This course is an introduction to the theory of differentiable manifolds. The topics covered include (not necesssarily in this order)

- Charts, atlases, definition of a differentiable manifold

- Differentiable maps, tangent bundle

- Immersions, submersions, submanifolds

- Partitions of unity

- Vector fields, differential equations on a manifold

- Differential forms, exterior differential calculus

- Lie derivative, Lie-Cartan calculus, Frobenius theorem

- Sard's lemma and applications

- Stokes' formula

- De Rham cohomology in maximal dimension, degree.

Analyse fonctionnelle, espaces vectoriels topologiques

Distributions, espaces de Sobolev

Théorie de Fredholm, théorie spectrale