Groups and Geometry

In this course we study classical groups (linear, unitary, orthogonal, symplectic), in their algebraic (reduction, conjugation classes...), geometric (actions, exponential application) and topological aspects.

Master the basic tools common to all branches of Geometry.

A Bachelor's degree in Mathematics.

Recommended prerequisites: the content of the two L3 courses "Groups and rings 1" and "Topology of metric spaces" of the Licence de Mathématiques of the University of Montpellier.

1. Groups and group actions: reminders, semi-direct product.
2. The general linear group _GLn : action on _Mn by equivalences, by similarities. Interpretation of the Gauss pivot and Jordan's theorem on C or R. Connectedness of _GLn, density in _Mn, adhesion of similarity classes. Action of _GLn on vector lines. The linear projective group; in dimension 2: homographies.
3. Unitary and orthogonal groups: matrix interpretation, relation with Hermitian forms, reduction, conjugacy classes. Topological properties. Isometry groups of regular polygons and polyhedra in dimension 2 and 3.
4. Matrix exponential and polar decomposition: real Hermitian and symmetric matrices, reduction of these matrices, square roots of positive definite Hermitian matrices. Polar decomposition for _Gln (C or R). Matrix exponential and polar decomposition, topological aspects.
5. Developments and applications, for example related to linear representations of finite groups (studied in UE Algebra I), or to the action of SL(2,R) on the Poincaré half-plane.

Hourly volumes:

            CM : 27

            TD : 27

            TP : 0

            Land : 0