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In this course, we will cover the following topics:

  • Definition of Riemann surfaces and holomorphic maps.
  • Examples: the Riemann sphere, complex tori, elliptic functions.
  • The identity theorem. Applications: discreteness of the fibers, open mapping theorem.
  • Holomorphic maps between compact Riemann surfaces, ramification theory.
  • Meromorphic functions with prescribed zeros and poles.
  • Holomorphic differential forms, residues, genus of a compact Riemann surface.
  • Quotients of Riemann surfaces. Statement of the uniformization theorem.
  • Some examples of modular curves.
  • Examples of monodromy representations.

This course is an introduction to the theory of differentiable manifolds. The topics covered include

- 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

Other topics which may be covered:

- Approximation results

- Sard's lemma and applications

- Frobenius theorem

- Stokes' formula

- De Rham cohomology in maximal dimension, degree.

Espaces vectoriels topologiques. Topologies faibles. Dualité.

Théorie des distributions

Espaces de Sobolev

Opérateurs compacts et théorie de Fredholm