : Anglais Date de publication : 30/03/15 Date de captation : 12/03/15 Sous collection : Research talks Format : quicktime ; audio/x-aac arXiv category : Differential Geometry ; Algebraic Geometry Domaine : Algebraic & Complex Geometry Durée : 00:57:25 Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2015-03-12_Ji.mp4 Informations sur la rencontre Nom de la rencontre : Analysis and geometry of resonances / Analyse et géométrie des résonancesOrganisateurs de la rencontre : Guillarmou, Colin ; Hilgert, Joachim ; Pasquale, Angela ; Przebinda, TomaszDates : 09/03/15 - 13/03/15 Année de la rencontre : 2015 URL Congrès : http://www.math.univ-metz.fr/~pasquale/C...Citation Data DOI : 10.24350/CIRM.V.18729403 Cite this video as: Ji, Lizhen (2015). Geometry and analysis of locally symmetric spaces of infinite volume. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18729403 URI : http://dx.doi.org/10.24350/CIRM.V.18729403

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