Groups and Geometry

In this course unit, we study classical groups (linear, unitary, orthogonal, symplectic) from their algebraic (reduction, conjugation classes, etc.), geometric (actions, exponential map), and topological perspectives.

Master the basic tools common to all branches of geometry.

A bachelor's degree program in mathematics.

Recommended prerequisites: the material covered in the two third-year courses, “Groups and Rings 1” and “Topology of Metric Spaces,” in the Bachelor’s program 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 via equivalences and similarities. Interpretation of the Gauss pivot and Jordan’s theorem over C or R. Connectedness of_GLn, density in_Mn, and adherence 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, for example related to linear representations of finite groups (studied in 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