# Mazhe

This is a big course of mathematics declined in two versions.

- Le Frido contient des mathématiques du niveau de l'agrégation. Il recouvre à peu près tout le programme.

- giulietta contains more or less everything I know in mathematics, including my research.
- manuel_du_contributeur.pdf contient des instruction pour la compilation du Frido, ainsi que des politiques éditoriales à l'attention de qui voudrait contribuer.

## Giulietta

The document Giulietta contains almost everything I know in mathematics. It includes

- A part (in French) about general mathematics at master level (Le Frido)
- A part (in French) containing the exercises and many corrections of the courses I gave at university.
- Higher level mathematics including research stuff -- my thesis is here (in English)

### General differential geometry

- Fibre bundles : vector, principal and associated bundles.
- Connexions on fibre bundle, covariant derivative.

### Lie groups, Lie algebra

- Lie groups and Lie algebra.
- Homogeneous and symmetric spaces.
- Root spaces, Iwasawa decomposition.
- Cyclic modules and representations.

### Quantum field theory

Very few physics. The aim is to understand why are we using groups, representations and principal bundle in quantum field theory.

- Link between fibre bundle and quantum field theory : why are particles irreducible representations of the Poincaré group ?
- Particles are modeled by sections of associated bundle : product of one representation of the Poincaré group and one representation of the gauge group. The interaction is given by a connexion on that bundle (with values in the Lie algebra of the gauge group).
- Clifford module and Dirac operator.
- Yang-Mills action.

### Non commutative geometry

- Compact quantum group.
- General non commutative geometry.
- WKB Deformation and quantization theory. Deformation of a manifold by action of a "deformable" group.

### Black hole in anti-de Sitter space

This is the argument of my thesis.

- Black hole in anti-de Sitter space. The singular part is defined as the closed orbits of the Iwasawa subgroup of SO(2,l-1) acting on the l-dimensional anti-de Sitter space.
- Dirac operator on the anti-de Sitter space.
- Deformation of the anti-de Sitter space.

### Other

- Symplectic geometry, Hamiltonian action.
- Bialgebra and Hopf algebra.
- Oscillatory integral.
- von Neumann algebra.