This master is oriented towards geometry, with a special focus on its interaction with group theory and Lie theory. Since the work of Poincaré, Klein and others, modern geometry hes been intimately related with the notions of manifold and group. For instance, the uniformisation theorem implies that every closed topological surface carries a metric of constant curvature. The isometry group of the universal cover of such a surface is a Lie group, hence the geometry of the surface can be recovered purely from group theory. More recently Gromov and others intriduced the study of countable groups as geometric objects. The broad goal of this course is to introduce and study in some depth these objects and to show how they relate with each others.
The first semester is mostly dedicated to understanding the construction of homogeneous spaces and the study of their geometric properties, in particular their curvature. This will be achieved through two courses, one on Riemannian geometry which studies geometry in a broader context, and one on Lie groups and algebras with a more algebraic slant. In parallel there will be a course on geometric group theory.
The second semester will be more focused on hyperbolic geometry: we will give constructions of Riemannian manifolds of negative curvature, and conversely study the existence and unicity of such metrics on smooth manifolds.
Schedule
Contact
For informations on this program please contact the organisers: Adrien Boulanger ou Jean Raimbault.For general information on the "Maths fondamentales" option, please contact Benjamin Audoux.
For general informaton on the master in Mathematics please contact Michel Mehrenberger ou Laurent Regnier.