Cours spécialisé (GD, GT)
Curvature bounds for metric spaces
Sébastien Boucksom
Contact : sebastien.boucksom à imj-prg.fr
Pas de notes de cours prévues.
Langue du cours : anglais
Présentation
Starting with the work of Aleksandrov in the 1950s, notions of upper and lower curvature bounds for metric spaces (Cartan-Aleksandrov-Topogonov spaces) have come to play a prominent role in differential geometry, in particular in the study of limits of Riemannian manifolds as metric spaces (Gromov-Hausdorff convergence). More recently, a notion of Ricci curvature lower bound for metric measure spaces (Riemannian Curvature-Dimension condition) has emerged, pioneered by the work of Lott-Sturm-Villani on optimal transport. The purpose of this course is to provide an introduction to this circle of ideas, with possible applications to Kähler geometry.
Contenu
- Riemannian geometry: sectional and Ricci curvature, space forms, Bishop-Gromov inequality
- Functional analysis: Sobolev and Poincaré inequalities
- Metric spaces: Hausdorff measure and dimension, Gromov-Hausdorff convergence, Gromov compactness
- CAT spaces
- RCD condition: from Bakry-Emery to Lott-Sturm-Villani
Prérequis
A familiarity with the basic concepts of differential and Riemannian geometry is recommended (as provided eg by the M2 course "Géométrie différentielle et riemannienne I,II")
Bibliographie
- M.Bridson, A.Haefliger. Metric spaces of non-positive curvature. Grundlehren der mathematischen Wissenschaften
- C.Villani. Optimal transport, old and new. Grundlehren der mathematischen Wissenschaften