Groups and Geometry

In this EU, we study classical groups (linear, unitary, orthogonal, symplectic) in their algebraic aspects (reduction, conjugacy classes, etc.), geometric aspects (actions, exponential map), 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 Bachelor's degree in Mathematics at 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 Gauss's pivot and Jordan's theorem on C or R. Connectivity of_Gln, density in_Mn, adherence of similarity classes. Action of_{GLn on} vector lines. The projective linear 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 dimensions 2 and 3.
4. Matrix exponential and polar decomposition: Hermitian and real 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 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