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.