Groups and Geometry

In this UE 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" from the Licence de Mathématiques at the Université de Montpellier.

1. Groups and group actions: reminders, semi-direct product.
2. The general linear group GLn_: action on _Mn by equivalences, by similitudes. Interpretation of Gauss's 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, relationship with Hermitian forms, reduction, conjugation classes. Topological properties. Isometry groups of regular polygons and polyhedra in dimensions 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, e.g. related to linear representations of finite groups (studied in EU 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